Advances in algebra : proceedings of the ICM Satellite Conference in Algebra and Related Topics

Advances in

Algebra

This page intentionally left blank

Proceedings of the ICM Satellite Conference in Algebra and Related Topics

Advances in

Algebra Editors

K P Shum The Chinese University of Hong Kong, Hong kong

ZXWan Chinese Academy of Science, Beijing, PR China

J PZhang Peking University, Beijing, PR China

WSWQorldScientific

New Jersey London Singapore Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224

USA o n c e : Suite 202, 1060 Main Street, River Edge, NJ 07661 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

ADVANCES IN ALGEBRA Proceedings of the ICM Satellite Conference in Algebra and Related Topics Copyright 0 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereoJ may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-260-7

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface In 1990, Hong Kong hosted the First Asian Mathematical Conference. Following the success of the conference, Hong Kong was again chosen to be the venue, this time for the first ever International Congress ofAlgebra and Combinatorics.We are honored to be the host again for this year’s ICM Satellite Conference in Algebra and Related Topics. We are grateful to have Professor R. Gonchidorazh, the former Vice-president of Mongolia Republic from Ulaanbaatar to give the opening address. Other distinguished guests this year include Dr. K. K. Wong, the Executive Council Member of the Hong Kong Airport Authority and Professor Ambrose Y. C. King, the Vice-Chancellor of the Chinese University of Hong Kong. All in all, we have over 250 eminent algebraists from all over the world participating in this year’s event. It has been no easy task organizing this event. In addition to the financial constraints, the bigger challenge facing us is the diminishingimportance of algebra as a subject within the academia. In recent years, research has gradually shifted towards applied fields that may yield quicker and more tangible results as compared to research in algebra topics. To many, algebra is fast becoming an outdated and oldfashioned discipline that has little direct applicationto the real world. They have failed to realize the importance of the subject in areas such as computer science and information technology. As algebraists, we must protect and defend the discipline. In this year’s conference, we will sadly miss the presence of Professor B. H. Neumann, who passed away in October 2002. He has been a keen supporter of the Society for the past 20 years -attending the conferences, and helping to promote the subject in the region. Many of us who knew Professor Neumann have dedicated their papers to him. There is also a memorial issue in this Proceedings dedicated to the late Professor Neumann. I would like to take this opportunity to thank the ICM Committee of the Chinese Mathematical Society. Without their dedicated support, it would have been impossible for Hong Kong to host such a large-scale conference. I would also like to thank the Dean of Science of the Chinese University of Hong Kong, Professor Lau Oi Wah who has seen that this is an important conference for the community and has wholeheartedly supported the event. Special thanks also go to Sir Q. W. Lee and Wei Lun Foundation for their continual support. It would not have been possible to host the many events in Hong Kong without his generous financial assistance. Last but not least, the organizing committee is indebted to the following parties. Without their support, this conference would not have become a reality. 1.

2. 3.

Beijing-Hong Kong Academic Exchange Centre Chung Chi College, The Chinese University of Hong Kong Chinese Mathematical Society, Beijing V

vi

The Croucher Foundation, Hong Kong K.C. Wong Education Foundation Ltd. 6. Lee Foundation, Singapore 7. The Science Faculty, Chinese University of Hong Kong 8. The Southeast Asian Mathematical Society 9. Wei Lun Foundation Ltd. 10. Wu Jie Yee Charitable Foundation Ltd.

4. 5.

K. P. Shum Organizer ICM Satellite Conference in Algebra and Related Topics, Hong Kong 2002 President Southeast Asian Mathematical Society 2002-2003

Dedicated to the memory of Professor B H Neumann

Professor Bernhard H. Neumann

Bernhard Hermann Neumann (15 October 1909-21 October 2002) Bernhard Hermann Neumann was born on 15 October 1909 in Berlin. He graduated from the University of Berlin and received his Doctorate in Philosophy in July 1932 at the age of 23. In August 1933 he left Germany for England where he became a research student, this time at Cambridge University, and received his Ph.D. there in 1935. He began his university career as an assistant lecturer at the University College in Cardiff (1937-1940) and a lecturer at the University College in Hull (1946-1948). He then moved to Manchester in 1948 where he later became a Reader. In July 1938 Bernhard married Hanna von Caemmerer. They were blessed with five children. Two of the children, Peter and Walter, are now well-known mathematicians. In 1962 Bemhard and Hanna moved to Australia, where he took up an appointment as the Foundation Chair in Mathematics at the Institute of Advanced Studies, Australian National University (ANU) in Canberra. Hanna was appointed a Professorial Fellow there, and shortly afterwards she became head of the then Pure Mathematics Department in the School of General Studies (now the Faculties) at ANU. Bernhard spent most his mathematical life in Australia. During that time he contributed much to mathematics and mathematical activities in the Asian countries including China, Hong Kong, India, South Korea, Singapore and Thailand. In 1959 Bernhard visited Tata Institute of Fundamental Research, India, and organized a series of 36 Lectures on Infinite Group Theory. The lectures were collected and then published in 1960 in a book entitled Lectures On Topics in the The0y of Infinite Groups. Since then he had established a mathematical network in Asia to promote the discipline in under developed countries. Kar Ping Shum from Hong Kong first met Bernhard in 1980 at Monash, Australia, again in 1985 and 1986 at Alberta in Canada. Bernhard then encouraged Kar Ping to organize an internationalmathematical conference.Kar Ping successfully organized his first international meeting on Algebra and Number Theory in August 1988. He continued to keep in touch with Bemhard and together they promoted to greater heights the level of mathematical awareness in the Asian region. Kar Ping has now become the main organizer for mathematical conferences in Hong Kong and Southeast Asia, which includes the First International Congress in Algebra and Combinatorics in 1997 and the Satellite Conference of ICM in 2002. Between 1982 and 1983 Bernhard also motivated AnnChi Kim from Pusan, South Korea, to organize the 1st International Conference on Theory of Groups in South Korea between 26-3 1 August 1983 -the first formal international conference held in South Korea since the founding ofthe Korean Mathematical Society in 1946. ix

X

AnnChi kept in close contact with Bemhard and eventually succeeded in organizing his second, third and fourth conferences on Theory of Groups in 1988, 1994 and 1998, respectively. The present research group on the Theory of Groups in Korea was founded with much leadership and motivation from Bernhard Neumann. Yuqi Guo from Kunming, People’s Republic of China, was also was motivated by Bemhard to organize his first International Conference on Semigroups and its Related Topics in Yunnan University on 8-23 August 1993. Bemhard also attended the Thailand International Conference on Geometry, 2-20 December 1991 and the Singapore International Group Theory Conference, 8-19 June 1987. Bemhard Neumann’s contributions to mathematics went far beyond his leadership especially for Asia. He was a dedicated supporter of the Southeast Asian Mathematical Society, having helped with the setting up of the Southeast Asian Bulletin of Mathematics and also the Algebra Colloquium (Beijing). He had served as the Honorary Editor for the Bulletin of the Australian Mathematical Society and also the Australasian Journal of Combinatorics. He was an Editorial Board member of the Houston Journal of Mathematics and Editorial Consultant for the magazine, Mathematical Spectrum. Bemhard was indeed a remarkable person. He was courageous, enthusiastic, energetic, gracious, humble, inspiring, peace-loving, sympathetic, warm-hearted and the list goes on. His contributions to the mathematical discipline, algebra especially, are beyond words. In his long and distinguished career he had won the love and respect of many people around the world. His contributions to mathematics will always be remembered and his prominent role in mathematics especially for Asia never forgotten. His absence from forthcoming conferences will be sadly missed by friends and colleagues.

AnnChi Kim and Kar Ping Shum

We are indebted to Mike Newman for allowing us to extract some phrases from an obituary he wrote in the GAZETTE, Australian Mathematical Society, Vol. 30, No.1 March (2003), 1-8.

Contents Preface

V

Dedicated to the memory of Professor B H Neumann

vii

Photograph

viii

Bernhard Hermann Neumann (15 October 1909-2 1 October 2002)

On the powers of a commutative nilpotent algebra B Amberg and L Kazarin

ix

1

On normalized table algebras generated by a faithful non-real element of degree 3 Z Arad and G-Y Chen

13

On finite groups in which subnormal subgroups satisfy certain permutability conditions A Ballester-Bolinches, R Esteban-Romero and M C Pedraza-Aguilera

38

Semigroups satisfying certain regularity conditions S Bogdanovii, M Cirii and M Mitrovii

46

Grobner-Shirshovbases for the braid semigroup LA Bokut, Y Fong, W-F Ke and L-S Shiao

60

4-valent plane graphs with 2-, 3- and 4-gonal faces MDeza, M Dutour and M Shtogrin

73

Graph Semigroups V Dlab and T Pospichal

98

General notions of independence K Gtazek

112

Finite groups with c-normal andf-hypercentral subgroups W-B Guo and KP Shum

129

Complemented minimal subgroups and the structure of finite groups X-Y Guo

140

xi

xii

Finite translation planes from the collineation groups point of view CY Ho

148

A polyhedral description of 3-manifolds AC Kim and Y-K Kim

157

Theorems on admissible subsets of a semigroup G Li, K p Shum and PY Zhu

163

Generalized Schreier variety and a criterion for non-existence of covering groups MRR Moghadam and AR Salemkar

174

Iso-, geno-, hyper-mathematicsand their isoduals constructed from open physical, chemical, and biological problems RM Santilli

185

Semisimple Clifford semirings MK Sen, SK Maity and K P Shum

22 1

Tight inverse semigroups BM Schein

232

Moor-Penrose generalized inverses of matrices over division rings Z-X Wan

244

Properties and characterizationtheorems of SAP-rings Z-X Wu and KP Shum

25 1

Parameterization of G*(2, Z ) on PL(F,) MAshiq and Q Mushtaq

264

Non Isomorphic trees with first three characteristicnumbers equal T Bier and T-H Han

27 1

Sectionally pseudocomplementedlattices and semilattices I Chajda and R Hala;

282

T-stable ideals of regular rings H-Y Chen, M-S Chen and J-Q Li

29 1

...

Xlll

Simple groups which are product of the linear fractional group with the alternating or the symmetric group MR Darafsheh

301

High-density close-closed loop burst error detecting codes BK Dass and S Jain

312

M-solid pseudovarieties and Galois connections K Denecke and B Pibaljommee

325

On regular ternary semirings TK Dutta and S Kar

343

Indecomposable decompositions of CS-modules JL Gomez Pardo and PA Guil Asensio

356

Hereditary rings, QF2 rings and rings of finite representation type CR Hajarnavis

367

Modified RSA cryptosystems over bicodes K Hashiguchi, K Hashimoto and S Jimbo

377

Solid burst error detecting cyclic codes S Jain

390

Survey of some recent results on CS-group algebras and open questions SK Jain, P Kanwar and JB Srivastava

40 1

On ,?*-unitary categorical inverse semigroups Z-H Jiang

409

New examples of noncommutative and noncommutative bialgebras J-Q Li and M-S Chen

424

Generalized con-cos groups AS Muktibodh

434

On some non-associative algebra using additive groups K-B Nam

442

XIV

On the homology bifunctors over sernimodules XT Nguyen

450

On simple subsemigroups of groups JS Ponizovskii

457

Interpolation theorems for the independence and domination numbers N Punnim

459

A note on serial multiplication modules JSanwong and S Srisook

465

*An inverse heat conduction problem A Shidfar and K Tavakoli

472

On a class of Azumaya Galois extensions G Szeto and L-YXue

477

Strongly eventually inverse semigroups whose lattice of full eventually inverse subsemigroups form a chain ZJ Tian and KM Yan

486

Radical theory: Main issues and recent developments R Wiegandt

494

Some problems and conjectures in modular representations J-P Zhang

506

*Erratum: This paper was previously published by Springer in the Southeast Asian Bulletin of Mathematics 26(3), 503-507, (2002) and should not have been reproduced in this publication. The Editors of this publication would like to express regret for this oversight.

On the powers of a commutative nilpotent algebra

Bernhard Amberg and Lev Kazarin*

DEVOTED TO THE MEMORY OF B.H. NEUMANN Abstract Let R be a commutative nilpotent algebra over the field F . If for some natural number k 2 3 the dimension of the factor Rk/Rk+' of the power series of R is bounded by 3 , then for each natural number j 2 k the dimension dim R3/R3+' is bounded by dim Rk/Rk+' . More can be said if the algebra R has only two generators.

AMS-Classification: 16N40, 13A10

1

Introduction

An associative algebra R is called nilpotent if R" = 0 for some positive integer m (see for instance [4]). Here the n -th power of an algebra R is the subalgebra R" of R generated by the set of elements of the form x1 x2 . . . x k with k 2 n, where 21, 22, . . . , xk E R . In particular R1 = R . The minimal natural number n such that the product of each n + 1 elements in R is equal zero, (i.e. Rn # 0 and Rn+l = 0 ) is called the nilpotency class of R . In the following we are interested in the dimensions di = di(R)= dim Ri/Riil of the factors of the power series of R . The following result shows that if d i is relatively small and i > 1 , then the numbers d j 5 di for each j 2 i . This property is useful in understanding the structure of a finite nilpotent algebra (see for instance [l],[2]).

Theorem 1.1 Let R be a n associative commutative nilpotent algebra over the field F such that d k 5 3 f o r some k > 2 and IF1 > 2 . T h e n dj 5 d k f o r each j 2 k . Moreover there exists a n element x E R \ R2 such that Rj = Rj-Ix + Rj+l . *The second author likes to thank the Deutsche Forschungsgemeinschaft (DFG) for financial support and the Department of Mathematics of the University of Maim for its excellent hospitality during the preparation of this paper.

1

2

The case k = 2 is more difficult to handle. Here we prove the following.

Theorem 1.2 Let R be an associative commutative nilpotent algebra over the field F such that d2 5 3 and the minimal number of generators of R is d(R) 2 3 . If the characteristic of F is not 2 , then one of the following holds. (i) For each j > 2 we have d j 5 d 2 . Moreover there exasts a n element x E R \ R2 such that Rj = Rj-lx Rj+l. (ii) There exists an element x E R \ R2 such that Rx C R3 . If F has even characteristic, then either (i) or (ii) holds or d3 _< 2 .

+

Theorems 1.1 and 1.2 may be applied to simplify some arguments used in [l]and [ 2 ] . The case d i 5 2 was considered in [5], [6]. Some previous results of this type can also be found implicitly in [3]. It is well-known that for a free commutative nilpotent algebra R with two generators, di = i 1 for each i > 1 (see [4]). Here we note the following.

+

Theorem 1.3 Let R be a n associative commutative nilpotent algebra over the field F such that d k 5 k f o r some k 2 2 . If the minimal number of generators of R is two, then d j 5 dk f o r each j > k .

2

Bilinear maps

In this section we consider a commutative nilpotent algebra R with minimal generating set X of €2. For each k > 1 consider the vector space Mk = Rk//Rk++' and let V = ( X ) N R / R 2 . Let the ordered set B1 = { q , v 2 , . . . ,v,} be a basis of the complement of the subspace Rk in the vector space R k - l , and let U = (B1) . Let the ordered set B2 = {wl,w2,. . . ,w,} be a basis of the complement of the subspace Rk+' in the vector space Rk , and let W = (B2). Clearly, we have U @ Rk = R"' and W @ Rk+' = Rk . We consider the bilinear map 1c, from V x U onto Rk/Rk+' determined by p ( x y ) = x y mod Rk+l for each pair of elements x E U , y E V . The map 8 : Rk/Rk++' -+ W is determined as a natural isomorphism of Rk/Rk+' onto W . We determine the map C$ from V x U to W as a product C$ = 8 p , so that the following diagram is commutative

uxv

4 +

In,

W

7 8 Rk/Rk+'

Define for every element g E V an n x m-matrix [g] = ( a i j ) , 1 5 i 5 n , l 6 j 5 m , aij E F with respect to the bases B1 and B2 where aij are coefficients which are determined by the following expression n

$(gwi) =

Cajiwj, i = 1 , 2 , . . . , m . j=1

3

It is clear that for each element g E R2 we have [g] = 0 . The changing of the bases corresponds to a changing of all matrices [g] via the related Gauss transformations. We may permute the rows and columns in all matrices [g] simultaneously, multiply the rows or columns by a non-zero element of F or add the rows or columns with given index to the rows or columns with another index. Note that the rank of a matrix corresponding to an element does not depend on the choice of the bases. In the following the rank of the element h of U is the rank of its matrix [h] with respect to some bases B1 and B2 as defined above. Let F [ X ] be the algebra of all commuting polynomials over the field F where X is a set of indeterminates, and let F o [ X ] be the subalgebra of all polynomials in F [ X ] with no constant terms. We may regard the algebra R as a factor algebra of F o [ X ] modulo some ideal. Therefore R has a natural basis consisting of a set of linearly independent monomials with degrees bounded by the nilpotency class of R . Clearly we have u _N h/r, _N F k [ x ] / L k for some subspace L k in Fk[X] where Fk[X]is a subspace of &''[XI consisting of all homogeneous polynomials of degree k . Obviously the basis of U can be chosen so that it consists of monomials of degree k . Choose an element x E V whose rank is maximal among all elements g E V . Without loss of generality (it is not supposed at the moment that the basis B1 consist of monomials) we may assume that

r x r -matrix with r = r a n k ( [ x ] .) If r = n then 1 [XI

=

[

=

)

(

where I is a unit

1

0 0 ...

*o. 0

[XI

O ... 0 1 ...

+

and xu1 = ~ 1 , 2 1 1 2= w2,. . . xu, = w, . Therefore Rk-lx Rk+' = Rk . We = Rk+' . Since Rk = show that in this case dk+l 5 n . Indeed, R k x R"' (wl, w2,. . . ,w,) Rk+l then R k x Rk+2= ( W ~ X , W Z X , .. . ,wnx) Rk+2 = R"'. Hence d k + l 5 n = d k . These considerations show that to prove that dim Rk+'/Rk+' 5 n we may assume that the maximal rank of each matrix corresponding to an element in V is less than n . We will now show that the other extreme possibility when the maximal rank of the element in V is 1 is impossible if dim W > 1 .

+

+

+

+

Lemma 2.1 Let R be a commutative nilpotent algebra over the field F and let V U , W be as above. If dim W = n > 1, then there exists an element x E V such that ranb[z]> 1 . Proof. Suppose that the rank of each element x E V does not exceed 1 . Choose a basis B1 consisting of monomials. Since dim W > 1 there are elements a, b E B1 and x , y E V such that ax = w1, by = w2,where the elements w1, w2 are linearly independent in W . Clearly x # y . Since

4

+

ranlc[x]5 1 and rank[y]5 1 we have ay = Aw2, bx = pwl . Now a ( x y ) = w1 Xw2 and b(x + y) = pw1 w2. Suppose that the elements a and b are linearly independent. Since rank[x y] 5 1 it follows that Xp = 1 . Therefore we may assume that ax = w1,ay = w2 by replacing X W Z by w2 if needed. Recall that the elements in B1 are monomials. Thus the element a can be ~ elements ci E X . It follows that written as a product a = clc2.. . C ~ C - of w1 = ~ 2 ~ 3. ck-lxcl, . . w2 = ~ 2 ~ 3. cI;-lycl .. . Hence the rank of the element c1 is at least 2 , as claimed. The next lemma simplifies some calculations.

+

+

+

Lemma 2.2 Let R be a commutative nilpotent algebra over the field F , and let V , U , W be as above. Let the element x in V have maximal rank. If IF1 2 r + 1 , then there exist bases B1 and B2 consisting of monomials such

that with respect to these the matrix [x] has the form [x]=

( 0’ MIF1’

where I is a unit r x r -matrix with r = ranlc([x]). Moreover, if M12(x) = 0 , then for each element y E V the corresponding matrix has the form [y] =

Proof. Let z E V be an element with maximal rank. Without loss of generality we may assume that the first r columns of this matrix are linearly independent. Then we may choose elements b l , b 2 , . . . ,b, E B1 such that w1 = blx, w2 = b 2 2 , . . . , w,- = b,x are linearly independent. Select a basis B2 of W having w1, w2,. . . ,w, as its first r elements. Then the matrix [z] is of the required form. Suppose that M12(x) = 0 and an element g E V has a matrix

Then there exists a column corresponding to the product by = z with b E B 1 such that z @ ( ~ 1 , 2 0 2 , .. .w T ). Without loss of generality we may assume that b = br+l and z = w,+1 E B2. The matrix [ y ] has a submatrix

(

)

+

M1l(y) of size ( r 1) x ( r + 1 ) . It is easy to see that there M;l ( Y ) 1 exists an element X E F such that the matrix [y + Ax] has a submatrix of rank r + 1 . This contradiction proves the lemma. In the case k = 2 we may always assume that the matrix corresponding to [y]’ =

the element z E V with maximal rank is of the form [ x ]=

(

)

where I

is a unit r x r -matrix with r = r a n k ( [ x ].) Indeed, if bj E B1 with j > r , then aijwi = cqjbix , so that (ELla i j b i - bj)x = 0 . In we have bjx = this case we may replace the element bj by (EL==,aijbi - b j ) = bi because the set Bi obtained from B1 replacing bj by b$ is also a basis of V = U .

XI=,

XI=,

5

3

Proof of Theorem 1.1

Let R be a commutative nilpotent algebra such that the dimension of W cv R k / R k + l is at most 3 , k 2 3 , B1 be a basis consisting of monomials with degree k - 1 of the vector space U cv R k - l / R k . By Lemma 2.1 there exists an element z E V such that the rank of the matrix [z] with respect to the bases B1 and B2 is at least 2 . It follows from the previous section that if Theorem 1.1 is not true, then rank[z]= 2 . We may assume that the first two elements bl and b2 of B1 satisfy w1 = blx,w2 = b 2 x , where { w l , w 2 , w g } = B 2 , Let w3 = by for some b E B1 and y E X . Suppose first that b is different from bl and b 2 . We may assume that b = b 3 , the third element of B1 . Attach to each element h E V the submatrix [h]' of the matrix [h], consisting of the first three columns of [h]. Then we have

'I.[

=

(

0 O1

a2 a1

0 0

0

) ( , [Yl' =

p 2Hl P

P22

p31

p32

1) 0

.

If p31 = ,832 = 0 , there exists an element X E F such that the rank of [y]' + X[x]' = [y Ax]' is equal 3 . This is a contradiction. Hence we may assume that 031 # 0 or p 3 2 # 0 . Suppose that p 3 1 # 0 . p31w3. It is easy to see that In this case we have b l y = ,Bllwl ,&w2 (bly,wg,wg) R"' = R" Thus we may choose our notation so that w1 = b l z , wg = bly, w3 = bzz . Clearly, if p 3 2 # 0 , then we also obtain a basis of the required form. . . with eleIf the element h in R is expressed as a product h = ~ 1 ~ 2. c, ments ci E X , then we define the set supp,(h) = (c1,c2,. . . c s } . Since there may be several expressions of this type, we consider a given decomposition of length s . If an element h does not have such a decomposition, let the set supps(h) be empty. It is clear that for each element h of B1 there exists a non-empty set S'uppk-1 ( h ) related to some decomposition. Define Supp, ( h ) as the union of all sets supps(h) for all decompositions of h of length s with respect to X . We claim that the set SUppk-l(b1) n SUppk-l(b2) is empty. Indeed, if Z E Suppk-i(b1) n S U p p k - i ( b 2 ) then we have W 1 = h 1 ~ ~ = , hlyz,W3 ~ 2 = h z x z , where bl = h l z , b2 = h2z and hl, h2 E R k P 2 .It follows that Rk = R"-'z Rk+' . In each case the rank of [z] is at least three, a contradiction. Hence Suppk-l(bl)nSuppk_l(bz)= 8. Let t E suppk-l(b1). Then we have w1 = alzt,w2 = alyt,wg = b2x for some a1 E R"' . Clearly, a15 and a l y are different from b2 . We may replace the basis B1 by a new basis Bi , which has a l z , a l y , b2 as its first elements. Indeed, it is obvious that these elements are linearly independent modulo Rk and are contained in R"' . Denote these elements by u1,u2,u3 respectively. The submatrices [t]and [z] corresponding

+

+

+

+

+

6

to these three elements in B1 are as follows

( ; ; 7 ) ( 2; ;3; 8) 1 0

[tI’=

P11

a1

Pl2

,[XI’=

,

where ai,PsjE F for all i E {1,2},1 I s I 3 , l I j L 2 . If ,631 = p 3 2 = 0 , there is an element A E F such that the determinant of the matrix [x]’+A[t]’ = [z At]’ is non-zero. Hence there exists an element in V whose matrix has rank 3 . Suppose that p31 # 0 . Then we have a1x2 = Pllwl P21w2 p31w3. It is clear that the elements w1 = ulzt,wz = a l y t and u1z2 are linearly independent modulo R“’. Hence we may consider the basis B2 consisting of these elements. Since SUPPk(W1) n SUPPk(W2) n SUPpk(W3) is non-empty, this is a contradiction. The case p32 # 0 is treated similarly. This proves the theorem.

+

+

+

Proof of Theorem 1.2

4

Consider first the case when the characteristic of the field F is different from 2 . It is possible to choose a basis of U N R2/R3 consisting of squares of elements in V . Indeed, if z , y are distinct elements in X , then xy = 1/2(x y)2 1 / 2 2 - 1/2y2. Suppose that x 2 and y2 are two elements in R which are linearly independent modulo R3 . If both systems x 2 ,xy and y2, xy are linearly dependent modulo R 3 , then xy = 0 mod R 3 . In this case the elements (x y)z and (x Y ) are ~ linearly independent modulo R3 . Therefore in each case we may choose elements z , y E X = B1 such that x 2 = w1,z y = w2 are the elements of the basis B2 of U . By the remark following Lemma 2.1 we may assume that the matrix [z] with respect to the

+

+

+

bases B1 and

B2

is of the following form:

[XI

=

( 0’ )

where I is a

2 x 2 unit matrix, and [z] is a 3 x m-matrix, where m = d ( R ) is the minimal number of generators of R . It follows from Lemma 2.2 that for each h E V we have [h]=

M1l(y)

M12(y)

,

where Mll(y) is an 2 x 2-matrix. Since

dim W = 3 , there exists an element t E V such that ut = W R . It follows from Lemma 2.2 that t E {z, y} and t = y . The following submatrices correspond to the first three elements z,y,z of the basis B1

0 0 0 where

y

p, S

+

+

are elements in F . Considering the matrices [x y]’ and [x ,B = S = 0 . It follows from the above considerations that

+ z]’ , we find that

7

the matrix corresponding to an element z E B1 \ {x,y} is the zero matrix. Hence V z C R3 . Thus, if the characteristic of the field F is different from 2 , we are done. Until1 the end of this section we deal with the case when the characteristic of the field F is 2 . There are the following two possibilities 1. There exist elements x , y E V such that the system {z2,xy} is linearly independent. 2. For each pair of elements x,y E V the system {x2,xy} is linearly dependent. In subcase 1 we may assume as before that the bases B1 = {x, y, z , . . .} and B2 = (w1, w2, w3) are such that the matrix corresponding to the element x is of the following form [XI =

(

1 0 0 ... 0 0 1 0 ... 0 0 0 0 ... 0

)

.

It follows from the above considerations, that we may assume IF1 = 2 . Suppose first that x2 = w1, zy = wg, y2 = 203 . We consider the submatrices [x]’, [y]’,[z]’ of the corresponding matrices with respect to the first three columns. We have

[y]’ =

=

det[y

i::: 2 1 0

=p

01

for some

EF

P1,/32

. Therefore [z]’ =

.

Since ranlc[x+y]’ 5 2 , this implies

3

(1;

0 p

72

for some

71,72,73

EF

.

From

+ z]’ = 0 , det[x + z]’ = 0 and det[x + y + z]’ = 0 it follows that (P + 1)Yl = 0, (P + 1)73+ P72 = 0, (P + 1)(71 + 7 2 + 73) = 0.

Suppose first that P # 1 . Since IF1 = 2 we have P = 0 . It follows from the equations above that for each element z E B1 \ {x, y} the matrix [z]= 0 . Hence V z C R 3 . We may assume now that P = 1 . The above equations yield 7 2 = 0 , so that

[%I.=(

01 1 0 0)’-]’=(

01 0 0‘ ) , [ . I . = (

0 01 701 ) .

0 0 0

0 1 1

0 1

where y1,y3 are elements in F . Since R2 = Rx+Ry+R3 = ( we have

R3 = R2x + R2y

+ R4 = ( w ~ x ,

73

~ 1w2, , w3)+R3,

+ R4.

W ~ XW , ~ XWiy, , W ~ YW , ~Y)

Let al = wlx = x 3 , a 2 = w2x = x2y = wly . Since yz = w2 mod R 3 , we have W ~ X +w3x E 0 mod R 4 ,so that W ~ G X W

+ w3 ~ X

and xz = 0 y2x E wly

8

w2y = a2 mod R 4 . Let a3 = y3. It is clear that (al,a2,a3) + R4 = R 3 . Suppose 71 = 73 = 0 . In this case z2 = 0 mod R3 and yz2 = (w2 + w3)z = 0 mod R 4 . Since w2z = xyz = 0 mod R 4 , we have that w3z = y2z = (yz)y EE (w2 w3)y = w2y + w3y = 0 mod R 4 . Hence a3 = w3y = w2y = a2 m o d R 4 . In this case dim R3/R4 5 2 . Suppose that 71 = 0 # 73. Then z2 = 73w3 and w3x = a2 = 0 mod R 4 . If 73 = 0 # 71, then w1x = a1 = 0 mod R 4 . In both cases we have dim R3/R4 5 2 . Next if 71 # 0 # 73, then ylwlx = 73w3x = ylal z 73a3 mod R 4 , and we arrive at the same conclusion. Suppose now that w1 = x2,w2 = xy, w3 = z2 , where the element z E V is different from x, y . Then the submatrices related to the basis B1 = {x, y, z . . .} are as follows

+

[Yl’ =

0 a 71 1 P 72 ( O 0 73)

I.[ 1

’

=

0 71 0 0 72 0 , ( 0 73 1 )

where the entries of these matrices are elements of F . If 73 # 0 , then a = 0 . Considering the matrix [y x]’ , we obtain that = 1 , and since the rank of the matrix [y+z]’ is less than 3, we have 71 = 0 . But in this case ranlc[x -t- y z ] = 3 , a contradiction. Suppose that 73 = 0 . Considering the matrix [z x] , we obtain y2 = 1 . Since the rank of the matrix [z y] is less than 3 we have 71 = a . From the inequality rank[x y z ] < 3 we obtain that ,b’ = 0 . Thus we have

+

+

+ +

[y]’ =

+

+

O a a ( 01 0 01 )

O a O

; ;),

[zl‘=(:

where a ,P E F . Let a1 = x3 = wlx, a2 = w2x = x2y, w3z = z3. Clearly wly = a2 , w2y = aal mod R 4 , w1z = w2z = 0 mod R 4 , w3y = z2y = z(aw1 w2) 3 0 mod R4. Since zy = awl w2 it follows that w2y a w l y = 0 mod R 4 . From zy2 = zw1 = 0 mod R4 we obtain that w2y = aal = aa2 mod R 4 . If a # 0 , then dim R3/R4 5 2 . On the other hand, if a = 0 , then w2x = x2y = xyz = 0 mod R4 . So in this case we also conclude that dim R3/R4 5 2 . Now we may assume that B2 = {wl = x2,w2 = xy,wg = z u } , where { z , u } n {x, y} = 8 . The submatrices [y]’, [z]’ with respect to the elements x, y, z , ‘LL of the basis B1 are as follows

+

[Yl’ =

+

(

0 1 0

P1

71

P2

72

0

0

:;) 0

, [z]’=

+

( : 5: ; :) 0

0

0

,

1

where all the entries of these matrices are elements of F . It follows from the inequality ranlc[x+z] 5 2 that 7 2 = 1, and p2 = 0 . Hence p1 = 0 . Since the rank of the element y z does not exceed 2 we have also that 71 = P1 = 0 . This implies that the rank of the element x y z is 3 , a contradiction.

+

+ +

9

Suppose now that the basis B2 consists of the set of elements {z2 = w1,xy = w2,yz = wg} . Since y2 E (w1, w2) , the submatrices with respect to the elements z , y , z from the basis B1 are as follows

[Yl’

=

0 0 0 01 ) ,

;

701).

0 0 71

( o1 P

Izl’=(;

+

where the entries of these matrices are elements of F . Since the ranks of J: y and z z are less than 3 , we obtain that ,8 = l,y2 = 0 . It follows from the inequality rank[y z] 5 2 that y1 = 0 . As before it is easy to show that dim R3/R45 2 . In subcase 2 we have the following. If a and b are elements in X such that the system { a 2 ,b 2 } is linearly independent, then it is easy to see that ab = 0 . However the system {(a+b)a,(a+ b)’} is linearly independent, a contradiction. Thus we may assume that each pair of squares of the elements of V is linearly dependent. Suppose that the matrix of some element h E V has rank 2 . There exist elements u,u E V such that uh and vh are linearly independent. Clearly, u and u are distinct from h by the above considerations. If h2 # 0 , then at least one of the systems { h 2 ,hu} or { h 2 ,hv} is linearly independent, a contradiction. Hence if the rank of the matrix of some h E V is two, then h2=0. Now we may assume that the bases B1 and B2 are such that B1 = {z, y, z , . . .} , Bz = {WI = zy,wz = z z , . . .} . It is clear that

+

+

[z]=

(

0 1 0 ... 0 0 0 1 ... 0 0 0 0 ... 0

1

.

If rank[y] = 1 , then we have

[y] =

(

1 a1 0 0 0 0

a2

0 0

... ... ...

am-l

0 0

1

,

where m = d ( R ) and ai E F for each i 5 m - 1. Since dim W = 3 there exists a pair of elements u , u E B1 such that uv = wg . Suppose that { u ,v}n{z, y, z } = 0. We may assume that B1 = {z, y, z, v,. . .} . The submatrix of the matrix [u]corresponding to the first four elements is as follows

[u]’ =

(::: :) 0

0

72

0

,

where the entries of the matrix [u]’ are in F . Since the ranks of [u] and [u+y] do not exceed 2 , it follows that 7 2 = 0 . This implies rank[u y z] = 3 , a contradiction.

+ +

10

Thus u E {x,y,z}. Clearly u is different from x and y . So we have u = z and vz = w3. Since rank[z] 5 2 we obtain z2 = 0 and v # z . If yz # 0 , then the rank of [z] is 3 and if yz = 0 , then [x z] has rank 3 , a contradict ion. Now we may assume that rank[y] = 2 . Suppose that xy = w1,xz = ~ 2 , 2 1 2 ,= w3,yh E (wl,w2) for each h E V and v,u E B1 \ {x,y,z}. Since y2 = 0 , the submatrices of the matrices [y]’,[u]’ corresponding to x, y, z , v are as follows

+

where the coefficients of [y]’, [u]’ are in F . It is obvious that if p2 # 0 , the matrix [u y] has rank at least 3 , a contradiction. If p2 = 0 , then either [y u] or [x y u] has rank 3 , which is also impossible. Therefore we may assume that u E {z,y, z } . If u = y and w $ {z,y,z}, then

+

+

+ +

+

In this case x y has rank 3 , a contradiction. There remains only the case yz = w3. Recall that xy = w1,xz = w2 and z 2 -- y 2 -- z 2 -- 0 . Now we compute the dimension of R 3 / R 4 . Since each product of three terms, containing some square is in R 4 , the basis for R3/R4 consists only of the element xyz . The theorem is proved.

5

Proof of Theorem 1.3

Let R be a commutative nilpotent algebra over the field F . It is convenient to regard R as an algebra over some extension field K of F . Indeed the algebra R1 = R @F K contains R as a subset. Observe that dimF R = dimK R1 and di(R)= di(R1)for each i 2 1 . Therefore we may assume that the ground field F for the algebra R is large enough. Let d(R) = 2 , d k = dim Rk/Rk+’ 5 k for some k 2 2 . We may regard R k / R k f lN W as a factor space of the vector space Fk[x, y] of all homogeneous commuting polynomials over z , y of degree k with some kernel L . It is easy to see that dim Fk[x,y] = k 1 . Attach to each polynomial h(z,y) E Fk[z,y] the polynomial &(t) = h(x,y)/yk E F[t] , where t = z/y . Let f = f ( z , y ) E Fk[x, y] be a non-zero polynomial in L . We may assume that the field F contains all roots of the polynomial d j . Clearly there are not more than k roots of 4f in each field containing F . We may suppose that IF1 > k . Then there exists an element a E F such that ( t - a ) does not divide 4 f ( t ). Choose polynomials 91, g2,. . .g k E F[t] such that g i ( t ) = (t for each i = 1 , 2 , . . . k . Obviously these elements are linearly independent and the subspace

+

11

M = (91, g2.. . .gk) C F [ t ] does not contain the polynomial 4p(t). Hence we have d i m ( g ~ , g z ., .. ,gk) = k and dim(gl,g2,. . . , g k , 4 f ) = Ic 1 . Now there exist polynomials fi(z,y) = gi(x/y)y'" E Fk[x,y] such that (f1, fz,. . .fk, f) = Fk[x,y]. In this case Fk[x,y]= L F)-1[x,y](z- a y ) . Therefore R'" = Rk-'z R"' for the element z E R corresponding to the coset L (x- a y ) . The theorem is proved.

+

+

6

+

+

Final remarks and a conjecture

Let V and W be vector spaces over a field of characteristic different from 2 , f be a symmetric bilinear map from V x V to W such that dim W < dim V . We conjecture that the one of the following conditions holds: 1. there exist x 6 V such that x # 0 and f(z,V) = 0 ; 2. there exist y E V such that y # 0 and f ( y , V) = W . Note that the restriction on the characteristic of the field in Theorem 1.2 is essential. It is easy to construct a corresponding counterexamples using methods as in the proof of Theorem 1.2. If the order of the ground field F of the commutative nilpotent algebra R is 2 and dim R j / R j f l = 3 , then in general there is no element x E R such that Ri = Ri-lx+Ri+' for each a 2 j . However such an element always exists in the algebra R1 = R @IF K if we consider a proper extension field K of F . Thus the inequality di 5 d j for each i 2 j also holds in the case IF1 = 2 .

References [l]B. Amberg, L. Kazarin, On the dimension of a nilpotent algebra, Mat. Zametki (Math. Notes) 70 (2001), 483-490 (in Russian).

[a] B.

Amberg, L. Kazarin, Commutative nilpotent p-algebras with small dimension, (to appear in Quaderni di matematica (Napoli)).

[3] R. Bautista, Units of finite algebras, Ann. Inst. Mat. Univ. Nac. Authoma, Mkxico, 16, No.2 (1976), 1-78 (In Spanish). [4] R.L. Kruse, T. Price, "Nilpotent rings", Gordon and Breach, New York, 1969. [5] C.Stack, Dimensions of nilpotent algebras over fields of prime characteristic, Pacific J. Math. 176 (1996), 263-266. [6] C. Stack, Some results on the structure of finite nilpotent algebras over fields of prime characteristic, Journ. Combinat. Math. Combin. Comput. 28 (1998), 327-335.

12

Address of authors: Bernhard Amberg Fachbereich Mathematik der Universittl Mainz D-55099 Mainz Germany e-mail: [email protected]

Lev Kazarin Department of Mathematics Yaroslavl State University 150000 Yaroslavl Russia e-mail: [email protected]

On Normalized Table Algebras Generated by a Faithful Non-real Element of Degree 3 *Zvi Arad Department of Mathematics and Computer Science, Bar-Ilan University, Ramat Gail 52900,Israel and Netanya Academic College, Kiryat Rabin, Netanya, Israel

tGuiyun Chen Department of Mathematics, Southwest China Normal University,

400715, Chongqing, P. R. China. Email:[email protected]

August 26, 2002 Dedicated t o the memory of Professor B H Neumann

Introduction The concept of “table algebra” was introduced by 2. h a d and H. Blau in [l]in order to study in a uniform way properties of products of conjugacy classes and of irreducible characters of a finite group.

DEFINITION. A table algebra (A, B ) is a finite dimensional, commutative algebra A with identity element 1 over the coniplex numbers C , and a distinguished base B = (61 = 1, bar ..., b k } such that the following properties hold ( where (hi, u ) denotes the coefficient of 6; in u € A, a written as a linear combination of B; and where R* denotes Rt U {0}, the set of non-negative real numbers): *The author is indebted to International Co-research Foundation of China Education Ministry for its fimncial support. ‘The author is indebted to Fred and Barbara Kort Sino-Israel Post,doctora.lProgramme for supporting my post-doctoral position at Ba,r-Ilan University, Emmy Noether Mathematics Institute, NSF of China for their financial support.

13

14

(I) For all i, j, m, bibj = 1 ,G+b, with S i j , a nonnegative real number. (11) There is an algebra automorphism (denoted by -) of A, whose order divides 2, such that bi E B implies that 6;E B. ( Then i is defined by

ti = b). (111) Hypothesis (11) holds and there is a function g:B x B positive reals) such that

-+

R+(the

where g(b;, bm) is independent of j for all i , j, m. B is called the table basis of ( A , B ) . We always use bl to denote base element 1, and Bfl to denote B \ { b l } . The elements of B are called the irreducible components of ( A , B ) , and nonzero nonnegative linear combinations of elements of B with coefficients in R+ are called components. If a = Ck=, Ambm i s a linear combination of elements of B (A, E R”) then Supp{a} = {bmlXm # 0} is called the set of irreducible constituents of a. An element a E A is called a real element if a = ti. Let ( A , B ) be a table algebra with IBI = Ic. Let T I , r2, ...,rli be any positive real numbers such that r1 == 1 and q = T ; for all i . Let 6::= rib; for all i . Then B’ := {bill 5 i 5 Ic} is called a rescaling of B. Two table algebras ( A , B ) and (A’, B’) are called isomorphic (denoted B 2 B’) when there exists an algebra isomorphism $I : A + A’ such that $ ( B )is a rescaling of B’; and the algebras are called exactly isornorphic(denoted B Ez B’) when $ ( B ) = B’. So B Sz B’ means that B and B’ yield the same structure constanats. Proposition 2.2 of [I] shows that if ( A , B ) is a table algebra, then there exists a basis B’, which consists of suitable positive real scalar multiples b’; of the elements b; of B , g(h:, b i ) = 1 for any bi, b$ E B’. Such a basis B’ is called a normalized basis. Now Supp{b:, b:, . . . t&} consists of the corresponding scalar multiples of Supp{bi,bi, . . . b i t } , for any sequence 21, 22, ..., it of indices. So in the proof of any proposition which identifies the irreducible constituents of a product of basis elements, we may assume that B is normalized. Suppose that B is normalized. I t follows fom (111) and [l]Sect. 2 that

15

A has a positive definite Hermitian form, with B as an orthonormal basis, and such that ( a ,bc) = (ab,c ) for all a , b, c E R B . It follows from [I] Sect. 2 that there exists an algebra homomorphism f from A to C such that f ( B )2 Rt. For an element h E B , f ( b ) is called the degree of b. If f ( a ) = f ( b ) for any a, be B’, then ( A , B ) is called homogeneous algebra. If f : b --+ ( b b , l ) , for any b E B is an algebra homomorphism, then ( A , B ) is called a standtud table algebra. Table algebras, as they were defined, may be considered as a special class of C-algebras introduced by Kawada IS] and Hoheisel 191. More precisely, a table algebra is a C-algebra where the structure constants are non-negative. Each finite group G yields two natural table algebras: the table algebra of conjugacy classes (denoted by ( Z C ( G ) ,Cl(G)))and the table algebra of generalized characters (denoted by (Ch(G),Irr(G)). Both table algebras arising from group theory have an additional property: their structure constants and degrees are non-negative integers. Such algebras were defined in [lo] as integral table algebras. Each integral table algebra may be rescaled to a homogeneous one [6], i. e., an algebra, whose non-trivial degrees are equal. The first result in this direction was obtained by Z. Arad and H. blau in [l]where they classified homogeneous table algebras of degree 1. The classification of homogeneous integral table algebras of degree 2 with a faithful element was obtained by H. Blau in [3]. The classification of standard homogeneous integral table algebras of degree 2 with L l ( B ) = 1 was complete in [19], where h ( B ) is the set of linear basis elements ,that is, the set of elements of degree 1. This research was continued in [7] where a complete classification of homogeneous integral table algebras with a faithful element of degree 3 was obtained. Another important class of ITA is comprised of so-called standard integral table algebras which axiomatize the properties of Bose-Mesner algebras of commutative association schemes. Each elements of a table algebra is contained in a unique table subalgebra which may be considered as a table subalgebra generated by this

16

element. So it is natural to start the study of integral table algebras from those which are generated by a single element. Normalized integral table algebras generated by an element of degree 2 were completely classified by H. Blau in [2].If a table algebra is generated by an element of degree three or greater, then its structure is more complicated. If a generating element is real, then we are faced with a classification of P-polynomial table algerbas which, in the case of being done, would imply powerful consequences for a classification of distance-regular graphs. In contrast to this, if a generating element is non-real and of small degree, then either a complete classification or important structure information may be obtained. For example, a complete classification of standard integral table algebras generated by a non-real element of degree 3 were found in ill], [12] under the additional assumption that there is no non-trivial basis element of degree 1. In [4], integral standard table algebras generated by a non-real element of degree 4 or 5 were investigated. The finite linear groups in dimension n 5 7 have been completely classified. See Feit [13]. Deep theory and properties of finite groups are heavily used for the proofs. For n = 2, 3, 4 see Blichfeldt [14]. For n = 5 see Brauer [15]. For n = 6 see Lindsey [16]. For n = 7 see Wales [17]. The purpose of this article is to describe the current situation of our research about classification of normalized integral table algebras (abbrev. as NITA) ( A , B ) generated by a faithful non-real element of degree 3 and without nonidentity irreducible elements of degree 1 or 2. In [20], we state the following Proposition.

Proposition H, Let (A, B) be a NITA generated by a non-real element B of degree 3 and without non-identity basis elements of degree 1 or 2. Then one of the following holds:

b3 E

HP1 : HP2 : HP3 : HP4 : HP5 :

b3b3 = 1 + 2c4, b3b3 = 1 c4 C4, b3b3 = 1 c3 d5, b3b3 = 1 c4 d4, b3b3 = 1+ c B ,

+ + + + + +

where where where where where

c4 E B i s real; c4 E B ; c3, d:, E B and c3, d5 are real; c4, d4 E B , c4 # d4 and c4, d5 are real; cs E B i s real.

17

As a consequence of this proposition we proved in [20] by elementary methods and without using finite group theory the following Main Theorem: Main Theorem. Let ( A , B ) be a NITA generated by a non-real element 63 E B of degree 3 and with,out non-iden,tity basis elements of degree 1 or 2. T h e n b3b3 = 1 C8 an>done of the following h.olds: (1) (A, B ) Nz (Ch(PSL(2,7)),I T T ( P S L (7~) ,) ; (2) bz = 63 bs, where bs E B i s nonreal; (3) bi = c3 bG, where c 3 , bs E B and c3 # b3, 63; (4) bi = c4 c5, where c4 E B and c5 E B.

+

+

+

+

Remark 1: The NJTA induced from the algebra of characters of 3 As is the unique NITA satisfying (2) induced from the algebras of characters of finite groups by Blichfeldt [14] and the Atlas of Finite Groups [18]. I

Remark 2: We constructed two NITA not induced from finite groups and salisfying ( 3 ) . One NITA has dimention 22 where s g is nonreal and the second NITA has dimension 32 where bs is nonreal. (The reader is adviced to put attension to that in the NITA of dimension 22 we set sf3instead of brj as stated in the main theorem). The equations of these NITA are too long to introduce in this paper . We conjecture that NJTA satisfying (4)dose not exist . For the purpose of the proof of the main theorem, we list the structure of the NITA induced from the algebra of characters of PSL(2, 7) below. It has a basis B = {bl = 1, b3, $3, bs, b7, cg}, where bs, b7 and C8 are real and the following equations hold:

18

Here we also list the structure of the algebra induced from characters of 3 . As, which is of dimension 17 with a table subset of dimension 7:

19

20

21

+

+

b ; ~ = 1 f 2b5 f 2 C 5 2 C 8 f 2b8 f 3bg This normalized table algebra is strictly isomorphic to the normalized table algebra ( C h ( 3 . AG),I r r ( 3 * As)) as one can prove using the Atlas [18].

Remark 3. In the following 2 examples of dimensions 22 and 32 satisfy the condition of our Main Theorem we set for the example of dimension 22: b&s = 1 f b8 and h: = T 3 f S6 instead Of b363 = 1 f C8 and b: = C 3 f h6 as stated in the main theorem. The reader is adviced to put attention to these changes of notations.

Two NITA Generated by a Non-real element of Degree 3 not derived From a Group Here are two emmples of NITA not induced from group theory, which has either 3 2 or 22 basis elements. The NITA of dimension 32 contains a subalgebra that is strictly isomorphic to the algebra of dimension 17 of characters of the group 3 . AG, we denote it as ( A ( 3 . As . a ) , B32), where its basis is B32.

For

B32

c = B3:! =

D

there are three table subsets: { 1) C C

C D C B , where

(1, b8, 2101 6 5 , C5r c8, Z9)r CU{Cx, c 3 , &, 23, Cg, c g , h, $6, Y5, gi5), nu ( h 3 , 63, 7-3, ZG, ZG, S6r b9, $9, 515, 515, t15r

d9i y3i 23,

53).

Froin the below equations one can check that B32 has the following table subsets: (1) C C C E = CU (7-3, S 6 , t15, d g , y 3 ) C: B32. The table subsets E and D are two maximal table subsets of B32.

22

The structure algebra constants of C.

The strucure algebra constants

23

the equations of C as shown above):

24

From the equations one can see that C and D are table subsets of B32. Furthermore the NITA subalgebra (< D >, 0) of dimension 17 is strictly isomorphic to ( C H ( 3 . As), Irr(3 . A s ) ) while our NITA of dimension 32 is not induced from a finite group G.

25

The following equations give out other products of basis elements in B32:

26

27

28

29

30

31

The interesting thing is that this example also contains table subsets.

where C and E are the same as defined in the previous example of dimension 32. In particular both examples of dimensions 22 and 32 contain each the same sub-table algebra of dimension 12 generated by the basis E. Also E is a maximal table subset of B22. The equations of products of the basis elements of C and E are as in the example of A(3 . A6 . 2, 32) described above.. b363 = 1 b8,

+ b$ = + = f = + t15, = 2t15 + s 6 f = b8 f + = + + T3

S6,

bat6

b8

b3f6

T3

x10,

b3b15

210

b3b15

b3~9

ti5

b3y9

C8

&I

f 29

b 3 ~ 3 dg,

b3Z3 = ~

g ,

d9, 65

f

~ 3 ,

+ $10,

C5

+

C8

f

Xg,

32

33

53d9

= 615

+ + y9. 63

34

Table 1: NITA ( A ( 3 .A6 . a), B 3 2 )

The powers of

b3

If we define in a table algebra ( A , B ) for R, S C B ,

R* S = { U S u p p ( R . S ) ( rE R, s

E

S},

then: Cb3 * Cb3 = Cri = Cc3 * CC3 = C, C is identity. (Cb3)2= CC3, Cba * Cc3 = (77-3, Cb3 * CC3 = C63 and Cb3Cr3 = CE3. For the definitions of abelian table algebra,, quotient of table algebra and its basis and the subtable algebra generated by an element b E B denoted by < b >= Bb see [l] and [lo]. Also the definitions of linear and faithful element b E B are found there. One can derived from our table that B32 = U ~ ~ l = B UEOBG b ~ and b3 is a faithful element of B32 generates B32. Also the quotient table algebra with basis B32/C is strictly isomorphic the table algebra induced by the cyclic group Z6 of order 6.

35

Table 2: NITA (A(7.5. lo),

B22)

One can derived from this table that B 2 2 = U:=,BbZ; = U ~ o , o S band $ b3 is a faithful element of B22 generates B 2 2 . Also the quotient table algebra with basis B22/C is strictly isomorphic to the table algebra induced by the cyclic group Z4 of order 4. The proofs related to our results are too long for the proceeding of the conference and we plan to send them for publication elsewhere.

References [l] Z.Arad and H. Blau. On table algebras and application to finite group theory, J. Algebra 138(1991),137-185 [2] H. I. Blau. Integral table algebra, affine diagrams, and the analysis of degree two, J. Algebra, 178(1995), 872-918

[3]H.I. Blau. Homogeneous integral table algebras of degree two, Algebra Colloq. 4(1997), 393-408 [4]Z. Arad and M. Muzychuk. Standard Integral Table Algebras Generated by a Non-real Element of Small Degree. Lecture Notes in Mathematics, Springer, 1773(2002)

36

[5] B. T. Xu. On a class of integral table algebra, J. Algebra., 178(1995), 760-78 [6] H. I. Blau and B. T. Xu. On homogeneous table algebras, J . Algebra, 199(1998), 393-408 [7] H. I. Blau, B. Xu, 2. Arad, E. Fisman, V. Miloslvsky and M. Muzychuk. Homogeneous integral table algebras of degree three: a trilogy. Memoirs of the AMS, 144(684)(2000)

[8] Y. Kawada. Uber den Dualitatssatz der vcharaktere nichtcommutative Gruppen. Proc. Phys. Math. SOC.Japan, 1942, 24(3), pp. 97-109 [9] G. Hoheisel, Uber Charatere. Monatsch. f. Math. Phys. 1939, 48, pp. 448-456 [lo] H. I. Blau. Quotient structures in C-algebra. J. of Algebra, v. 177(1995), pp. 297-337 [ll] 2 . Arad, H. Arisha, E. Fisman, M. Muzychuk. Integral standard table

algebras with a faithful nonreal element of degree 3, J . of Algebra 231(2), (2000), 473-483 [12] H. I. Blau. Integral table algebras and Bose-Mesner algebras with a faithful nonreal element of degree 3, J. of Algebra 231(2)(2000), 484545

[13] W. Feit. The current situation in the theory of finite simple groups, Actes, Congrbs intern. math., 1970. Tomel, pp. 55-93 [14] H. F. BIichfeldt. Finite Collineation Groups. University of Chicago Press, Chicago (1917) [15] R. Brauer. Uber endliche 1ineareGruppen von Primzahlgrad, Math. Ann., 169(1967), pp. 73-96 [16] J. H. Lindsey. On a projective reperesentation of Hall-Janko group, Bull. AMS, 74(1968), p. 1094

37

[17] D. B. Wales. Finite linear groups of degree seven I, Cana. J. Math., 21(1969), pp. 1025-1041

[18] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, New York, London, Oxford, 1985

[19] Z. Arad, E. Fisman and M. Muzychuk. Standard integral tabel algebra generated by elements of degree two. (submitted) [20] G. Y. Chen, Z. Arad. On Normalized Table Algebras Generated by a Faithful Non-real Element of Degree 3. (to be submitted)

ON FINITE GROUPS IN WHICH SUBNORMAL SUBGROUPS SATISFY CERTAIN PERMUTABILITY CONDITIONS A. BALLESTER-BOLINCHES Departament d'Algebm, Universitat de Valdncia; Dr. Moliner, 50; E-46100 Burjassot (Valdncia, Spain) E-mail: Adolfo.Ballester4uv.es

R. ESTEBAN-ROMERO AND M. C. PEDRAZA-AGUILERA Departament de Matembtica Aplicada, Universitat Politdcnica de Valdncia; C a m i de Vera, s/n; E-46022 Valdncia (Spain) E-mail: restebanQmat.upv.es and [email protected]. es Some characterisations of the classes of finite groups in which normality (respectively, permutability and Sylow permutability) are given in this survey.

Dedicated to the memory of Professor B H Neumann 1

Introduction

All groups considered in this paper will be finite. A group G is said to be a T-group when every subnormal subgroup of G is normal in G, that is, when normality is a transitive relation. These groups have been widely studied (see, for example, A subgroup H of a group G is said to be permutable (or quasinormal) in G when H K = K H for all subgroups K of G. It is clear that normal subgroups are permutable. Recently, some authors have been interested in the groups G in which permutability is a transitive relation, that is, if H is permutable in K and K is permutable in G, then H is permutable in G. These groups are called PT-groups. By a theorem of Kegel (Satz 1 of 4), permutable subgroups are subnormal. Hence PT-groups are exactly those groups in which every subnormal subgroup permutes with every other subgroup. Therefore every T-group is clearly a PT-group. These two classes are both subclasses of a wider class: the class of PST-groups. Recall that a subgroup H of a group G is said to be S-permutable in G when H permutes with every Sylow subgroup of G. A group G is said to be a PST-group if every subnormal subgroup of G is S-permutable in G. Kegel showed that Spermutable subgroups are subnormal. Therefore PST-groups are the groups in which S-permutability is a transitive relation, that is, if H is S-permutable in K and K is S-permutable in G, then H is S-permutable in G. This class of groups has been studied, for example, in.516,7,8>9>10J1 A non-modular p-group is a typical example of a PST-group which is not a PT-group, and a modular pgroup which is not Dedekind is a typical example of a PT-group which is not a T-group. There are, in essence, three different ways of approaching the question of characterising T - , PT-, and PST-groups. We shall present a general perspective of these three lines.

38

39 2

Characterisations based on the normal structure

The classical theorem of Gaschutz of 1957 shows that soluble T-groups are exactly the soluble groups G with an abelian normal Hall subgroup L of odd order such that G I L is a Dedekind group and such that the elements of G induce power automorphisms in L. In 1964, Zacher l2 stated the corresponding theorem for soluble PT-groups by replacing ‘Dedekind’ by ‘nilpotent modular’ in Gaschutz’s theorem. Finally, in 1975, Agrawal obtained the characterisation of soluble PSTgroups in a way similar to Gaschutz’s and Zacher’s characterisations by changing ‘Dedekind’ by ‘nilpotent’ in Gaschutz’s characterisation. An immediate consequence of these theorems is that the classes of soluble T-, PT-, and PST-groups are subgroup-closed, a fact which is not evident from the definition. Note that the differences between all three classes is just the Sylow structure. The following result (Theorem 2 of 7 , supports this claim: Theorem 1. Let G be a group. 1 . Suppose that p i s a p r i m e number and that H is a n S-permutable p-subgroup

of G. If the Sylow p-subgroups of G are modular (respectively, Dedekind), then H is permutable (respectively, normal) in G. 2. A s s u m e that H is a n S-permutable subgroup of G. If the Sylow subgroups of G are modular (respectively, Dedekind), t h e n H i s permutable (respectively, normal) in G. Local approaches to Agrawal’s theorem are presented in.7 3

Characterisations based on the Sylow structure

As we mentioned in the previous sections, the Sylow subgroups give the difference between the above classes in the soluble case. Consequently, it seems natural to look for characterisations of the these classes in terms of psubgroups and their normalisers ( p a prime). Robinson introduced in 1968 the class C, of groups G in which every subgroup of a Sylow psubgroup P of G is normal in the normaliser NG(P). He showed that a group G is a soluble T-group if and only if G satisfies C, for all primes p. Thirty-one years later, Beidleman, Brewster, and Robinson extended this concept to permutability as follows: G satisfies X, when every subgroup of a Sylow psubgroup P of G is permutable in the normaliser N,(P). They proved in that G is a soluble PT-group if and only if G satisfies X, for all primes p. However, if we say that G is a Y,*-group by changing ‘permutable’ by ‘S-permutable’ in the definition of X,,we have that there are groups, like the symmetric group of degree 4,satisfying y; for all p , but not being PST-groups. Hence a local approach to PST-groups should be slightly different. The abnormality of the Sylow normalisers in the group proves easily that the property C, is inherited by subgroups. The property X, is also inherited by subgroups, as Beidleman, Brewster, and Robinson noted in.’ However, their proof of

’

40

this result follows after an intensive study of the property and its influence on the structure of the group. In fact, this is the last theorem of their paper and is a consequence of a much stronger result. the first and second authors proved the following pnilpotence In Theorem 1 criterion in groups with modular Sylow psubgroups: Theorem 2. Let p be a prime and let G be a group with a modular Sylow p-subgroup P . T h e n G i s p-nilpotent if and only if N G ( P ) is p-nilpotent. As an immediate consequence of this criterion, we have that property X, is inherited by subgroups, because X,-groups have modular Sylow psubgroups. This allows US to simplify the proofs of many of the already existing theorems. For example, one can see that in Beidleman, Brewster, and Robinson’s paper, a lot of effort is put into proving that every group satisfying X , for every prime p must be soluble. However, this assert follows now directly from the fact that X , is closed under taking subgroups. A similar definition for S-permutability must be subgroup-closed, as there are no structure restrictions on Sylow psubgroups: We say that a group G is a Ypgroup when for all psubgroups H and S of G such that H 5 S, H is S-permutable in N G ( S ) . With this definition and the previous remarks, we have that the only differences between the properties C,, X,, and Y p are just the structure of Sylow psubgroups: Dedekind in the C,-case, modular in the X,-case (Theorem 3 of 7). It is also proved in Theorem 4 of that a group G is a soluble PST-group if and only if G satisfies Yp for all primes p . In Theorem 5 the following result has been proved: Theorem 3. A group G is a Yp-group i f and only i f G i s either p-nilpotent, or G has abelian Sylow p-subgroups and G satisfies C,. This theorem has revealed itself to be useful to prove some interesting results on PST-groups. For instance, in Theorem H of it has been proved that a soluble group G is a PST-group if and only if every subnormal subgroup of G permutes with every Carter subgroup of G and the subnormal subgroups are hypercentrally embedded in G. As an application of Theorem 3, its is proved in Corollary 2 of l3 that the permutability with the Carter subgroups can be removed: Theorem 4. A soluble group G i s a PST-group af and only i f every subnormal subgroup of G is hypercentrally embedded. A local version of this theorem for psoluble groups with Y p is also given in Theorem B of. 13. Another application of Theorem 3 is the following Agrawal-like characterisation of p-soluble groups with property Y,, proved in Theorem A o f I 3 Let p be a prime and let Ep, 6, be the saturated formation of all pnilpotent groups. For each group GIwe denote by G ( p ) the E,rG,-residual of G, that is, the smallest normal subgroup of G such that G / G ( p ) is pnilpotent. Theorem 5. A group G is a p-soluble group satisfying Y p i f and only i f 1. either G is p-nilpotent,

OT

2. G(p)/O,t ( G ( p ) ) is a n abelian normal Sylow p-subgroup of G/O,t ( G ( p ) ) such that the elements of G / O , f ( G ( p ) ) induce power automorphisms in

41

G(P)lOPl ( G W ) . This theorem allows us to give a shorter proof of Agrawal’s theorem and, hence, of Zacher’s and Gaschutz’s theorems. On the other hand, Bryce and Cossey studied in the soluble universe a new local version of T-groups. Given a prime p, they defined the class Up’of p-supersoluble groups G whose pchief factors constitute a single isomorphism class of G-modules. They also introduce the class Tp of groups G in which every p’-perfect subnormal subgroup of G is normal in G. They show that a soluble group G is a Tp-group if and only if G is a U,*-group and all Sylow p-subgroups of G are Dedekind groups. They characterised soluble T-groups as the groups G satisfying Up’for all p and whose Sylow subgroups are Dedekind (Theorem 2.3 of ’). Bearing in mind that the difference between all three classes of groups in the soluble universe is the Sylow structure, it seems natural to think of Up’ as a local version of PST-groups. Alejandre, the first, and the third authors studied in l4 the class U,*for a single prime p. They defined PST,-groups, for a prime p, as psoluble groups G in which every p’-perfect subnormal subgroup of G permutes with every Hall p‘-subgroup of G. They showed the equivalence between both concepts (Corollary 2 of 14) and that soluble PST-groups are exactly the groups satisfying Up’ for all primes p (Theorem 8 and Corollary 2 of 14). They also showed that soluble PT-groups are the groups satisfying Up’for all primes p whose Sylow subgroups are modular (Corollary 4 of 14). On the other hand, in Theorem 9 of,? the first and second authors prove that a psoluble group satisfies Y p if and only if it is a PST,-group. 4

Characterisations based on embedding properties

In this section we are interested in subgroup embedding properties 0 such that a group G belongs to a certain class of groups if and only if every subgroup of G (or every subgroup in a family of subgroups of G) is @-embedded in G. Peng l5 and Robinson showed that a group G is a soluble T-group if, and only if, every psubgroup of G is pronormal in G. More recently, Bianchi, Gillio Berta Mauri, Herzog, and Verardi introduced in l6 the following embedding property: a subgroup H of a group G is an ‘H-subgroup of G when for every g E G, N c ( H )nHg 5 H . They show that a group G is a soluble T-group if, and only if, every subgroup of G is an ‘H-subgroup of G. This embedding property is closely related to the weak normality introduced by Muller in:17 a subgroup H of a group G is weakly normal in G when from Hg 5 N G ( H )it follows that g E N G ( H ) . It is clear that every ‘H-subgroup of G is weakly normal, but there exist weakly normal subgroups that are not ‘H-subgroups. If H is a weakly normal subgroup of a group G and H is a normal subgroup of a subgroup K of G, then N G ( K )5 N G ( H ) . This fact is the key for the proof of the main theorem in l6 and is an embedding property introduced by Mysovskikh in:18 A subgroup H of G is said to satisfy the subnormalaser condition in G when for every subgroup K of G such that H K it follows that N G ( K )I N G ( H ) . The goal of the paper,” by the first and second authors, is to study all possible

a

42

connections between the latter embedding properties and to use them to characterise soluble T-groups. The following result is proved: Theorem 6. Let G be a group. The following statements are pairwise equivalent: 1. G is a soluble T-group,

2. every subgroup of G is weakly normal in G, and 3. every subgroup of G satisfies the subnormalaser condition. The study of local properties related to the above embedding properties in the line of Bryce and Cossey’s work are useful in the proof of the previous theorem. This draws our attention to the study of psubgroups, where p is a prime. Curiously, in these subgroups pronormality, weak normality and subnormaliser condition are equivalent properties. One might look now for permutable and S-permutable versions of weak normality and subnormaliser condition useful to characterise the classes of soluble PT- and PST-groups. In the paper,” the first and second authors define these properties. We say that a subgroup H of a group G is weakly S-permutable when the following condition holds:

Hg),then H is S-permutable in If g E G and H is S-permutable in (H, ( H ,9 ) . We say that a subgroup H of a group G satisfies the S-subpermutiser condition when the following condition holds:

If H is S-permutable in a subgroup K of G and z is an element of G such z), then H is S-permutable in (H, z). that K is S-permutable in (H, It is clear that weakly normal subgroups are weakly S-permutable, and that the subnormaliser condition implies the S-subpermutiser condition. A well-known result of Wielandt shows that the following three statements are equivalent for a subgroup H of a group G: 1. H is subnormal in G,

2. H is subnormal in ( H ,Hg) for every g E G, and

3. H is subnormal in (H,g) for every g E G. In Theorem A of,2oWielandt’s result is extended in the following way: Theorem 7. A subgroup H of a group G is S-permutable in G if, and only if, H is S-permutable in (H,g) for every g E G. As a consequence, we have that if a subgroup H satisfies the S-subpermutiser condition in a group G and H is subnormal in a subgroup K of G, then H is Spermutable in K , and that every weakly S-permutable subgroup satisfies the Ssubpermutiser condition. On the other hand, it is possible to characterise the property yp in terms of the S-subpermutiser condition:

43

Theorem 8 (Theorem B of equivalent:

20).

If G is a group, the following statements are

1. G is a &-group, and 2. every p-subgroup of G satisfies the S-subpermutiser condition.

We must note that, contrarily to what happens to normality and 7-groups, these conditions are not equivalent to every psubgroup being weakly S-permutable in G. This theorem allows us to give the following characterisations of soluble PSTgroups: Theorem 9 (Theorem C of 20). Let G be a group. The following statements are equivalent: 1. G is a soluble PST-group,

2. every subgroup of G is weakly S-permutable, 3. for every prime p , every p-subgroup of G is weakly S-permutable in G ,

4. every subgroup of G satisfies the S-subpermutiser condition in G, and 5. f o r every prime p , every p-subgroup of G satisfies the S-subpermutiser condition in G.

These results can be easily extended to permutability and PT-groups. 5

U; as a class of groups

The important role played by the class U; in the study of T - ,PT-, and PST-groups makes interesting the study of its behaviour as a class of groups. It is clear that U; is closed under taking homomorphic images, but Up*is not a formation, that is, U; is not closed under taking subdirect products. In fact, we have: Theorem 10. The class of all p-nilpotent groups is the largest formation contained in Up*. Since a soluble PST-group is a U,*-group for all primes p (Theorems 6 and 8 of 14), we have: Corollary 1. The class of all nilpotent groups is the largest formation contained in the class of all soluble PST-groups. Up’is not, in general, saturated (see 21 or 22). However, the following result is true: Theorem 11. Let G be a group. The following statements are equivalent:

I . f o r every subgroup H of G, H / @ ( H )is a U;-group, and 2. G is a U,*-group.

44

The above theorem can be applied to obtain some interesting results on T - , P T and PST-groups. Let P S T o (respectively, P T o , TO)be the class of groups G such that G/@(G) is a PST-group (respectively, PT-group, To-group). Theorem 12. 1. The class of all soluble PST-groups is the largest subgroupclosed class contained in the class of PSTo-groups. 2. The class of all soluble PST-groups is the largest subgroup-closed class contained in the class of PTo-groups.

3. The class of all soluble PST-groups is the largest subgroup-closed class contained in the class of To-groups. Acknowledgments This work has been supported by Proyecto BFM.2001-1667-C03-03, from MCyT (Spain) and FEDER (European Union).

References 1. R. A. Bryce and J. Cossey. The Wielandt subgroup of a finite soluble group. J. London Math. SOC.,40(2):244-256, 1989. 2. W. Gaschutz. Gruppen, in dennen das Normalteilersein transitiv ist. J. reine angew. Math., 198:87-92, 1957. 3. D. J. S. Robinson. A note on finite groups in which normality is transitive. Proc. Amer. Math. SOC.,19:933-937, 1968. 4. 0. H. Kegel. Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math. Z., 78:205-221, 1962. 5. R. K. Agrawal. Finite groups whose subnormal subgroups permute with all Sylow subgroups. Proc. Amer. Math. SOC.,47(1):77-83, 1975. 6. M. Asaad and P. Csorg6. On T*-groups. Acta Math. Hungar., 74(3):235-243, 1997. 7. A. Ballester-Bolinches and R. Esteban-Romero. Sylow permutable subnormal subgroups of finite groups. J. Algebra, 251(2):727-738, 2002. 8. J. C. Beidleman, B. Brewster, and D. J. S. Robinson. Criteria for permutability to be transitive in finite groups. J . Algebra, 222(2):400-412, 1999. 9. J. C. Beidleman and H. Heineken. Finite soluble groups whose subnormal subgroups permute with certain classes of subgroups. Preprint. 10. D. J. S. Robinson. Minimality and Sylow-permutability in locally finite groups. Preprint. 11. D. J. S. Robinson. The structure of finite groups in which permutability is a transitive relation. J . Austral. Math. SOC.Ser. A, 70:143-159, 2001. 12. G. Zacher. I gruppi risolubli in cui i sottogruppi di composizione coincidono con i sottogrupi quasi-normali. Atti Accad. Naz. Lincei Rend. cl. Sci. Fis. Mat. Natur. (8), 37:150-154, 1964. 13. A. Ballester-Bolinches and R. Esteban-Romero. Sylow permutable subnormal subgroups of finite groups 11. Bull. Austral. Math. SOC.,64(3):479-486, 2001.

45 14. M. J. Alejandre, A. Ballester-Bolinches, and M. C. Pedraza-Aguilera. Finite soluble groups with permutable subnormal subgroups. J. Algebra, 240(2):705721, 2001. 15. T. A. Peng. Finite groups with pronormal subgroups. Proc. Amer. Math. SOC., 20:232-234, 1969. 16. M. Bianchi, A. Gillio Berta Mauri, M. Herzog, and L. Verardi. On finite solvable groups in which normality is a transitive relations. J. Group Theory, 3(2):147-156, 2000. 17. K. H. Miiller. Schwachnormale Untergruppen: Eine gemeinsame Verallgemeinerung der normalen und normalisatorgleichen Untergruppen. Rend. Sem. Mat. Univ. Padoua, 36:129-157, 1966. 18. V. I. Mysovskikh. Investigation of subgroup embeddings by the computer algebra package GAP. In Computer algebra in scientific computing-CASC’99 (Munich), pages 309-315, Berlin, 1999. Springer. 19. A. Ballester-Bolinches and R. Esteban-Romero. On finite 7-groups. To appear in J. Austral. Math. SOC. 20. A. Ballester-Bolinches and R. Esteban-Romero. On finite soluble groups in which Sylow permutability is a transitive relation. Preprint. 21. M. Asaad and A. A. Heliel. Finite groups in which normality is a transitive relation. Arch. Math. (Basel), 76:321-325, 2001. 22. R. W. Van der Waall and A. Fransman. On products of groups for which normality is a transitive relation on their Frattini factor groups. Quaestiones Math., 19:59-82, 1996.

SEMIGROUPS SATISFYING CERTAIN REGULARITY CONDITIONS*

STOJAN BOGDANOVIC Faculty of Economics, University of NiS, Trg Kralja Aleksandra 11, P. 0. Box 121, 18000 NiS, Serbia E-mail: sbogdanOpmf.ni.ac. yu

MIROSLAV CIRIC Faculty of Sciences a n d Mathematics, University of NiS, ViSegradska 33, P. 0. Box 224, 18000 NiS, Serbia E-mail: ciricmObankerinter.net

MELANIJA MITROVIC Faculty of Mechanical Engineering, University of Nag, Beogradska 14, 18000 NiS, Serbia E-mail: me1iOmasfak.masfak.ni.ac. yu To the memory of Professor B. H. Neumann

S. Bogdanovid, M. CiriC, P. Stanimirovid and T. Petkovid in [4] proved that every regularity condition on semigroups is equivalent to one of the 14 conditions given here in Figure 1. Semigroups satisfying some of these conditions were studied only from the ideal-theoretical point of view, by S. Lajos and G. S z h z in [ll].The purpose of this paper is to describe the structure and give some new ideal-theoretical characterizations of these semigroups.

1. Introduction The notion of the regularity in semigroups and rings was introduced by J. von Neumann in 1131, 1936, and his work initiated investigation of many other types of the regularity. Left, right and complete regularity were introduced and studied by A. H. Clifford in [6],1941, and R. Croisot in [8], 1953, 'This work is supported by grant MM-1227 of the Ministry of Science and Technologies of Republic of Serbia

46

47

intra-regularity by R. Croisot, 1953, and J. Callais [5], 1961, defined and studied left and right quasi-regularity. Semigroups called here intra-quasiregular (known also as semisimple semigroups) were from various aspects investigated by W. D. Munn in [12], 1955, and G. Szdsz in [15], 1974. G. Szdsz in [16, 171, 1976, studied left, right and intra-reproduced elements and semigroups, whereas S. Lajos and G. SzQz in [ll],1975, investigated the so-called ( p ,q, r)-regularity. All types of the regularity of semigroups determined by equations of the form a = amzan, with m, n 2 0 , m+n 2 2, were studied by R. Croisot in [8], 1953, who proved that any of them is equivalent to the regularity, left, right or complete regularity (see the book by Clifford and Preston [7], Sect. 4.1). A similar problem, concerning all types of the regularity determined by equations of the form a = aPxaqyar, withp, q, r 2 0, was treated by S. Lajos and G. Szdsz in [ll],1975. In [4], S. Bogdanovib, M. Cirib, P. StanimiroviC: and T. PetkoviC considered a more general problem and determined all types of the regularity of semigroups and their elements determined by linear equations, where linear equations are defined as follows. For a given alphabet A , by A+ the free semigroup over A is denoted. Let X be a countable alphabet whose elements are called variables, and let c be a symbol such that c $! X , called a constant. By L we denote the set of all words u 6 (X U { c } ) + satisfying the following conditions: - the constant c appears at least once in u, - at least one of the variables appears in u, - any variable appears at most once in u. Note that any word u E L is linear with respect to variables. We write u(c,2 1 , . . . ,xn)instead of u to emphasize that {XI,.. . ,x,} is the set of all variables appearing in the word u. Let u ( c ,z1,. . . ,x,) E L, let S be a semigroup and a, a l , . . . ,a, E S. By u ( a , a l , . . . ,a,) the element of S determined by u ( a ,a l , . . . ,a,) = u p is denoted, where 9 : (X U { c } ) + + S is a homomorphism satisfying the conditions ccp = a and xicp = ai, for every i E [l,n]. In other words, u ( a ,a l , . . . ,a,) is an element of S obtained replacing c by a and xi by ai in u and considering the multiplication in S instead of the concatenation in the free semigroup ( X U {c})+. For u(c,2 1 , . . . , z,) E L , an expression of the form c = u(c,5 1 , . . . ,x,) is called a linear equation, regularity equationa, or simply an equation. For The equation is called linear because the word u is linear with respect t o variables, and it is called a regularity equation because it defines one kind of regularity of elements

a

48

a semigroup S and a E S , an expression of the form a = u ( a ,z1,. . . ,zCn) is called an equation in S . This equation is said to be solvable in S if there exist al,. . . , a , E S such that a = u ( a ,al,. , . , a n ) . For an equation c = u ( c ,21,. . . ,z,), by the regularity condition determined by this equation an expression of the form (( c = u ( c ,2 1 , . . . , 2,))) is meant. For a semigroup S we say that it satisfies the regularity condition ( ( c= u(c,z1,.. . ,z,))) if the equation a = u(a,z1,.. . , z n )is solvable in S for every a E S. We say that a regularity condition (( c = u(c,X I ,. . . ,xn))) implies a regularity condition (( c = v(c, y1,. . . ,),y )) if every semigroup satisfying ((c = u ( c ,2 1 , . . . ,z,))) satisfies also ((c = v(c,y l , . . . ,y,))), and that (( c = u(c,zl,. . . ,x,))) is equivalent to (( c = v(c,y1,. . . ,y,))) if for every semigroup S , S satisfies (( c = u ( c ,z1,. . . ,z n ) ) )if and only if it satisfies (( c = 2,( c ,Y 1, . * . 7 Ym ) )) . , S. Bogdanovib, M. Cirib, P. Stanimirovid and T. Petkovib in [4] proved that every regularity condition is equivalent to one of the 14 conditions given in the following figure:

Figure 1.

Figure 1 also represents the implication diagram for these conditions. This diagram can be extended by conjunctions of some of the above regularity conditions. Namely, we will say that a semigroup S satisfies the regularity condition (( c = u ( c ,XI,.. . ,z n ) ,c = v(c, y1,. . . ,y,))) if it satisfies both of the conditions (( c = u ( c ,21,.. . ,z,))) and (( c = ~ ( cy1,. , . . ,ym))).

of a semigroup.

49

Combining some of the above quoted regularity conditions we obtain the following diagram, in which the names of some classes of semigroups determined by the given regularity conditions are given below or over the corresponding condition. (( c = I C Y )) intra-reproduced

((c = XC)) left reproduced

((c = ZCYC)) left quasi-regular

intra-regular

(( c = C Z C Y )) right quasi-regular

\ / (( c = .c2 )) left regular

((c=cZ,))

right regular

completely regular Figure 2.

The same name from Figure 2 we use for a semigroup satisfying the regularity condition (( c = u(c,51,. . . , 2,) )) and an element a of a semigroup S for which the related equation a = u ( a ,z1, . . . ,2,) is solvable in S. Semigroups satisfying the framed regularity conditions in Figure 2 were studied by S. Lajos and G. Sz&z in [Ill from ideal-theoretical point of view. The purpose of this paper is to describe the structure and give some new ideal-theoretical characterizations of these semigroups. 2. Preliminaries

A semigroup S is called n-regular if for every a E S there exists n E N such that an is a regular element. In the same way, using other names given in Figure 2, we define left, right, completely and intra-7r-reproduced semigroups, left, right, completely and intra-quasi-r-regular semigroups, and left, right, completely and intra-n-regular semigroups.

50

Recall from [l]that a subsemigroup B of a semigroup S is a bi-ideal of B . For a E S , B ( a ) = { a } U {a'} U aSa is the least bi-ideal containing a, and it is called the principal bi-ideal of S generated by a. Recall also that a semigroup S is called globally idempotent if S' = S (i.e. every element of S is decomposable).

S if B S B

For undefined notions and notations we refer t o the books [l]and [9]. Next we prove several auxiliary assertions. Lemma 2.1. Let a semigroup S be a semilattice Y of semigroups S,, a E Y . If S satisfies anyone of the regularity conditions from Figure 3, then S, satisfies the same condition, for every 0 E Y .

Figure 3.

Proof. We will prove only the assertion concerning the condition (( c = zcycz)). The remaining cases can be proved similarly. Let S satisfies the condition (( c = zcycz)) and let a E Y be an arbitrary element. For any a E S, there exists z, y , z E S such that a = xayaz, and we have that z E Sp, y E S, and z E 5'6, for some p , y , b E Y . By a E S, and a = xayaz it follows that ap = a y = a6 = a , so a = z a y a z = z ( z a y a z ) y ( z a y a z ) z= = (z'ay)a( zyz)a(yaz') = (z'ay) (zayaz)( z y z ) a ( y a z ' ) = = (z%yz)a( yaz' yz) a(yaz') E

s,as,as,.

Therefore, S, satisfies the condition (( c = zcycz)). The following lemma establishes an interesting connection between intra-quasi-regular and intra-regular, left quasi-regular and left regular, and right quasi-regular and right regular elements.

51

Lemma 2.2. The following conditions o n a semigroup S are true:

( a ) S has an intra-quasi-regular element i f and only if it has a n intraregular element. ( b ) S has a left quasi-regular element i f and only if it has a left regular element. ( c ) S has a right quasi-regular element i f and only if it has a right regular element.

Proof. ( a ) Let a be an intra-quasi-regular element of S, i.e. a = m y a z , for some z, y , z E S. Then y a z = y ( z a y a z ) z = ( y z ) a ( y a z 2 )= ( y z ) ( z a y a z ) ( y a z 2 ) = ( y z 2 a ) ( y a z ) 2 zE S ( y a z ) 2 S ,

so we have that yaz is an intra-regular element of S. The converse is clear. Further, let a be a left quasi-regular element of S , i.e. a = z a y a , for some z, y E 5’. Then Ya = Y ( W a ) = ( y z ) a ( y a ) = ( y z ) ( z a y a ) y a ) = (Yz2a)(Ya)2E s(Y42, so yu is a left regular element of S. The converse is evident. The assertions (b) and (c) can be proved similarly.

0

It is well-known that an element a of a semigroup S is regular if and only if the principal left ideal L ( a ) (or the principal right ideal R ( a ) ) has an idempotent generator. In a similar way here we characterize left, right and intra-quasi-regular elements.

Theorem 2.1. Let a be any element of a semigroup S . Then the following assertions are true:

( a ) a is intra-quasi-regular i f and only i f the principal ideal J ( a ) of S has an intra-regular generator. (b) a is left quasi-regular i f and only i f the principal left ideal L ( a ) of S has a left regular generator. ( c ) a is right quasi-regular i f and only i f the principal right ideal R ( a ) of S has a right regular generator.

Proof. ( u ) Let a be an intra-quasi-regular element. Then a = z a y a z , for some z, y , z E S , so J ( a ) = J ( y u z ) . By Lemma 2.2 it follows that y a z is an intra-regular element, so we have proved that J ( a ) is generated by an intra-regular element.

52

Conversely, let J ( a ) be generated by an intra-regular element b. Then J ( a ) = J ( b ) and b = pb2q, for some p , q E S , by which it follows that a E J ( b ) = J(pb2q)=C Sb2S. On the other hand, by b E J ( a ) it follows that b2 E SaSaS. Therefore, a E SaSaS, which is to be proved. The assertions ( b ) and (c) can be proved similarly. By LQReg(S),IQReg(S) and I R e g ( S ) we denote respectively the sets of all left quasi-regular, intra-quasi-regular and intra-regular elements of a semigroup S .

Theorem 2.2. A semigroup S is left quasi-7r-regular if and only if it is intra-quasi-7r-regular and IQReg(S) = LQReg(S). Proof. Let S be left quasi-7r-regular. Then it is also intra-quasi-.rr-regular and LQReg(S) C_ IQReg(S). To prove the opposite inclusion consider an arbitrary a E IQReg(S). Then a = zayaz for some z,y,z E S , so a = (zay)nazn,for every n E W. On the other hand, since S is left quasi7r-regular, then there exist n € N and p , q € LQReg(S) such that ( m y ) " = p(zay)nq(rcay)n.Now a = (zay)nazn= p(zay)"q(zay)nazn= p(zay)nqa E SaSa, so a E LQReg(S). Thus, LQReg(S) = IQReg(S),which is to be proved. The converse is obvious. 3. Semigroups satisfying the conditions (( c (( c = zczyc, c = cxc2 y))

= zc'yc)) and

Before we state and prove the main result of this section we prove the following:

Theorem 3.1. The following conditions on a semigroup S are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii)

(Va, b E S ) a E SbSa;

S is simple and left quasi-regular; S is simple and left quasi-7r-regular; S is simple and left reproduced; S is simple and left .rr-reproduced; every left ideal of S is a simple semigroup; S is simple and every left ideal of S is an intra-regular semigroup.

53

Proof. It is evident that (i)e$(ii), (ii)+(iii), (i)+(iv), (iv)+(v) and (vi)+(vii) hold. (iii)+(ii). By Theorem 2.2, S is intra-quasi-7r-regular and IQReg(S) = LQReg(S). On the other hand, S is simple, so S = IReg(S) = IQReg(S). Therefore, S = LQReg(S) so S is left quasi-n-regular. (v)+(iv). Let a E S . Since S is simple, then a = xay, for some x , y E S , so a = xnayn, for every n E N. On the other hand, S is left 7r-reproduced, so there exist n E N and z E S such that x" = zxn. Now a = xnayn = zxnayn = x u , so a is left reproduced. Thus, (iv) holds. (iv)+(i). Consider arbitrary a, b E S . Since S is left reproduced, then a = xu, for some x E S , and since S is simple, then x = ybz, for some y , z E S. Therefore, a = xu = ybza E SbSa, which is to be proved. (i)+(vi). Consider an arbitrary left ideal L of S and a , b E L. Then by the hypothesis (i) it follows that a E Sb2Sa C SLbSL

C LbL.

Therefore, L is a simple semigroup. (vii)+(ii). If L is a left ideal of S and a E L, then a E La2L L 2 , so L = L2. Therefore, every left ideal of S is globally idempotent, so S is left quasi-regular, by the dual of Lemma 1.1 of [5]. 0 Every semigroup satisfying anyone of the equivalent conditions of Theorem 3.1 will be called left strongly simple. Dually we define a right strongly simple semigroup, whereas by a strongly simple semigroup we mean a semigroup which is both left and right strongly simple. In fact, strongly simple semigroups are the ones which satisfy anyone of the equivalent conditions of the following consequence of Theorem 3.1 and its dual. Corollary 3.1. The following conditions on a semigroup S are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii)

(b'a,b E S ) a E aSbS n SbSa; S is simple and completely quasi-regular; S is simple and completely quasi-7r-regular; S is simple and completely reproduced; S is simple and completely 7r-reproduced; every one-sided ideal of S is a simple semigroup; S is simple and every one-sided ideal of S is an intra-regular semigroup.

54

Now we are ready to prove the main theorem of this section.

Theorem 3.2. The following conditions on a semigroup S are equivalent: (i) (ii) (iii) (iv) (v) (vi)

S satisfies the condition (( c = zc2yc)); S is a semilattice of left strongly simple semigroups; S is intra-regular and left quasi-regular; S is intra-regular and left quasi-n-regular; every left ideal of S is an intra-regular semigroup; every left ideal of S is an intra-quasi-regular semigroup.

Proof. (i)j(iii). This implication is evident. (iii)+(ii). Let S be intra-regular and left quasi-regular. Then it is a semilattice Y of simple semigroups S,, a E Y . By Lemma 2.1 we have that for every a E Y , S, is simple and left quasi-regular, i.e. it is left strongly simple. Therefore, (ii) holds. (ii)+(i). Let S be a semilattice of left strongly simple semigroups S,, a E Y , and let a E S be an arbitrary element. Then a E S,, for some a E Y , and since S, is left strongly simple and a,a' E S,, according to Theorem 3.1 we have that a E S,a2S,a Sa'Sa. Thus, S satisfies the condition (( c = zc' yc)). (iv)+(iii). Let S be intra-regular and left quasi-n-regular. By Theorem 2.1 it follows that LQReg(S) = IQReg(S). On the other hand, S = I R e g ( S ) C IQReg(S), so we conclude that S = IQReg(S) = LQReg(S). Therefore, S is left quasi-regular. (iii)+-(iv). This implication is obvious. (i)+-(v). Let L be a left ideal of S and let a E L. Then by (i) we have that a = xa2ya and a' = ua4va2,for some x,y , u , v E S , and now

a = za'ya = z(ua4va2)ya= (zua2>a2(va2ya) E La'L, so we have proved that L is an intra-regular semigroup. (v)+-(vi). This implication is trivial. (vi)+(i). Consider an arbitrary a E S and the principal left ideal L = L ( a ) = Sla. By (vi), L ( a ) is an intra-quasi-regular semigroup, which yields

a E LaLaL = SlaaS'aaSla Thus, (i) holds.

Sa'Sa. 0

Other ideal-theoretical characterizations of semigroups satisfying the condition ((c= zc'yc)) can be found in S. Lajos and G. SzLsz [ll].

55 Using Theorem 3.2 and its dual, the following can be proved.

Corollary 3.2. The following conditions o n a semigroup S are equivalent: (i) (ii) (iii) (iv) (v) (vi)

S satisfies the condition (( c = zc2yc,c = czc2y)); S is a semilattice of strongly simple semigroups; S i s intra-regular and completely quasi-regular; S is intra-regular and completely quasi-7r-regular; every one-sided ideal of S is an intra-regular semigroup. every one-sided ideal of S is an intra-quasi-regular semigroup.

4. Semigroups satisfying the condition (( c = czc'yc)) Before we give a theorem which characterizes semigroups satisfying the condition (( c = czc2yc)), we prove the following:

Theorem 4.1. The following conditions o n a semigroup S are equivalent: (i) (ii) (iii) (iv)

(Va, b E S ) a E aSbSa; S i s simple and regular; S i s simple and 7r-regular; every bi-ideal of S i s a simple semigroup.

Proof. The equivalence of the conditions (i), (ii) and (iii) was proved by the authors in [3] (Theorem 2.2), so it remains to prove the equivalence of the conditions (i) and (iv). (i)+(iv). Let B be a bi-ideal of S and let a , b E S . By (i) we have that a E aSb3Sa, which yields a E aSb3Sa = (aSb)b(bSa) ( B S B ) b ( B S B )C B b B . Thus, we have proved that B is a simple semigroup. (iv)+(i). Consider arbitrary elements a , b E S and the principal bi-ideal B = B ( a ) = { a } U { a 2 }UaSa. By the hypothesis, B is a simple semigroup, and a , aba E B , so we have that

a E B a b a B C aS'abaS'a 2 aSbSa. Therefore, (i) holds. Now we are ready to state the main result of this section.

56

Theorem 4.2. The following conditions (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

orl.

a semigroup S are equivalent:

S satisfies the condition (( c = cxc2yc)); S i s a semilattice of simple regular semigroups; S i s intra-regular and regular; S is intra-regular and r-regular; every left ideal of S is a right quasi-regular semigroup; every right ideal of S i s a left quasi-regular semigroup; every bi-ideal of S i s a n intra-regular semigroup; every bi-ideal of S i s a n intra-quasi-regular semigroup.

Proof. The equivalence of the conditions (ii), (iii) and (iv) can be easily proved using Theorem 4.1, Lemma 2.1 and the well-known fact that a semigroup is intra-regular if and only if it is a semilattice of simple semigroups. (iii)=+(i). Let a E S. By the hypothesis (iii), a = xa2y and a = aza, for some x , y, z E S , so a = aza = azaza = azxa2yza. Therefore, (i) holds. The converse implication, (i)=+(iii),is evident. (i)=+(v).Let L be a left ideal of S and let a E L. By the hypothesis (i), a = axa2ya, for some x , y E S , and since xu, y a E S L L , we have that a = axa2ya = a(xa)a(ya)E aLaL, what means that L is a right quasi-regular semigroup. ( v ) j ( i ) . Let a E S and let L = L ( a ) = Sla be the principal left ideal of S generated by a. Then L is a right quasi-regular semigroup, i.e. a E aLaL = a(S1a)a(S1a),and we can easily check that a E aSa2Sa. Similarly we can prove the equivalence of the conditions (i) and (vi). (i)=+(vii).Let B be a hi-ideal of S and a E B . By (i) we obtain that a = axa2ya and a2 = a2ua4va2,so

a

= axa2ya = ax(a2ua4va2)ya =

= (axa2ua)a2(ava2ya) E ( B S B ) a 2 ( B S B5 ) Ba2B,

and hence, B is an intra-regular semigroup. The implication (vii)+(viii) is evident. (viii)+(i). Let a E S and let B = B(a) = { a } U { a 2 } U aSa be the principal hi-ideal of S generated by a. Then B is an intra-quasi-regular semigroup, i.e. a = xayaz, for some x , y , z E B , and since x E as1, y E as1n S l a and z E S l a , we can easily verify that a E aSa2Sa, which is to be proved. 0 For other ideal-theoretical characterizations of semigroups satisfying the condition ((c = cxc2yc)) we refer to S. Lajos and G. S z b z [ll].

57

The equivalence (iii)%(iv) in Theorem 4.2 can be generalized as follows.

Theorem 4.3. An intra-quasi-regular semigroup S is 7r-regular i f and only i f i t is regular. Proof. Let S be intra-quasi-regular and 7r-regular and let a E S. Then a = x a y a z , for some x, y, z E S , so a = ~ ~ a ( y a zfor ) ~every , k E N. Since S is n-regular, there exist 1 E N and u E S such that (yaz)l = ( y ~ z ) l u ( y a z ) ~ , so a = 51 a(ya.2) 1

= 5 1 a(yaz) 1u(yaz) 1 = au(yaz)1 = apaq,

where p = uy and q = z(yaz)'-'. Now we have that a = (ap)nLaqm,for every m E N, and since S is n-regular, then (up)" = ( ~ p ) ~ u ( a p for ) ~ ,some n E N and u E S , so a = (ap)naqn = (ap)nv(ap)naqn = (ap)nua.E aSa.

Therefore, a is regular. The converse is evident.

0

Note that Theorem 4.3 also generalizes the related statements for simple and 0-simple semigroups proved by the authors (Theorem 2.2 of [3]), and independently by P. Jones (private communication). 5. Left and completely regular semigroups

Various structural characterizations of left regular semigroups were given by S. Bogdanovib and M. Cirib in [2]. Here we give some new characterizations of these semigroups, in terms of properties of their left ideals.

Theorem 5.1. The following conditions on a semigroup S are equivalent: (i) S i s left regular; (ii) every left ideal of S is a left quasi-regular semigroup; (iii) every left ideal of S i s a left reproduced semigroup.

Proof. (i)+(ii). Let L be a left ideal of S and let a E L. By the left regularity of S we have that a = xu2 for some 5 E S , so a=Za2-

- 53 a4 -- (53a)aaa E LaLa.

Hence, L is a left quasi-regular semigroup. (ii)+(iii). This implication is evident.

58 (iii)+(i). Consider an arbitrary a E S and the principal left ideal L = L(a) = S'a. By the hypothesis, L is a left reproduced semigroup, so a E La S'aa. By this we easily conclude that a E Sa2. Thus, S is a left regular semigroup.

0

Similarly we prove the following theorem.

Theorem 5.2. T h e following conditions o n a semigroup S are equivalent: (i) S is completely regular; (ii) S i s left (resp. right) regular and right (resp. left) quasi-regular; (iii) S i s left (resp. right) regular and right (resp. left) quasi-7r-regular;

(iv) every left (resp. right) ideal of S i s a right (resp. left) regular semigroup; (v) every left (right) ideal of S is a completely quasi-regular semigroup; (vi) every bi-ideal of S i s a left (right) quasi-regular semigroup.

References 1. S. BogdanoviC, Semigroups with a system of subsemigroups, Institute of Mathematics, Novi Sad, 1985. 2. S. BogdanoviC and M. CiriC, A note on left regular semigroups, Publ. Math. Debrecen 48 / 3-4 (1996), 285-291. 3. S. BogdanoviC, M. CiriC and M. MitroviC, Semilattices of nil-extensions of simple regular semigroups, Algebra Colloquium 1 O : l (2003), 81-90. 4. S. BogdanoviC, M. CiriC, P. StanimiroviC and T. PetkoviC, Linear equations and regularity conditions on semigroups, (submitted). 5. J. Calais, Demi-groupes quasi-inversifs, C. R. Acad. Sci., Paris, 252 (1961), 2357-2359. 6. A. H. Clifford, Semigroups admitting relative inverses, Annals of Math. 42 (2) (1941), 1037-1049. 7. A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. 1, Amer. Math. SOC.,Providence, R. I., 1961. 8. R. Croisot, Demi-groupes inversifs et demi-groupes reunions de demi-groupes simples, Ann. Sci. Ecole Norm. Sup. 70 (3) (1953), 361-379. 9. J. M. Howie, Fundamentals of semigroup theory, London Math. SOC.monographs, New Series, Clarendon Press, Oxford, 1995. 10. N. Kuroki, On semigroups whose ideals are all globally idempotent, Proc. Japan. Acad. 47 (1971), 143-146. 11. S. Lajos and G. SzAsz, Generalized regularity in semigroups, Dept. Math. K. Marz Univ. of Economics, Budapest, DM 75-7 (1975), 1-23. 12. W. D. Munn, On semigroup algebras, Proc. Cambridge Phil. SOC.5 1 (1955), 1-15. 13. J. von Neumann, On regular rings, Proc. Nat. Acad. Sci., USA 22 (1936), 707-713.

59 14. G. SzLz, Uber Halbgruppen, die ihre Ideale reproduzieren, Acta Sci. Math. (Szeged) 27 (1966), 141-146. 15. G. S z b z , Semigroups with idempotent ideals, Publ. Math. Debrecen 21 / 1-2 (1974), 115-117. 16. G. S z b z , On semigroups in which a E SakS for any element, Math. Japonica 20 (4)(1976), 283-284. 17. G. S z h z , Decomposable elements and ideals in semigroups, Acta Sci. Math. (Szeged) 38 / 3-4 (1976), 375-378.

GROBNER-SHIRSHOVBASES FOR THE BRAID SEMIGROUP L. A. BOKUT*,Y.FONG, W.-E KE, AND L:S. SHIAO

Dedicated to the memory of Professor B H Neumann 1. INTRODUCTION

In this paper, we continue to use the language of Grobner-Shirshov bases for some semigroups presented by defining relations (in papers [31,321 some groups have been treated as semigroups). Traditionally in this situation one use some other languages like elementary transformations language ([ 1, 2]), Newman’s Diamond lemma language ([64]), the rewriting systems language ([34], [46]), or something else ([56]). For example, P. M. Cohn used diamond lemma for semigroups as early as in his 1956 papers ([41, 421). After that, the first author used the same language to prove some embedding theorems for semigroups [8,9]. Also the first author used a language of rewriting systems in order to prove that multiplicative semigroups of some rings embeddable into groups [12, 131 (see also [16, 201). However, to formulate our main results in terms of Diamond lemma or rewriting systems would be more difficult than using Grobner-Shirshov bases. The Grobner (or standard) and Grobner-Shirshov (noncommutative Grobner or standard) bases methods have a prehistory and a history (here word “bases” has the same meaning as it is in the Hilbert Basis Theorem). As for the prehistory one may mention the Euclidian algorithm, the Gauss elimination algorithm, the Hilbert Bases Theorem, the Poincare-Birkhoff-Witt theorem, some papers by P. Gordon (1900), F. S. Macaulay (1927), G. Hermann (1926), and W. Grobner (1939) for commutative algebras (see [45]), by M. H. A. Newman [64] and P. M. Cohn [43] for semigroups and noncommutative algebras, by A. I. Zhukov [71], and E. Evans [47,48] for nonassociative algebras and quasi groups. The history started with papers by A. I. Shirshov [69] and H. Hironaka [51], and a thesis by B. Buchberger [35] under the supervision of W. Grobner. Shirshov’s paper was published in Russian and only recently has been translated into English (see [69]). It has been missing in the West (see, for example, “History of Grobner bases” in the book [45]). Shirshov treated the most difficult case of Lie polynomials using so called Lyndon-Shirhsov monomial basis of the free Lie algebra [68, 401. The case of noncommutative polynomials is just a specialization of the case of Lie polynomials (here we use an analogy with an expression from [45] that the case of commutative polynomials is a specialization of the case of noncommutative polynomials). Shirshov’s method was been used in the same year in [7], and was explicitly presented in [14] and [15] by the first author. Hironaka’s paper is called the landmark (see [45]) and has become very important in algebraic geometry (this can be seen by the fact that Hironaka received a Fields Medal in 1970 with this very paper). Actually Hironaka had been dealing with the case of power series rather than the case of polynomials. He suggested the name “standard bases” that is widely used now. Buchberger’s results were first published in [36], and then in [37]. Since then Buchberger’s Grobner Bases Method has become the most popular. See for example the proceedings of the 1998 conference on 33 years of Grobner Bases [39]. *Supported in part by the Russia’s Fund of Fundamental Research.

60

61

In the context of universal algebras, the well-known Knuth-Bendix rewriting algorithm was developed in [57]. As a matter of fact, Shirshov's algorithm [69] and Buchberger's algorithm [35,36] are both Knuth-Bendix type of algorithms. An explanation of the KnuthBendix algorithm for semigroups and groups maid in [46] is very closed to the Newman's Diamond Lemma 1641. An alternative approach to the Grobner-Shirshov bases method can be found in a paper by G. Bergman published in 1978 [5]. Bergman's paper become widely popular due to his version of the Newman's Diamond Lemma. During the past few years the Grobner-Shirshov bases method has been used to study some important examples of associative algebras (quantum enveloping algebras of the type A,, Hecke algebras of type A,), Lie algebras (simple Lie algebras of types A,, B,, C,, D,, G2, F4, E6, E7, and Eg, Kac-Moody algebras of type A;), B;), CA'), DL')), Lie superalgebras (simple Lie superalgebras of types A,, B,, C,, and D,), modules (irreducible modules over simple Lie algebras of type A,, Specht modules over Hecke algebras of type A,), and groups (Grobner-Shirshov bases for Coxeter groups of types An, Bn, and 0,) [251-[311, [53]-[55], and [66], [67]. Also this method has been applied to associative conformal algebras [21,22]. In a recent paper [32], we applied the Grobner-Shirshov bases method to study some groups by P. S. Novikov [65] and W. W. Boone [33] with the algorithmically unsolvable word problem. Using the very language of Grobner-Shirshov bases, we revised some papers [lo, l l, 17, 181 (see also [19, 281) in which the rewriting systems language (in line of HNN-extension theory) was used. Thus, one can use no group theory in proving Novikov's and Boone's theorems on unsolvability. Some other activities on Grobner-Shirshov (noncommutative Grobner, standard) bases may be found in papers by T. Mora [63], V. N. Latyshev [59], V. Ufnarovski [70], A. A. Mikhalev and A. A. Zolotyh [62], A. A. Mikhalev and E. Vasilieva [61], V. P. Gerdt and V. V. Kornyak [50]. A great interest has been emphasized on the word and conjugacy problems for braid groups Let us mention a pioneering paper by E. Artin [3], and papers by E. Artin [4], A. A. Markov [60], A. F. Garside [49], J. Birman, K. H. KO, S.-J. Lee [6]. In these papers, especially in 1491 and [6] ,a fundamental role of a semigroup of positive words of B,+1 is proved to be crucial. Let us denote this semigroup by Br,f+,, and call it the braid semigroup of type n 1. It has the following presentation in Artin generators ai, 1 5 i 5 n:

+

ai+1aja;+1 = aiai+lai,

aiai = aja;,

1 5 i 5 n,

i - j 2 2.

In this paper, we give a Grobner-Shirshov basis for Br-;+'. Also we begin the study a more general concept of Artin semigroups. Namely, the braid semigroup Br:+l is an Artin =A$. semigroup of type A,, i.e., With the support of these examples, we formulate a hypothesis on Grobner-Shirshov bases of any Artin semigroup. It is a modification of a hypothesis on Grobner-Shirshov bases of Coxeter groups [31]. Finally, let us mention that a semigroup BBr;+, of positive braids ([6]) has the following presentation with the Birman-Ko-Lee generators a,, (1 5 t < s 5 n 1):

+

atsarq = arqars,

atsasr

( t - r)(t - q)(s- r ) ( S - 4 ) > 0, II 1 2 t > s > r 2 1.

= a1rat.q = asrat,,

+

62

It would be interesting to fine a Grobner-Shirshov basis of this semigroup also. It may have some connections with a new public-key cryptosystem [%I. 2. GROBNER-SHIRSHOV BASES

Let X be a linearly ordered set, k be a field, k ( X ) be the free associative algebra over X and k. On the set X * of words we impose a monomial well order > (i.e. a well order that agrees with the concatenation of words). Any polynomial f E k ( X ) has the leading word f . We say that f is monic if f occurs in f with coefficient 1. By a composition of intersection (f,g)w of two monic polynomials f and g relative to some word w, such that w = f b = ag, deg(f) deg(g) > deg(w), one means the following polynomial

+

( f d w = f b - a‘?. By composition of including (f,g)w of two monic polynomials f and g, where w = f = agb, one means the following polynomial (f,& = f - agb. In this case the transformation

.fe( f , d w = f - w b is called the elimination of the leading word (ELW) of g in f. In the above two cases, the word w is referred to as an ambiguity of the polynomials (relations) f and g. A composition (f,g)w is called trivial relative to some S c k ( X ) (denoted by (f,g)w E 0 mod (S,w)), if (f,g),+ = xcliaitibi for some ti E S, ai, bi E X * , and aicbi

< w.

In particular, if ( f , g ) w goes to zero by the ELW’s of S then ( f , g ) w is trivial relative to S. We say for some polynomials f 1 and f2 that f1 f2 mod (S,w) if f1 - f2 z 0 mod (S,w). A subset S of k ( X ) is called a Griibner-Shirshov basis if any composition of polynomials from S is trivial relative to S. By ( X I S), the algebra with generators X and defining relations S, we mean the factor algebra of k ( X ) by the ideal generated by S. The following lemma goes back to the Poincare-Birkhoff-Witt (PBW) theorem, the Diamond Lemma of M. H. A. Newman [64], the Composition Lemma of A. I. Shirshov ([69]) (see also [14], [15], where this Composition Lemma was formulated explicitly and in a current form for Lie algebras and associative algebras respectively), the Buchberger’s Theorem ([35]), (published in [36]), and the Diamond Lemma of G. Bergman [5] (this lemma was also known to P. M. Cohn, see, for example, [43, 441 and some historical comments to Chapter “Grobner bases” in [45]):

Lemma 2.1 (Composition-Diamond Lemma). S is a Grobner-Shirshov basis if and only if the set { u E X* I u # afb,for any f E S } of S-reduced words consists of a linear basis of the algebra ( X I S ) . While S is fixed we will simply refer to a S-reduced word as a reduced words. Actually we would like to introduce a new term for this basis-the PBW-basis for the algebra relative to a Grobner-Shirshov basis of relations of this algebra. It would reflect the fact that, as we have mentioned, the Composition-Diamond lemma goes back to the PoincareBirkhoff-Witt (PBW) Theorem as well. Actually PBW Theorem is a particular, and the

63

most important case, of Composition-Diamond lemma when S = {xixj -xjxi - Cxfj,i > j } with [xixj] = Xxfj, i > j as the multiplicative table of some Lie algebra. In this case, the PBW-basis is given byPBW(S) = {xi,xi2...xik,il 5 i z . . . 5 ik}. If a subset S of k ( X ) is not a Grobner-Shirshov basis then one can add to S all nontrivial compositions of polynomials of S, and continue this process (infinitely) many times in order to have a Grobner-Shirshov basis F o m p that contains S. This procedure is called the Buchberger-Shirshov algorithm [69, 35, 361. A Grobner-Shirshov basis S is called minimal (or reduced) if any s E S is a linear combination of S\{$}-reduced words. Any ideal of k ( X ) has a unique minimal Grobner-Shirshov basis. If S is a set of “semigroup relations” (that is, polynomials of the form u - v, where u,v E X*),then any nontrivial composition will have the same form. As a result the set P o m p also consists of semigroup relations. Let A = smg(X I S) be a semigroup presentation. Then S is a subset of k ( S ) and one can find a Grobner-Shirshov basis Pomp. The last set does not depend on k, and consists of semigroup relations. We will call P o m p a Grobner-Shirshov basis of A. It is the same as a Grobner-Shirshov basis of the semigroup algebra kA = ( X I S). Let {ui} be a set of words in an alphabet. By V ( u i )we understand any word in {ui}. Let {wi} be some other set of words with a surjection ui M wi. Then by V ( w i )we understand the same word V, but in the set {wi}. 3 . A HYPOTHESIS

Here we will formulate a general hypothesis on Grobner-Shirshov bases for any Artin semigroup. We will present Coxeter group W relative to Coxeter matrix M in the following way. Let S be a linear ordered 1-elements set, M = (msg)be 1 x 1 Coxeter matrix. Then W = smg(S I Vs,s’ E S,s2 = land

(SS’)~S#

= 1,s #$’,for some finite mss!).

Let us define for finite m, m(s,s’) = ss’. . .

(m- 1) (s,s’) = ss’. . .

(there are m alternative letters s, s’), (there are m - 1 alternative letters s,s’),

and so on. For example, if rnsst = 2, then rn(s,s’) = ss‘, ( m- l)(s,s’)= s. If m,,! = 3, then m(s,s’) = ss’s, (rn - l)(s,s‘)= ss’.Using this notations, the defining relations of W can be presented in the form (3.1)

s2 = I , rn(s,s’) = m(s’,s), s > d ,

for all s,s’E S and finite m.

Definition 3.1. The following semigroup will be called the Artin semigroup W+ relative to the Coxeter matrix M: (3.2)

W+ = smg(s1,. .. ,sn I ms5J(s,s’)= mss!(s‘,s), s > s’, s,s’ E S,mssl < m ) .

Also we will use the following notation (m,i)(s,s‘)to denote the result of removing first i letters in m(s,s’),where 1 5 i 5 rn. Let us say that two words in S are equivalent if they are equal modulo the commutativity relations (3.1). To be more precise, it means that they are equal in the so-called free partially commutative semigroup (algebra) generated by S with commutativity relations (3.1)

64 [31]. Two relations a = b and c = d of W are called equivalent if a, b are equivalent to c, d respectively.

Hypothesis. A Grobner-Shirshov basis T of W+ includes the initial relations (3.2)and the relations which are equivalent to the following:

,

.. . ( m- ik) (S2k- 1 , S2k)m( S2k+ 1, S2k+2) = m(s’, s)(m, io)(s2 $1 ) .. ’ (m, ik-1 ) (SZk, S2k- 1) (m, ik)(S2k+2, S2k+l)l

(3.3) (m - io) (s, s’)(m.-

i l l (s1 s2)

7

where S

> S’, Si < S2, . .. ,S2k-1 < S2k, S2k+l < S2k+2, and

(m- i k ) b2k(m- ik-1)

= m(32k-1,

1 7 S2k)rn(S2k+2, S2k+l)

(S2k-3, SZk-2)m(S2k,SZk-l)

,

S2k) (m,i k ) (S2k+2 S2k+l),

= m(S2k-3

j

S2k-2)

(m, ik-1)

(S2k, S2k-1)

>

(m- il)(sl, s 2 ) m ( s 4 , ~ 3 )= m(s1 , s 2 ) ( m 1i i ) ( ~ 4 , ~ 3 ) , (m- io) (s, s’)m(sz,s1) = m(s,s’)(m,io) ( s 2 , s i ) . In the above expressions, = means the equality in the free semigroup. Let us fix both sides of (3.3). If A = B has the form of (3.3), then a transformation A+B

will be called apositive chain, while B + A will be called a negative chain. Any other relation of T is equivalent to the following one:

x =Y,

(3.4)

where X goes to some X’ by a series of negative chains,

x+x1+...+Xk=xr, (in the free partially commutative semigroup), X’ = AY1, and Y = BY1, where A 3 B is a positive chain. Remark 3.2. Let us compare this hypothesis with our hypothesis on Grobner-Shirshov bases of Coxeter groups [31]. Relations (6.2) in [31] is just the relations (3.3) with io = i l = ... = ik = 1 and with an extra condition that all neighbor pairs (s,~’),(s1,s2), . . . , (szk+l, SZk+2) are different. In particular, we do not expect negative chains for Coxeter groups at all.

4. BRAIDSEMIGROUPS Let A; = B r l be the braid semigroup with 2 generators a1 and a 2 such that a2a1a2 = We will assume that a1 < a 2 . Here, for words u and v, the order u < v is using the deg-lex order (first we compere words by the degree (length) and then lexicographically). ala2al.

Lemma 4.1. The reduced Grobner-Shirshov basis of Br3f consists of the initial relation and relations (4.1)

of a form as in (3.3).

aaa:a2a1

= a1aza:a;-1,

I? 2

65

Proot First of all, (4.1) follows from Garside's formulas [49, Lemma 21: ~ 1 A= 2 A 2 ~ 2 , ~ 2 A=2A241,

where A2 = ala2al. Secondly, (4.1) has the form of (3.3):

( m - l)(a2,a1)(m - 2)(a1,a2)(m - 2)(a1,a2).. . ( m- 2)(a1,a2)m(a1,a21 = m ( a 1 , ~(m, ) 1) (a2,al)(m, 2 ) ( a 2 , 4 . . . (m,2)(az,al)(m, 2)(az,a1). Now we need to prove that any composition of relations from the lemma is trivial. There are some compositions of intersections relative to the following ambiguities w: a2a1a2a1a2,a2a:a2a1a2,

12 2,

1 a2a1a2qa2a1, l k a 2 q 42a 1a2a1,

12 2, 12 2and k 2 2.

Let us check the last composition:

(f,&

2 1-1 k-1 l k = (a2a:a2al- ala2ala2 )a1 a2a1 - a2al(a2qaZal - a 1 a 2 4 4 - 9

= a2a;+'a2a:a;-' = ala2a:a~ala;-' G

- ala2a:a:-'a~-la2al - ala2ala2 2 1-1 a1 k-1 u2u1

0. 0

All other compositions can be done in a similar way. Let us introduce the following notations which are analogous to that of [31]:

a"r,- a'@ - 1 . .

.aj,

i>j,

a"It - a'I , a"r 1 + l = 1, aij(p) = .p"uf~l..

i>j,

.u?-J,

aii (p) = up", @i+l

(P)= 1,

where p = (PO,...p i - j ) , all pl 2 1, 1 = 1,2,...,i - j . We will write p of pl is greater that 1. Otherwise we will write p = 1. It is easy to see the fcllowing equality in Br:: ui,-1ui+lj-l

(4.2)

= ai+lj-lai+l,,

j

aij-l(P)ai+lj-l = ai+lj-lai+lj(p),

> 1, if at least one

5 i+ 1. j

I i+ 1.

In fact, the first equality of (4.2) has the following form: (4.3)

( m - I)(ai,ai+l)(m-

l)(ai-i,ai)

...( m -

l)(aj,aj+l)m(aj-l,aj)

= m(ai+l, ai)(m,1)(at-1 ai) . ..(my 1)(aj+l,a j ) (m,1) (aj,aj-l)

The second equality of (4.2) has an analogous form:

(4.4)

(m-2)(ai,ai+l)p0-l(m-

l)(Ui,Ui+l)

= m ( ~ i + l , ~ i ) (2)(ai+l m, ,ai)Po-'

...(m-2)(uj-l,uj) p ' .-1 m(aj-1,aj) ...(m,l ) ( ~ j , ~ j - 1 ) ( ~ , 2 ) ( ~ j , ~ j - l ) ~ i - j + ~ - ' I-'

Also we will use the following Garside's formula: (4.5)

a,ai+lj

= ai+lja,+l,

j

5 i, j 5 s 5 i.

66

This formula can be written as m(a,,a;+l)... ( m - l)(as,as+l).. . ( m- l)(as+i,aj) = (m,l)(ai+i,as)...(m,l)(a~+i,as)...m(aj,as+i). By W ( a j , ...,ai) = W (j , i) we shall mean any word in alphabet a,,aj+l,. . . ,ai if j 5 i , and empty word if j > i .

Theorem 4.2. A Grobner-Shirshovbases of A,f = Brz+l consists of the following relations: (4.6) (4.7)

ai+laiV(l,i- l)W(j,i)a;+lj = a;ai+laiV(l,i- l)a;jW’(j+ l , i + l ) , asak = aka,, s - k 2 2,

where 1 5 i 5 n - 1, 1 5 j 5 i + 1, W’ = W(aj+l,. ..,ai+l), and W begins with ni not empty.

if it is

Remark 4.3. In particular, the following relations are in the previous list: ai+iaiai+i = aiai+lai, a. a!a. a. - a . a . 1+1

1

If1

I

-

I

If1

(V = 1,j = i + l ) ,

.2,1-1 ; ;+I,

12 2 , (V = I , w = af-’ , j = i ) ,

ai+iafai-ij(p)ai+ij = aia;+la~ai-l,afl:aij+l(p),

I 2 2,i- 12 j ,

(V = 1 , = af-’a;-Ij(p)). ~ These relations have the form of (3.3) because of Lemma 4.1 and (4.4). In general, if W is not empty, the relation (4.6) has the form of (3.4). This follows from above and (4.5). Also the right hand side of (4.2) can be made into the left hand side by using the eliminations of leading words of relations (4.6).

Proof: Let S is the set of relations (polynomials) from (4.6) and (4.7). We need to prove that any composition of polynomials from S is trivial modulo S. First of all let us deal with compositions of including. It is easy to see that any composition of including of polynomials of (4.7) in a polynomial of (4.6) is trivial. Indeed, the only possible compositions of this including should correspond of some including of asak E VW, s - k 2 2, or else of aiak E a;V, i - k 2 2. So we need to figure out any composition of including relations of the type (4.6). Let f be a relation (polynomial) of (4.6) and g be the following relation:

+

=ai,ail+iailVi(l,ii- l)a;ljlWi(jl l.il+ I), ail+lailVi(l,il- l)~i(ji.h)ail+ijl where j l 5 i l 1 and Wi begins with ail if not empty. Let u and u1 be the leading monomials o f f and g respectively. Suppose that u1 is a subword of u. The following possibilities may occur: (1) i = i l . Thus V = V1, W = W1 and j1 2 j . Then

+

=

( f , g ) U= aia;+iaiVaij, W’ajl-lj - a;ai+laiVai,W’ 0, for W’ = W[ = W(ajl+l, ...,ai+l). ThereforeW’aj,-l, ajl-ljW’.

=

(2) i = i l + l . T h e n U I = aiai-iVi(l,i

- 2)Wl(jl,i - l)a;j,.

It means that V(1,i- 1) =ai-iVl(l,i-2)W~(ji,i-

1) and

W ( j , i ) =aijlW2(j,i),

67

and we have

The first case is clear. For the second case, we see that

and

Now we need to check compositions of intersection.

68

It is easy to see that all compositions of intersection with (4.7) are trivial. Therefore we only need to consider compositions of intersection of (4.6) relative to the following ambiguities w: (1) Intersection with a single letter. We have w = a, +1 a,V(l,i- l)W(j,Ofli +U fl 7 --iVi(l,; - 2)Wl ( j i , j - l)a!h, and note that and

Therefore,

- (a,+ia,V(U

= 0.

(2) Intersection with two letters. We have w = a,-+ifl,-V(l , ;' - l)W(j, i)ai+ij+2aj+iajVi(l, j - l)Wi (ji , j) and note that

and Therefore,

1,7+1)

= 0.

69

(3) Intersection with more than two letters. We have

+ l , i + 1 ) ~ 2 ( 1 , i l -l ) W l ( j l , i l ) a i l + l j l - a i a i + l a i V ( I , i - I ) a i j ( W ’ ( j + l , i + I)ail+l)V2( 1 , i l - I ) ~ i , j ,

aiai+laiV(I,i - I)aijW’(j

.Wi(jl+ l,il+

1)

z 0. We have checked all compositions and the theorem is proved.

0

Remark 4.4. In view of [49], it is important to have an efficient algorithm to know whether begins with the fundamental word A in some presentation, where a word v of

A = A l A Z . . .An,& = a i . . . ~ ,2i 1.

Let us formulate the following statement that easily follows from the previous theorem.

Proposition 4.5. Let v E PBW(Br;+,). Then v begins with A in some presentation if and only if

v = A l V l ( l ) A z V 2 ( 1,2). ..&-1Vn-1(1 in the free semigroup.

,TI - l)Anu

REFERENCES [ l ] S . I. Adjan. Defining relations and algoritmicalproblems forgroups andsemigroups. Proceedings of Steklov Math. Institute, 85,1958. Moscow:Nauka. [2] S . I. Adjan, V. G. Durnev, Algorithmic problems for groups andsemigroups. Uspehi Math. Nauk. 55 (2000), 3-94. [3] E. Artin, Theory der ZV,Abh. math. Sem. Hamburg Univ., (1926),47-72. 141 E. Artin,Theory of Braids, Ann. Math. 48 (1947), 101-126. [5] G. M. Bergman, The diamond lemmafor ring theory, Adv. in Math. 29 (1978). 178-218. [6] J . Birman, K . H. KO,S . J . Lee, A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (1998). 322-353. [7] L. A. Bokut, Embedding Lie algebras into algebraically closed Lie algebras. (Russian) Algebra i Logika Sem. 1 (1962),47-53. [8] L. A. Bokut. Some imbedding theorems for rings and semigroups. I. (Russian) Sibirsk. Mat. 2. 4 (1963), 500-518. [9] L. A. Bokut, Some embedding theoremsfor rings and semigroups. 11. (Russian) Sibirsk. Mat. Z. 4 (1963), 729-743.

70 [lo] L. A. Bokut, On apropeity of the Boone groups. Algebra i Logika Sem. 5 (1966), no. 5.5-23. [I 11 L. A. Bokut, On a properly of the Boone groups. II. Algebra i Logika Sem. 6 (1967) no. 1,15-24. [ 121 L. A. Bokut, Groups of fractions of multiplicative semigroups of certain rings. I, II, III. Sibirsk. Mat. Z. 10 (1969), 246-286; ibid. 10 (1969). 744-799; ibid. 10 1969 800-819. [13] L. A. Bokut, Theproblem ofMalcev. Sibirsk. Mat. 2. 10 (1969), 965-1005. [14] L. A. Bokut, Unsolvability of the word problem, and subalgebras offinitely presented Lie algebras. Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1173-1219. [15] L. A. Bokut, Imbeddings into simple associative algebras. Algebra i Logika 15 (1976), 117-142,245. [16] L. A. Bokut, Malcev’s problem and groups with a normal form. With the collaboration of D. J. Collins. Stud. Logic Foundations Math., 95, Word problems, I1 (Conf. on Decision Problems in Algebra, Oxford, 1976). pp. 29-53, North-Holland, Amsterdam-New York, 1980. [17] L. A. Bokut, New examples of groups with the standard basis. Dokl. Akad. Nauk SSSR 277 (1984), no. 2, 277-280. [18] L. A. Bokut, The centrally symmetric Novikov group. Sibirsk. Mat. Zh. 26 (1985). no. 2, 18-28,221. [19] L. A. Bokut, On the Novikov and Boone- Borisov groups. Illinois J . Math. 30 (1986). no. 2,355-359. 1201 L. A. Bokut, Abstract semigroups and groups offormal series of dependent variables. Siberian Advances in Mathematics. Siberian Adv. Math. 6 (1996). no. 3, 1-26. [21] L. A. Bokut, Y. Fong and W.-F, Ke, Grobner-Shirshov bases and composition lemma for associative conformal algebras: an example. Contemporary Math. 264 (2000), 63-90. [22] L. A. Bokut, Y.Fong and W.-F. Ke, Compositton-Diamond lemma for associative conformal algebras. J. Algebra, to appear. [23] L. A. Bokut, Y. Fong, W.-F. Ke, and P. S . Kolesnikov, Grobner and Grobner-Shirshov bases in algebra and conformal algebras, Fundamental and applied mathematics, 6 (2000), 669-706. [24] L. A. Bokut, S.-J. Kang, K.-H. Lee and P. Malcolmson, Grobner-Shirshov bases for Lie superalgebras and their universal enveloping algebras. J. Algebra. 217 (1999), no. 2,461495. [25] L. A. Bokut and A. A. Klein, Serre relations and Grobner-Shirshov bases for simple Lie algebras. I, II. Internat. J. Algebra Comput. 6 (1996), 389400 and 401412. I261 L. A. Bokut and A. A. Klein, Grobner-Shirshov bases for exceptional Lie algebras. I.. In “Ring theory. Selected papers from the conference held in Miskolc, July 15-20, 1996, ” (L. Rowen., ed). J. Pure Appl. Algebra 133 (1998). 51-57. North-Holland Publishing Co., Amsterdam, 1998. [271 L. A. Bokut and A. A. Klein, Grobner-Shirshov basesfor the exceptional Lie algebras E s , E7, E8. Algebras and cornbinatorics (Hong Kong, 1997). 37-46, Springer, Singapore, 1999. [28] L. A. Bokut, and G . P. Kukin, Algorithmic and combinatorial algebra. Mathematics and its Applications, 255. Kluwer Academic Publishers Group, Dordrecht, 1994. I291 L. A. Bokut, and P. Malcolmson, Grobner-Shirshov bases for quantum enveloping algebras. Israel J. Math. 96 (1996). 97-113. [30] L. A. Bokut and P. Malcolmson, Grobner-Shirshov bases for relations of a Lie algebra and its enveloping algebra. Algebras and combinatorics (Hong Kong, 1997), 47-54, Springer, Singapore, 1999. [31] L. A. Bokut and L . 4 . Shiao, On Grobner-Shirshov bases of Coxeter groups, to appear in Commun. in Algebra. [32] L. A. Bokut and L.-S. Shiao, Grobner-Shirshov basesfor Novikov and Boon groups, [331 W. W. Boone, The wordproblem. Ann. Math. 70 (1959). 207-265. [341 R. V. Book and F. Otto, String-rewrting systems. Springer-Verlag, 1993. [351 B. Buchberger. An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal. (German). Ph. D. thesis, University of Insbruck, Austria, 1965. I361 B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations. (German). Aequationes Math. 4 (1970), 374-383. [371 B. Buchberger, A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bull. 10 (1976), 19-29. 1381 B. Buchberger, G . E. Collins, R. Loos and R. Albrecht (editors). Computer algebra. Symbolic and algebraic computation. Second edition. Springer-Verlag,Vienna, 1983. [391 B. Buchberger and F. Winkler (editors). Grobner Bases and Applications, London Mathematical Lecture Note Series 251. Cambridge University Press, Cambridge, 1998. [401 K.-T. Chen, R. H. Fox and R. C. Lyndon, Free dtxerential calculus. IK The quotient groups of the lower central series. Ann. of Math. 68 (1958), 81-95. 1411 P. M. Cohn, Embedding in semigroup with one-sided division. Journal London Math. SOC.31 (1956), 169181.

71 I421 P. M. Cohn, Embedding in sesquilaterial division semigroups. Journal London Math. SOC.31 (1956), 1811981. [431 P. M. Cohn, Some remark on rhe invariant basis property Topology. 5 (1966) 215-228. [MI P. M. Cohn, Review of “The diamond lemma for ring theory” by George M. Bergman, (Adv. in Math. 29 (1978), 178-218.) Amer. Math. SOC.Math. Review 81b:16001. [451 D. Eisenbud. Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. [461 D.B.A. Epstein et al. Word processing in groups. Jones and Barlett, Boston, London, 1992. I471 T. Evans, The wordpmblem for abstract algebras, J. Lond. Math. SOC.26 (1951), N, 1. 64-71. I481 T. Evans, On multiplicative systems dejined by generators and relations. I. Normal form fheorems, Proc. Cambrige Phil. SOC.47 (1951), 637-645. [491 A. F. Garside, The braid group and other groups, The Oxford Quarterle Journal of Math. 20 (1969), 235254. 1501 V. P. Gerdt and V. V. Kornyak, Construction ofjinitelypresented Lie algebras and Superalgebras Li algebras and superalgebras, J . Simbolic Comput. 21 (1996), 337-349. 1511 H. Hironaka, Resolution of singularities of an algebraic variety over ajield of characteristic zero. I, 11. Ann. of Math. 79 (1964), 109-203; ibid. 79 (1964) 205-326. [521 K. A. Kalarkoti, Decision problems in group theory, Proc. London Math. SOC.44 (1982), 312-332. [531 S.-J. Kang, K.-H. Lee, Grobner-Shirshov bases for representation theory, J. Korean Math. SOC.37 (2000), 55-72. [541 S.-J. Kang, K.-H. Lee, Grobner-Shirshov basis theoryfor irreducible -modules, J . Algebra 232 (2000), 1-20. 1551 S.-J. Kang, 1.3. Lee, K.-H. Lee and H . Oh, Hecke algebras, Specht modulev and Grobner-Shirshov bases, to appear. 1561 0. G . Kharlampovich, M. V. Sapir, Algorithmic problems in varieties, Inter. J . Algebra Computation, 5 (1995), 379-602. [57] D. E. Knuth and P. B. Bendix, Simple word problems in universal algebras, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 263-297. Percamon, Oxford, 1970. K. H . KO,New public-key cryplosysrem using braid groups. Advances in Cryptology-CRYPT0 2000, LNCS 1880 (2000) 166-183 (with 5 others). V. N. Latyshev, Canonization and sfandart bases offiltered structures. In Lie Algebras, Rings and related Topics, Fong Yuen, A. A. Mikhalev, E. Zehnanov (Eds.), 61-79, Springer-Verlag, Hong Kong, 2000. A. A. Markov, An introduction to the algebraical theory of braids. (Russian). Proceedings of the Steklov Mast. Ins. RAS, 16 (1945). A. A. Mikhalev and E. Vasilieva, Slandard bases of ideals of free supercommutative polynomial algebras (E-Grobner bases). In Lie Algebras, Rings and related Topics, Fong Yuen, A. A. Mikhalev, E. Zehnanov (Eds.), 61-79, Springer-Verlag, Hong Kong, 2000. A. A. Mikhalev and A. A. Zolotykh, Standard Grobner-Shirshov bases of free algebras over rings. I. Free associative algebras. Internat. J . Algebra Comput. 8 (1998). 689-726. T. Mora, An introducfionto commufativeand noncommutative Grobner bases. Second International Colloquium on Words, Languages and Combinatorics (Kyoto, 1992). Theoret. Comput. Sci. 134 (1994), 131-173. M. H. A. Newman, On theories with a combinatorial dejinition of “equivalence. AM. of Math. 43 (1942), 223--243. P. S . Novikov, On algoritmic unsolvability of the word problem in the theory of groups. Moscow:Nauka, 1955 (Trudy MIAN, 44). E. N. Poroshenko, Grobner-Shirshov bases for Kac-Moody algebras of type Ail’. Comm. in Algebra, 30 (2002), 6,2617-2637. E. N. Poroshenko, Grobner-Shirshov bases for Kac-Moody algebras of type Ci” Dh”. Vestnik NGU 2 (2002), 1,58-70. A. I . Shirshov, On free Lie rings, (Russian) Mat. Sb. N. S. 45 (87) (1958). 113-122. A. I . Shushov, Some algorithm problems for Lie algebras, (Russian) Sibirsk. Mat. Z. 3 (1962). 292-296. (Translation appears in ACM SIGSAM Bull. 33 (1999).) V. Ufnarovski, Introducfionto noncommutative Grobner bases theory. In Grobner bases and applications, London Math. SOC.Lecture Note Ser. 251, Cambridge Univ. Press, 1998.259-280. A. I. Zhukov, Reduced systems of defining relalions in nonassociative algebeas, Mat. Sb. 27 (1950). 267280. ”

72 NOVOSIBIRSK 630090, RUSSIA; AND CHANG JUNG CHRISKWAYJEN, TAINAN71 1, TAIWAN E-mail address: bokutQmath.nsc .ru

SOBOLEV INSTITUTE OF MATHEMATICS, TIAN UNIVERSITY,

DEPARTMENT OF MATHEMATICS, NATIONAL CHENG-KUNG UNIVERSITY, TAINAN, TAIWAN E-mail address: yf ongQmail .ncku .edu .tw DEPARTMENT OF MATHEMATICS, NATIONAL CHENG-KUNG UNIVERSITY, TAINAN, TAIWAN E-mail address: wfkeQmail.ncku.edu.tw CHANG JUNG CHRISTIAN UNIVERSITY, KWAY]EN, TAINAN 71 1, TAIWAN E-mail address: 1sshiaoQmail.cju.edu.tw

4-valent plane graphs with 2-, 3- and 4-gonal faces Michel DEZA CNRS/ENS, Paris and Institute of Statistical Mathematics, Tokyo, Mathieu DUTOUR ENS, Paris and Hebrew University, Jerusalem, * Mikhail SHTOGRIN t Steklov Mathematical Institute, Moscow, Russia. December 22, 2002 Dedicated to the memory of Professor B H Neumann

Abstract Call i-hedrite any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only andpz+pS = i. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of a n alternating link, whose components correspond to its central circuits. Call an i-hedrite irreducible, if it has no rail-road, i.e. a circuit of 4-gond faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges. We present the list of all i-hedrites with a t most 15 vertices. Examples of other results: (i) All i-hedrites, which are not 3-connected, are identified. (ii) Any irreducible i-hedrite has a t most i - 2 central circuits. (iii) All i-hedrites without self-intersecting central circuits are listed. (iv) All symmetry group of i-hedrites are listed. Mathematics Subject Classification. Primary 52B05, 52B10; Secondary 05C30, 05C10. Key words. Plane graphs, Eulerian graphs, alternating links, point groups. *Researchof the second author was financed by EC’s IHRP Programme, within the Research Training Network “Algebraic Combinatoricsin Europe,” grant HPRN-CT-2001-00272. t Third author acknowledges financial support of the Russian Foundation of Fundamental Research (grant 02-01-00803) and the Russian Foundation for Scientific Schools (grant 00-15-96011)

73

74

1

Introduction

See [Grun67] for terms used here for plane graphs. It is well-known that the p-vector of any 4-valent plane graph satisfies to 2pz + p 3 = 8 Ci25(i - 4)pi. Some examples of applications of plane 4-valent graphs are projections of links, rectilinear embedding in VLSI and Gauss crossing problem for plane graphs (see, for example, [Liu98]).

+

Call an i-hedrite any plane 2-connected 4-valent graph, such that the number of its j-gonal faces is zero for any j , different from 2 , 3 and 4, and such that pz = 8-2. So, an n-vertex i-hedrite has (p2,p3,p4)= (8-2,22-8,n+2-2). Clearly, (2; pz,p3) = (8; 0,8), (7; 1,6), (6; 2,4), (5; 3,2) and (4;4,O) are all possibilities. An 8-hedrite is called octahedrite in [DeSt02]; in fact, this paper is a follow-up of [DeStOa]. In a way, this paper continues the program of Kirkman ([Kir85] p. 282) of classification of projections of alternating links. pj

2

Central circuits partition

In this Section, we consider a connected plane graph G with all vertices of even degree, i.e. an Eulerian graph. Call a circuit in G central if it is obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one; such circuit is called also traverse ([GaKe94]), straight ahead ([Harb97]), [PTZ96]), straight Eulerian (Chapter 17 of [GoRoOl]), cut-through ([Jeo95]), intersecting, etc. Clearly, the edge-set of G is partitioned by all its central circuits. Such CC-partition can be considered (see, for example, [Harb97]) for any drawing on the plane of any Eulerian (in general, not planar) graph, so that edges are mapped into simple curves with at most one crossing point. Denote by CC(G) = (...,a:;, ...; ...,b?, ...) its CC-vector, where ...,ui,... and ...,bj, ... are increasing sequences of lengths of all its central circuits, simple ones and self-intersecting, respectively, and ai, pj are their respective multiplicities. Clearly, u i q + Cjb j p j = 2n, where n is the number of vertices of G. For a central circuit C , denote by Int(C) := (CO;...,c?, ...), the intersection vector of C , where co is the number of self-intersections of the circuit C and ...,ck, ... is decreasing sequence of sizes of its intersection with other central circuits, while the numbers Yk are respective multiplicities. Two central circuits intersect in an even number number of vertices. The length of a central circuit is twice the number of its points of self-intersection plus the sum of its intersections with other circuits, so the length of a central circuit is even. We will say that an i-hedrite is pure if any of its central circuits simple, i.e. has no self-intersections. Easy to check that any pure i-hedrite has an even number n of vertices. In fact, any vertex in this case belong t o the intersection of exactly two central circuits. Call an Eulerian graph G balanced, if all its central circuits of same length have the same intersection vector. Any 8-hedrite with n 5 21 is balanced, but there is unbalanced 22-vertex 8-hedrite1 which is 8-hedrite 14-7of Table 2 inflated along a central circuit of length 8. We do not find unbalanced 5-hedrite or 7-hedrite

xi

75 for n 5 15. The first unbalanced 6-hedrite is 12-12. Any 4-hedrite is balanced (Theorem 5). For a plane graph G , denote by G* its plane dual and by Med(G) its medial graph. The vertices of Med(G) are the edges of G , two of them being adjacent if the corresponding edges share a vertex and belong to the same face of the embedding of G in the plane. So, Med(G)=Med(G*). Clearly, Med(G) is a 4-valent plane graph and, for any i-hedrite G , Med(G) is an i-hedrite with twice the number of vertices of G and all 2-, 3-gonal faces being isolated. The medial of smallest 8-hedrite 6-1,7-hedrite 7-1,g-hedrite 4-1,s-hedrite 3-1, 4-hedrite 3-1 are, respectively, 8-hedrite 12-4, 7-hedrite 14-9, 6-hedrite 8-3, 5-hedrite 6-2, 4-hedrite 4-1. The operation of taking the medial is a particular case of the Goldberg-Coxeter construction for the parameters ( k , 1) = (1,l) ([Gold37], COX^^], [DD03]).

3

Intersection of central circuits

The following Theorem is a local version (for “parts” of the sphere) of the Euler - 4)pi for pvector of any 4-valent plane 3-connected formula 2pz p3 = 8 Ci25(i graph P. For any 4-valent planar graph P , a patch A is a region of P bounded by q arcs (paths of edges) belonging to central circuits (different or coinciding), such that all q arcs form together a circle. A patch can be seen as a q-gon; we admit also 0-gonal A, i.e. just the interior of a simple central circuit. Suppose that the patch A is regular, i.e. the continuation of any of bounding arc (on the central circuit to which it belongs) lies outside of the patch. See below two examples of patch.

+

+

Let $(A) := p i , ... be the p-vector enumerating the faces of the patch A. The curvature of the patch A is defined as c(A) = C k Z l ( 4- k)p’,. So, the k-gon can be seen as, respectively, positively curved, flat, or negatively curved, if k < 4, k = 4, or k > 4.

Proposition 1 (a) If A be a regular patch, then c(A) = 4 - q, moreover: (i.1) c(A) = 4 if and only if A is bounded b y a simple central circuit. (i.2) c(A) = 0 if and only if A is a rectangle formed b y 4-gons put together. (ii) Any patch A is the union of regular patches Al, . . . ,A,; one has c ( A ) = (4 - 41) . . . (4 - q p ) , where each patch A, is bounded b y qi arcs. (iii) If a graph G is the union of patches A l , . . . , A p ,then 8 = c(Al)+. . .+c(A,).

+ +

76

Proof. (i) is a restatement in our terms of Theorem 1 of [DeStOP], (il) and (i2) are easy consequences. The properties (ii) and (iii) follow from the definition of the curvature of a patch and from Euler formula. Proposition 2 (i) A n y central circuit of a 4-hedrite has no self-intersection vertices. (ii) A t least one central-circuit of a 7-hedrite self-intersects. Proof. In fact, if a central circuit of a 4-hedrite self-intersects, then we have an 1-gonal regular patch. The equality of above Theorem becomes 2pL + pk = 3, an impossibility since p$ 5 p3 = 0. Take a central circuit containing an edge of the unique 2-gon, then the sequence (possibly empty) of adjacent 4-gons will necessarily finish by a 3-gon, or this 2-gon; both cases yield a self-intersection. Let us call graph of curvatures of an i-hedrite G, the graph (possibly, with loops and multiple edges) having as vertex-set all 2-gons and 3-gons of G. Two vertices (say, 2- or 3-gonal faces F and F' of G) of this i-vertex graph are adjacent if there is a pseudo-road connecting them. A pseudo-road is sequence of 4-gons, say, Fl,. . . ,f l , such that putting F o = F and Fl+l = F', we have that any F k with 1 _< k 5 1 is adjacent to Fk-l and F k + l on opposite edges (cf. the definition of a rail-road in next Section). Clearly, in the graph of curvatures, the vertices corresponding to 2- and 3-gons, have degree 2 and 3, respectively. Proposition 3 Let C1, C2 be any two central circuits of an i-hedrite. Then they are disjoint i f and only if they are simple and there exist a ring of 4-gons separating them. Proof. In fact, if both C1 and C2 are simple circuits, Theorem is evident: the curvature of the interior of a patch is 4 and so, two circuits are separated by 4-gons only. Suppose that C1 is self-intersecting. Then it has at least three regular patches and each of them has curvature at most 3. The circuit C2, being disjoint with C1, lies entirely inside one of those patches, say, A . So, all its 3-gons and 2-gons, except, possibly, those from its exterior patch, lie in A . So, c(A) 2 5, since the exterior patch of C2 has curvature at most 3. It contradicts to the fact that A has curvature at most 3.

Remark 1 Consider a 4-ualent plane graph G having only one central circuit; then, the set of faces of G can be partitioned into two classes C1, C2 i n chess manner. Every uertex v is contained in two faces F and F'. Also, the unique central circuit can be given an orientation, which induces an orientation on the set of edges. The vertex v is incident to two edges of F, el and e2, and to two edges of F', e; and e;. If el, e2 have both arrow pointing to the uertex or both arrow pointing out of the uertex, then the same is true for e; and eh, and then, we say that v belongs to Class I. Class I and its complement, Class 11, form a bipartition of the set of vertices of the knot; reversing orientation of the central circuit or interchanging C1 and C2 does not change the bipartition.

77

If the graph consists of p , p 2 2, central circuits GI,. . . , C,, then, one can put orientations on every central circuit and get a bipartition of the set of vertices. But in that case the bipartition will depend on the chosen orientations.

4

Adding and removal central circuits

The deleting of a central circuit C in an i-hedrite G consists of removal of all edges and vertices contained in C. It produces a 4-valent plane graph P’ having only k-gonal faces with k 5 4. But since cases Ic = 0 , l are possible, we do not always obtain an i-hedrite. The cutting of an i-hedrite G consists of adding another central circuit to it. The faces of the new 2’-hedrite G’ with 8 2 i’ 2 i comes from the cutting of faces of G. This operation is only partially defined, since arbitrary cutting can produce k-gons with k > 4. The cutting of a 4-gon in several 4-gons (two, if the face is traversed only once) is possible only if the 4-gon is traversed on opposite edges. This corresponds to the notion of shore-zone in [DeSt02]. A cutting changes CCpartition of an i-hedrite only in the following way: new central circuit C is added and all others central circuits remain unchanged, except that the length of each of them increases by one for any intersection with C. Call a rail-road a circuit of 4-gons, possibly self-intersecting, in which every 4gon is adjacent to two of its neighbors on opposite edges. A rail-road is bounded by two “parallel” central circuits. The deleting of one of those central circuits (in other words, collapsing rail-road into one central circuit) is called reduction. The cutting produces a rail-road if and only if it is an inflation along a central circuit C , i.e. replacing it by (thin enough) rail-road. A t-inflation along a central circuit C is replacing this central circuit by t - 1 parallel (thin enough) rail-roads. A tinflation of un i-hedrite is new i-hedrite obtained from original one by simultaneous t-inflation along all of its central circuits. A t-inflation of G is G if t = 1, and it is just inflation of G if t = 2. An i-hedrite is called irreducible if it contains no rail-road. It is called maximal irreducible if it cannot be obtained from another 2’-hedrite by a cutting.

Remark 2 Let C be a central circuit of G with CC(G)= (...,a?, ...; ..., b p , ...), and let Int(C) = (co;cT,..., c?). The t-inflation of G denoted b y Gt has CC(Gt) = (..., tafa,, ...; ..., tb;’J, ...); if C‘ is one of t parallel copies of C , then Int(C’) = (cg; c y , ..., CFY., (2co)t-l).

5

Connectivity of i-hedrites

For any integer m 2 2 denote: by the 2m-vertex 6-hedrite, such that each 2-gon is adjacent to two 3-gons; by 1 ~ , 2 ~ the + l (2m+ 1)-vertex 5-hedrite, such that two 2-gons share a vertex and remaining 2-gon is adjacent to two 3-gons;

78

m.3

m.2

n

i-hedrite

m=4

I Group [ CC-vector

n

Table 1: All i-hedrites, which are not 3-connected by I4,zm+2 the (2m+ 2)-vertex 4-hedrite, such that four 2-gons are organized into two pairs sharing a vertex; by J4,Zm the m-inflation of only one central circuit of 4-hedrite 2-1; they are projections of composite alternating links 21#21#27.. . #2: ( m times), which we denote by m x 2:. See in Table 1 the first occurrences (for 2 5 m 5 5) of those graphs, followed by their symmetry groups and CC-vectors.

Lemma 1 Any i-hedrite is 2-connected. Proof. Let G be an i-hedrite and assume that there is one vertex v, such that G - {v} is disconnected in two components C1 and Cz. Then two edges from v will connect to a vertex w of C1 and two edges from Y will connect to a vertex w' of Cz, because, otherwise, the exterior face is m-gonal with m > 4. See below the corresponding drawing.

79

But the vertex w will disconnect the graph and so, iterating the construction, we obtain an infinite sequence 211,. . . ,v, of vertices that disconnects G. This is impossible since G is finite. Any i-hedrite (moreover, any Eulerian graph) has at least one Eulerian circuit of edges; so, there is no cut-edge. But a cut-vertex appears already for some Eulerian (3-vertex) and 4-valent (4-vertex) plane graphs.

Theorem 1 Anyi-hedrite, which is not 3-connected, is one o f I ~ , 2 15,zrn+l, ~, I4,2m+2, J4,Zrnfor some rn 2 2. Proof. Let G be an i-hedrite and assume that it is not 3-connected. Then there are two vertices, say, v and v‘, such that G - {v, of} is disconnected in two components, say, C1 and C2. Amongst the 4 edges from v (respectively v’), the edges {el,. . . , e,} (respectively {ei, . . . ,e:,}) go to C1. Two numbers s and s’ can takes values 1, 2 or 3; we will consider all possible cases. Assume that s = 1 and s’ = 1, then the edges e and e‘ must be distinct, since, otherwise, C1 is the empty graph. Moreover, e and e’ have no common vertices, since, otherwise, G would not be 2-connected. So, v and vf are connected by e and e‘ to a vertex w and w‘,respectively. Since face-size is at most 4, the vertices v and v‘, (respectively, w and w’) are linked by two edges (see Figure 1). Two points w and w’can either be connected by two edges and we are done, or disconnect the graph. In the latter case, we can iterate the construction. Since the graph is finite, the construction eventually finish and we get a graph J4,2rn. If s = 1 and s‘ = 3, then by a similar reasoning, one gets again a graph J4,zrn. Assume that s = 2 and s‘ = 2. One has {el, e2) n {ei, e;} = 0, since, otherwise, one can attribute an edge to C2 and get the case s = 1 and s’ = 1. So, one has, say, el n ei = {wl} and e2 n ea = {w2}, and the following two possibilities (see Figure 1): either w1 = w2 (this corresponds to {el,e2} and {ei,ea} being the edges of two 2-gons) and we are done, or w1 # w2. Assume now that w1 # w2; two points w1 and w2 can either be connected by two edges and we are done, or disconnect the graph. In the latter case, we can iterate the construction. Since the graph is finite, the construction eventually finish. If we do the same construction on the other side, then we get a similar structure and the graph is of the form 14,zrn+2,IF,,~~+Ior 16,2rn with m 2 2. Assume now that s = 2 and s’ = 1. The edges el, e2 and e; are all distinct, since, otherwise, the vertex v disconnects the graph. So, v’ is connected by e; to a vertex w‘. Now, either v’ or w’ is connected to v, since, otherwise, we would have a 5-gonal face. If w’ is connected to 21, then the pair {w’, v} disconnects the graph. This construction is infinite (see Figure 1);so, we get a contradiction.

80

(a) The case s = 1, s' = 1

(b) The case s = 2, s' = 2

(c) The cases = 2, s' = 1

Figure 1: The three cases of Theorem 1

Theorem 2 (a) If G is an i-hedrite with two adjacent 2-gons, then this is a 4-hedrite J4,2m with m 2 2. (ii) If G is an a-hedrite with two 2-gons sharing a vertex, then it is either 4-hedrite 4-1, or an I4,2,,+2, or 5-hedrite 3-1, or an I5,2,,+1 with m >_ 2. 2-1 or a

Proof. The proof for (i), (ii) is similar to the cases (s,s')=(l,l), ( 2 , 2 ) of Theorem 1. Theorem 3 (a) an n-vertex 4-hedrites exists i f and only if n 2 2, even. (ii) an n-vertex 5-hedrites exists if and only if n 2 3, n # 4. (iii) an n-vertex 6-hedrites exists if and only if n 2 4. (iu) an n-vertex 7-hedrites exists if and only if n 2 7 . (v) an n-vertex 8-hedrites exists if and only if n 2 6, n # 7. Proof. The case (v) is proven in [Grun67], page 282. The case (i) is trivial; take, for example, the serie J4,zm. The series IG,~,,, I5,2,,+1 for any m 2 2, give 6-hedrites and 5-hedrites with even and, respectively, odd number of vertices. For 5 - , 6- and 7-hedrites, we get by (t - 1)-inflation along a central circuit of length 4 in corresponding i-hedrite 6-1, 5-1 and 7-1, series with 4t + 2, 4t + 1 and 4t + 3 vertices. By (t - 2)-inflation along such central circuit in 7-hedrite 8-1, we get a serie of 7-hedrites with 4t vertices. By (t - 1)-inflation along central circuit in 6-hedrite 11-4,we get serie of 6-hedrites with 8t + 3 vertices. By t-inflation along central circuit of length 8 in 6-hedrite 15-10,we get serie of 6-hedrites with 8t + 7 vertices.

81

Inscribing consecutively 4-gons in the 4-gon, which is adjacent only to 3-gons, in 7-hedrite 9-1 and 10-2, we get series of 7-hedrites for the remaining cases of 4t 1 and 4t 2 vertices. In case of 5-hedrites, it remains to prove existence for the case n = 4t > 1. We obtain existence in the sub-cases n = m x 4b, where m 2 3 and not divisible by 4, n = 8 x 4b, and n = 16 x 4b, respectively: by binflation of some m-vertex 5-hedrite; we showed their existence, by binflation of 5-hedrites 8-1, by b-inflation of any 16-vertex 5-hedrite (for example, one coming from 10-2 by inflation along central circuit of length 6). Our computation (see the last Section) present all i-hedrites with at most 15 vertices.

+

+

6

Irreducible i-hedrites

Theorem 4 A n y irreducible i-hedrite has at most i - 2 central circuits and equality is attained for each a, 4 5 i 5 8.

Proof. For i = 8, the Theorem is proved in [DeSt02]. We will show, using suitable cutting, that this result implies the Theorem for i < 8. Let us start with the simplest case of 7-hedrites. Consider a simple circuit S in its curvature graph, which contains the vertex corresponding to unique 2-gon. Remind that the vertices in the curvature graph correspond to 2- or 3-gons, while edges correspond to pseudo-roads. So, the simple circuit S corresponds to the circuit of faces of G, containing our 2-gon, some 4-gons and, possibly, some of six 3-gons; see the picture below.

Suppose that the 7-hedrite G is irreducible and has k central circuits. By adding the central circuit C , which is shown by dotted lines on picture above, we produce an 8-hedrite (since, the 2-gon is cut by C in two 3-gons), which is still irreducible and has k 1 central circuits. So, k 1 5 6, by Theorem 3 of [DeSt02]. For remaining cases of i-hedrites with i = 4,5,6, the proof is similar. In each case, we consider all possible distribution of 2-gons by simple circuits in the graph of curvatures and, for each such circuit, we add suitable number of new central circuits. All possibilities are presented on Figure 2: two for i = 6, three for i = 5 and two for i = 4. In the last case, there are no 3-gons and so, simple circuits in the curvature

+

+

82

(a) The two cases for 6-hedrites

(b) The three cases for 5-hedrites

(c) The two cases for 4hedrites

Figure 2: The construction of Theorem 4 for 6-, 5-, 4-hedrites

(a) The first case

(b) The second case

Figure 3: The two cases of cutting of irreducible 8-hedrite graph contain only even number of 2-gons by local Euler formula of Theorem 1. Note that the case of 4-hedrites is obvious by Theorem 5 of [DeSt02]. Lemma 2 Let G be an irreducible 8-hedrite and C a central circuit, which is incident to three 3-gons on one side. Then one can add another central circuit to G, so that the resulting graph is still irreducible.

Proof. From every one of the three 3-gons, say, T I , T2, T3, one can define two pseudo-roads from the sides of the 3-gon, which do not belong t o the central circuit. Each such pseudo-road defines an edge, say, el, e2, e3 in the graph of curvatures and so, a triangle in that graph. Then, either two triangles Ti and Tj are linked by a path, which does not involve the edges e k , or they are not linked by such a patch. In both cases we can cut the 8-hedrite according to Figure 3 and obtain another 8-hedrite, which is still irreducible. See below example of an irreducible 7-hedrite (its CC-vector is (lo2,12'; 20), its symmetry group is CzV)with the maximum number of central circuits.

83

For all other i, there is an example of irreducible i-hedrite with i circuits (see Theorem 6), which is, moreover, pure.

-

2 central

Conjecture 1 An irreducible i-hedrite is maximal irreducible if and only if it has i - 2 central circuits.

7

Classification of pure irreducible i-hedrites

The easiest case, i = 4, of i-hedrites admits following complete characterization:

Theorem 5 (i) Any 4-hedrite can be obtained from some 4-hedrite with two central circuits b y simultaneous tl- and tz-inflation along those circuits; it is irreducible if and only if tl = tz = 1. (ii) Any 4-hedrite with two central circuits is defined b y its number of vertices n and by shift j , 0 5 j 5 n / 4 with g c d ( n / 2 , j ) = 1, vertices between the pair of boundary 2-gons on the horizontal circuit (see, for example, 4-hedrite 8-1) and the remaining pair of 2-gons. Remark that several different values of shift can yield the same graph. (iii) Any 4-hedrite is balanced. Proof. (i) and (ii) are proved in [DeSt02], while (iii) is obvious for 4-hedrite with two central circuits and remain true under tl- and tz-inflation. The shift j = 0 corresponds to 2-1 and its only-on-one-circuit m-inflations Denote by K4,4m, J4,n=2m. The shift j = 1 corresponds to 4-1 and I4,+=Zm+2. for any m 2 2, any 4m-vertex 4-hedrite obtained from 4-1 by m-inflation of only one its central circuit; so, its CC-vector is ( 4 m , 4 m ) ,its symmetry is Dad and it is reducible. Clearly, any K4,n=4mhas the maximal shift j = n / 4 . Theorem 6 A n y pure irreducible i-hedrite is, either any 4-hedrite with two central circuits, or a 5-hedrite 6-2, or one of 6-hedrites 8-6, 14-20, or one of the following eight 8-hedrites: 6-1, 12-4, 12-5, 14-7, and (see Figure 5) 20-1, 22-1, 30-1, 32-1.

Proof. Let G be a pure irreducible i-hedrite having r central circuits. If one deletes a central circuit, then, in general, 1-gon can appear. It does not happen for G, since it would imply a self-intersection of a central circuit. So, the result of deletion of a central circuit from G produces an pure irreducible i-hedrite with r - 1 central circuits.

84

Figure 4: No pure irreducible i-hedrite can be obtained by cutting of the above 4-hedrite First, if r = 2, then the Theorem 5 from [DeSt02] gives that such G are exactly 4-hedrites with two central circuits; all of them are classified in Theorem 5. We prove the Theorem by systematic analysis of all possible ways to add to G (for T = 2,3,4,5) a central circuit, in order to get a pure irreducible i-hedrite with r i1 central circuits. Let r = 2 . Then G can be only one of two smallest 4-hedrites. In fact, if G is another 4-hedrite1 then, because of classification Theorem 5, it has a form as in Figure 4. New central circuit should cut both 2-gons on opposite edges, since, otherwise, there is a rail-road. But Figure 4 shows, on example for n = 6, that a self-intersection appears if two central circuits intersect in more than four vertices. So, the only possible 4-hedrites with two central circuits, which can be cut in order to produce irreducible pure i-hedrite are 4-hedrites 2-1 and 4-1. All cases are indicated below.

4-hedrite 2-1

4-hedrite 4-1

5-hedrite 6-2

6-hednte 8-5

8-hedrite 6 - 1

6-hedrite 8-5

Now, all irreducible pure i-hedrites with three central circuits a.re 5-hedrite 6-2, 6-hedrite 8-5 and 8-hedrite 6-1 (i.e. the projections of links 6;, 82 and 62). In fact, we apply the same procedure to those three i-hedrites; see picture below:

85

Nr.20-1 (S5)

D2d

Nr.22-1 D2h (83,102)

Nr.30-1 0 (lo6)

Nr.32-1 D 4 h (104,122)

Figure 5: Pure irreducible i-hedrites with 5 or 6 central circuits

5-hedrite 6-2

8-hedrite 6-1

Shedrite 12-5

8-hedrite 12-4

6-hedfite 14-20

8-hedrite 12-5

8-hedrite 14-7

1 bhedrite 14-20

Next, all irreducible pure i-hedrites with four central circuits are 8-hedrites 12-4, 12-5, 14-7 and 6-hedrite 14-20. By the same method, one can see that there are exactly two pure irreducible i-hedrites with five central circuits and two with six central circuits (see Figure 5).

Remark 3 Any pure a-hedrite comes from a pure irreducible a-hedrite with, say, j central circuits b y simultaneous tl-, . . . ,tj-inflation along those circuits; at is irreducible if and only if tl = . . . = t3. -- 1.

8

Symmetry groups of i-hedrites

We consider below the maximal symmetry groups of plane graphs; these groups are identified with the corresponding point groups.

86

Theorem 7 We indicate here the lists of symmetry groups of i-hedrites, together with the smallest number of vertices, for which they appear:

(a) The only symmetry groups of 4-hedrites are point subgroups of Ddh, which contain D2 as point subgroup, a.e. D4h(n = 2), Dd(n = lo), &h(n = 4), D2d (n = 6), D2 (n = 12). (ii) The only symmetry groups of 5-hedrites are: C,(n = lo), Cz(n = 8), C,(n = 7), Cz,(n = 5), D3(n = 15), &(n = 3). (iii) The only symmetry groups of 6-hedrites are: &(n = 4), D2h(n = 6) and all their point subgroups, i.e. D2(n = 12), C2h(n = 10), C2,(n = 5), Ci(16 5 n 5 30), C2(n = 6), C, (n = 9), C1(n = 9). (iv) The only symmetry groups of 7-hedrites are point subgroups of C2,, i.e. C2,(n = 7), C2(n = I I), C, (n = 8), C1(n = I I).

(v) The only symmetry groups of 8-hedrites are: C,(n = 16), Cs(n = 14), C2(n = 12), CZV(n = II), Ci(22 5 n 5 46), CZh(22 5 n 5 26), S4(22 5 n 5 60), D2(n = lo), &(n = 14), &(n = 22), D3(n = 18), &(n = 12), &(n = 9), 0 4 ( n = 18), Ddd(n = 8), D4h (n = lo), o ( n = 30), oh(n = 6). Proof. For 4-hedrites1 see [DeStOP]. Any transformation stabilizing a 2-gon, can interchange its two edges and two vertices. So, the stabilizing point subgroup of a 2-gon can be C2,, C,, C2 or CI only. The unique 2-gon of a 7-hedrite has to be preserved by the symmetry group; so, all possibilities are: C2,, C,, C2, C1. Every symmetry of an i-hedrite induces a symmetry on its 2-gons and 3-gons. Since the stabilizer of a 2-gon, 3-gon has maximal size 4, 6, this imply that the order of the symmetry group of an i-hedrite is bounded from above by 41Sym(8 - i)I = 4(8 - z)! and 61Syrn(2i - 8)l = 6(2i - S)!. So, in particular, the order of symmetry group of an 6-hedrite is at most 8. I f f is an element of order three, then it fixes each of two 2-gons. Since the stabilizer of a 2-gon does not contain an element of order three, no such f exists. I f f is a rotation of order 4, then f 2 is a rotation of order 2 stabilizing each 2-gon; so, the axis of f goes through the two 2-gons. This is a contradiction. By a search in the Tables of the groups one can see that the only possibilities are: C1, C,, Cz, Ci, CZ,, C2hr D2, D2h, D2d. But there exists a 6-hedrite for any of such symmetries, in Figure 6 and subsection 9.3 below. For 5-hedrites, since there are two 3-gons1the maximal order of the group is 12. The oddness of the number of 2-gons excludes central symmetry, axis of order 4 , and groups D z , DZh, D2d. If G is a 5-hedrite with a %fold axis then this axis goes through the two 3-gons, say, T'1 and T2. If one consider a belt of 4-gons around Tl, then, after a number p of steps, one will encounter a 2-gon and so, by symmetry, three 2-gons. So, we will have the following possibilities for p = 1:

87

There is only one way to extend this graph to a 5-hedrite and the obtained extension has symmetry at least D3. so, the group is D3, D3h or D3d. An 8-hedrite G has k-fold axis of rotation with k = 2, 3 or 4. If k = 3, then the axis of the rotation goes through two 3-gons, say TI and T8. If one consider, around 3-gon Tl, a belt of 4-gons, then, after some number p of steps, one will encounter a 3-gon and so, by symmetry, three 3-gons, say, {T2,T3,T4). Adding, if necessary, belts of 4-gons1one will encounter the last three 3-gons, say, {Ts,T6,T7). The patch formed by the six triangles T2,. . . ,T7 has symmetry D3 a t least and so, G has also this symmetry. Consequently, the symmetry of G is D3, D3h or D3d. If k = 4, then the axis of the rotation goes through a vertex or a 4-gon. Assume, for simplicity, that this axis goes through a vertex; then, by repeating above reasoning, one obtains two orbits of 3-gons, say, { T I T2, , T3,T4} and {Ts,T6,T7,Ts), under 4-fold symmetry and the symmetry group is D4, D4h, D4d, 0 or oh. So, one obtains the above list of 18 possible point groups. All these groups , in Figure 6 appear in subsection 9.5 (groups oh, D4d1D3h, D2, D4h, c2vD3d, (groups D2d, D 4 , D3, C2h, s4) and in Figure 5 (groups D2h, 0 ) ;for all groups (except, possibly, C,, C2h and S4) those examples of 8-hedrites have smallest number of vertices.

c,,cl,

c,,

cz),

Remark 4 The simplest case, i = 4 , of i-hedrites admits following characterization for each of its five possible groups. It has symmetry D2h (respectively, D2d) if and with even m only if it is an t-inflation, for some t 2 1 , m 2 2, of J4,zm or 14,2m+2 (respectively, of K4,4m or I4,Zm+2 with odd m). A n y 4-hedrite with group D4 or D4h has 2(k2+12) vertices for some k 2 1 2 0 (the group D4h corresponds to the case 1 = k or 0); it comes from the smallest 4-hedrite 2-1 b y Goldberg-Coxeter construction (see [Gold37], [Cox71] and [DDO3]). All other 4-hedrites have symmetry D2; i n shift terms, they are exactly those, for which interchange of central circuits changes the value of shift. We expect, that a 7-hedrite with the highest symmetry C,, exists for any n 2 10.

9

Small i-hedrites

Here and below all links are given in Rolfsen’s notation (see the table in [Ro176] and also, for example, [Kaw96]) for links with at most 9 crossings and knots with 10 crossings, or, otherwise, in Dowker-Thistlewhaite’s numbering (see [Thi]),if any. We write N if the projection in the pictures and Table below is different from the one given in corresponding cases above.

88

5-hedrite 15-1 D3 smallest (30)

6-hedrite 30-1 C; (8;26’)

8-hedrite 14-4 D2d smallest (14’)

8-hedrite 14-2 C, smallest (6; 22)

8-hedrite 16-1 Cl smallest (6,8; 18) f

8-hedrite 18-1 Dq smallest (18’)

8-hedrite 18-2 0 3 smallest (36)

8-hedrite 60-1 (16; 264)

8-hedrite 26-1 C2h (8’; 36)

8-hedrite 46-1 Ci (10;82)

S4

Figure 6: Some i-hedrites with special symmetry groups

89

We give on the pictures below all i-hedrites with at most 12 vertices, indicating under picture of each its symmetry, CC-vector and corresponding alternating link. If an i-hedrite is 2-connected but not 3-connected, then we add a symbol *just after the number. If an i-hedrite is reducible, i.e. has a rail-road, then we add mention “red.”. All i-hedrites with 13, 14 and 15 vertices are listed in Table 2. Only three reducible i-hedrites with n 5 15 have self-intersecting railroad: 5hedrites 12-3, 14-6 and 6-hedrite 13-11. On the pictures below, in order to express better the (maximal) symmetry of an i-hedrite, we put: (i) a double arrow, in order to represent an edge passing at infinity, (ii) a quadruple arrow, in order to represent a vertex at infinity.

9.1

4-hedrites

Nr.4-1 D 4 h 4: (42)

D2h Nr.4-2’ 2 x 21 (2’,4) red.

Nr.8-1* Dzh 8; (8’)

Nr.8-2 D2d 8: (42,8) red.

W Nr.6-2* D2h 3 x 2f (Z3,6) red.

-

Nr.lO-2*

-

Nr.8-4* D2h 4 x 2: (24,8) red.

Nr.12-1* D 2 h ???? (122)

Nr.12-2 Dz ???? (S2, 12) red.

Dad (102)

Nr.12-3 D 2 d ???? (43,12) red.

Nr.8-3 D 4 h 8: (44) red.

Nr.lO-3* D 2 h 5 x 2: (25,10) red.

Nr.12-4 D 2 h ???? (43,6’) red.

90

Nr.12-5* D2h 6 x 22 (26, 12) red

9.2

5-hedrites

Nr.7-1* C2, 75 (14)

Nr.7-2 C, 72 (4;lO)

Nr.9-1 C2 938 (18)

Nr.9-2* Cz, 918 (18)

Nr.10-3 Cz, (4’,6’) red.

Nr.11-l* N

11236

Cz, (22)

Cz

Nr.7-3 C2, 7: (4’;s)

Nr.8-1

Nr.10-1 C2, (4’; 12) red.

Nr.10-2 C1 10i5 (6;14)

Nr.11-2 11124

Cz (22)

-

815

(16)

91

Nr.11-4 C, ll& (42;14) red.

Nr.12-3 ????

9.3

Nr.11-5 C, 11247 (4’, 6; 8) red.

Nr.12-2 C2, ???? (122)

D3h

(12’) red.

6-hedrites

-

Nr.6-2* D2h 6: (S2)

Nr.9-2 C1 933 (18)

92

Nr.9-3 C, 9;s (4; 14)

Nr.9-4 Cz, 9& (4'; 10) red.

Nr.10-1 Cz, 10120 (20)

f

Nr.10-6 CZ 10i6 (6;14)

Nr.10-7 Cz 10i3 (4; 16)

Nr.11-5 C z ???? (8;14)

Nr.11-6 C, ???? (8,14)

Nr.lO-8 C2h lo:36 (4;s')

-

93

Nr.12-3 12499

CZ (24)

Nr.12-5

Qh Nr.12-9

Nr.12-6 C2 121167 (24)

Nr.12-7 C1 12626 (24)

Nr.12-8 C1 ???? (6; 18)

Nr.12-10 C, ???? (8,16)

Nr.12-11 D2d ???? (42;16) red.

Nr.12-12 Cz, ???? (8;s')

C, (24)

121102

????

C2

(10,14)

Nr.12-13 C, ???? (6;8,lO)

Nr.12-14 D2h ???? (4',S2) red

9.4

7-hedrites

-

@ Nr.10-1

Nr.7-1 C2, 7; (4;lO)

Nr.8-1 C, 8f3 (4; 12)

Nr.9-1 934

C, (18)

10121

C,

(20)

94

Nr.10-3 C, lo& (8,12)

N

El

Nr.11-4 ll&

Nr.12-3 C, ???? (10,14)

9.5

Cz,

Nr.12-2 C1 ???? (6; 18)

(4'; 14) red.

Nr.12-4 Cz, ???? (6';12)

Nr.12-5 C, (4'; 16) red.

'???

8-hedrites

Nr.9-1 Dsh 940 (18)

Nr.10-2 D4h (4', 6') red.

Nr.12-3 Cz ???? (6;18)

Nr.11-1 Cz, ll:zo (6'; 10)

Nr.12-4 Oh ???? (64)

Nr.10-1 Dz lo& (6;14)

Nr.12-2 12868

A

Nr.12-5 D3h ???? (64)

Dz (24)

95

Nr.

I

14-3'

13-1' 13-2 13-3 13-4 14-1 14-2 14-3 144 145 14-6 14-7 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 15-9 15-10 ~

~ ~

~

13-1 13-2 13-3 13-4 13-5 13-6 13-7 13-8 13-9 13-10 13-11 13-12 13-13 13-14 14-1 14-2 14-3 14-4 14-5 14-6 14-7 14-8 14-9 14-10 14-11 14-12 14-13. 14-14 14-15 14-16 14-17 14-18 14-19 14-20 14-21 -

Group

I CC-vector 4-hedrites 2', 14 red. hedrites 26 26

14 6; 8,12 28 6; 22 8; 20 82; 12 43; 16 red. 6'; 8' red. 43, 8' red. 30 30 30 30 12,18 14,16

I

alt.knot

133097 134054

1416368

1583814 1554593 1583824 1520161

131739 133586 131345 133811 131485 133957 132957

43; 14 red. 42,8;10 red.

~

-

28 28 28 28 6; 22 6; 22 10; 18 10; 18 10,18 14' 142 12,16 12,16 42; 20 red. 8; 10' 16

62,82 63, 10 red.

1417173 1417079 148767 1417734 1417148 1417309 145570

4'; 10' red. 42,8; 12 red. 30 30 30 30 30 30 30 6;24 6;24

CZh

Cz, Cz Cz Cz

Ci Ci Ci Ci Ci Ci Cz,

1539533 1566949 1583008 1545248 1520975 1564488 1545357

8;22 8;22 8;22

C1

CI C, C1 Czv Cz. C,

10,20 10,20 8;10,12 6',8; 10 63;12 red.

-

13-1 13-2 13-3 13-4 13-5 13-6 13-7 14-1 14-2 14-3 14-4 14-5 14-6 14-7 14-8 14-9 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 15-9 15-10 15-11 15-12 ~

8'; 10

28 28 28

14-22 14-23 15-1 15-2 15-3 15-4 15-5 15-6 15-7 15-8 15-9 15-10 15-11 15-12 15-13 15-14 15-15 15-16 15-17

II

8; 22

8'; 14 43; 18 red. 43, 8; 10 red. hedrites 26 26 26 26 26 26 26 8; 18 12,14 4'; 18 red. g2, 10 red.

11

~

Ci C1 Ci C, Cz. C,

c1

c1 61

Cl

Ci

c-, Cz C, Cz, Cz Ci CI C, Ci Ci

c1 c1

Cz, Ci

Czu Ci,

1

~

~

13-1 13-2 14-1 14-2 14-3 14-4 14-5 14-6 14-7 14-8 15-1 15-2 15-3 15-4 15-5

26 6;20 10,16 10,16 6';14 6';14 28 28 28 28 6;22 10,18 14' @;16 @;16 30 30 30 6;24 6;24 6;24 10,20 10,20 14,16 14,16 6';18 43;18 red. 8-hedrites

133861 133769

1413725 1410841 145714 1414207

1582225 1560207 1580242

133478 1417895

C, Dz Dz,~ Cz

Dz

6;22 6;22 14' 6';16 8;1OZ

~

~

Table 2: All i-hedrites with 13. 14 and 15 vertices

1582477

96

References [Cox711 H.S.M.Coxeter, Virus macromolecules and geodesic domes, in A spectrum of mathematics; ed. by J .C.Butcher, Oxford University Press/Auckland University Press: Oxford, U.K./Auckland New-Zealand, (1971) 98-107. [DDF02] M.Deza, M.Dutour and P.W.Fowler, Zigzags, Rail-roads and Knots i n Ful lerenes, submitted. [DeGr99] M.Deza and V.P.Grishukhin, 11-embeddable polyhedra, in: Algebras and Combinatorics, Int. Congress CAC ’97 Hong Kong, ed. by K.P. Shum et al., Springer-Verlag (1999) 189-210. [DHL02] M.Deza, T.Huang and K-W.Lih, Central Circuit Coverings of Octahedrites and Medial Polyhedra, Journal of Math. Research & Exposition 22-1 (2002) 4966. [DeSt02] M.Deza and M.Shtogrin, Octahedrites, Symmetry, Special Issue “Polyhedra and Science and Art”, 2002. [Dut] M.Dutour, PlanGraph, a gap package for Planar Graph, in preparation. [DD02] M.Dutour, M.Deza, Zigzag Structure of Simple Bzfaced Polyhedra, submitted. [DD03] M.Dutour, M.Deza, Goldberg-Coxeter Construction for convex polyhedra, in preparation. [GaKe94] M.L.Gargano and J.W.Kennedy, Gaussian graphs and digraphs, Congressus Numerantium 101 (1994) 161-170. [GoRoOl] C.Godsi1 and G.Royle, Algebraic Graph Theory, Graduate Texts in Mathematics 207, Springer-Verlag, Berlin - New York, 2001. [Gold371 M.Goldberg, A class of multisymmetric polyhedra, Tohoku Math. Journal, 43 (1937) 104-108. [Grun67] B.Griinbaum, Convex polytopes, Interscience, New York, 1967. [Griin72] B.Grunbaum, Arrangements and Spreads, Regional Conference Series in Mathematics 10, American Mathematical Society, 1972. [GrunMo63] B.Griinbaum and T.S.Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canadian Journal of Mathematics 15 (1963) 744-751. [Harb97] H.Harborth, Eulerian straight ahead cycles in drawings of complete bipartite graphs, Bericht 97/23, Institute fur Mathematik, Tech. Universitat Braunschweg, 1997.

97

[Heid981 O.Heidemeier, Die Erzeugung von 4-regularen, planaren, simplen, zusammenhangenden Graphen mat vorgegebenen Flachentypen, Diplomarbeit, Universitat Bielefeld, Fakultat fur Wirtschaft und Mathematik, 1998. [Jeo95] D. Jeong, Realizations with a cut-through Eulerian circuit, Discrete Mathematics 137 (1995) 265-275. [Kaw96] A.Kawauchi, A survey of knot theory, Birkhauser, 1996. [Kir85] T . Kirkman, The enumeration, description, and construction of knots with fewer than 10 crossings, Trans. Roy. SOC.Edin. 32 (1885), 281-309. [Kot69] A.Kotzig, Eulerian lines in finite 4-valent graphs and their transformations, in: Theory of Graphs, Proceedings of a colloquium, Tihany 1966, ed. by P.Erdos and G.Katona, Academic Press, New York (1969) 219-230. [Liu98] Liu Yanpei, Embedding in Graphs, Kluwer, Dodrecht, 1998. [PTZ96] T.Pisanski, T.Tucker and A.Zitnik, Eulerian Embedding of Graphs, University of Ljubljana, IMMF Preprint Series 34 (1996) 531. [Ro176] D.Rolfsen, Knots and Links, Mathematics Lecture Series 7, Publish or Perish, Berkeley, 1976; second corrected printing: Publish or Perish, Houston, 1990. [Sha75] H.Shank, The Theory of Left-Right Paths, in: Combinatorial Mathematics 111, Proceedings of 3rd Australian Conference, St Lucia 1974, Lecture Notes in Mathematics 452, Springer-Verlag, Berlin - New York (1975), pp. 42-54. [Thi] M.Thistlewaite, Homepage, h t t p : //www .math . u t k . edu/"morwen.

GRAPH SEMIGROUPS

VLASTIMIL DLAB a n d TOMAS POSPfCHAL School of Mathematics and Statistics Carleton University Ottawa, ON K1S 5B6, Canada v d l a b Q m a t h . c a r l e t o n . c a , tpospichQmath.carleton.ca

The main objective of this paper is the representation theory of finite semigroups, over finite fields in particular. In our approach, which follows methods of the r e g resentation theory of finite groups, we are going to define a class of semigroups (Definition 2.1) that are shown to be quotients of path semigroups. The representation theory of finite semigroups is controlled by the representations of those graph semigroups which are free (Definition 2.2) in the following sense. Let S be a finite semigroup and K a field. Then there is a free graph semigroup F = F ( E ,G) such that the category of all (finit+dimensional) K-representations of S is equivalent to a subcategory of the category of all (finite-dimensional) representations of F .

For a finite or algebraically closed field K , the semigroup algebra KS is Morita equivalent to a quotient algebra K Q I I , where K Q is the path algebra over the quiver Q = ( E , G ) and I is an admissible ideal (ideal of relations). Our examples in Section 4 show that in many situations the ideal I is generated by monomials, and thus the category of all representations of S is equivalent to the category of all representations of a graph semigroup T = T ( E ,G ) . Of course, this is trivially true in the case that the ideal I is equal to zero, i.e., the algebra KS is hereditary. It is an open question under what conditions, given a finite semigroup S and a field K , there is a graph semigroup T such that the categories of K-representations of S and T are equivalent.

Keywords: graph semigroups, semigroup algebras, representations of semigroups

Dedicated to the memory of B. H. Neumann

1. Introduction The main objective of this paper is the representation theory of finite semigroups, over finite fields in particular. In our approach, which follows methods of the representation theory of finite groups, we are going to define a

98

99

class of semigroups (Definition 2.1) that are shown to be quotients of path semigroups. The representation theory of finite semigroups is controlled by the representations of those graph semigroups which are free (Definition 2.2) in the following sense. Let S be a finite semigroup and K a field. Then there is a free graph semigroup F = F(E,G)such that the category of all (finitedimensional) K-representations of S is equivalent to a subcategory of the category of all (finite-dimensional) representations of F. For a finite or algebraically closed field K , the semigroup algebra KS is Morita equivalent to a quotient algebra K Q I I , where K Q is the path algebra over the quiver Q = (ElG) and I is an admissible ideal (ideal of relations). Our examples in Section 4 show that in many situations the ideal I is generated by monomials, and thus the category of all representations of S is equivalent to the category of all representations of a graph semigroup T = T(E,G).Of course, this is trivially true in the case that the ideal I is equal to zero, i.e., the algebra KS is hereditary. It is an open question under what conditions, given a finite semigroup S and a field K , there is a graph semigroup T such that the categories of K-representations of S and T are equivalent.

2. Basic definitions and results Throughout the paper, we shall follow standard terminology of the theory of semigroups. We shall assume, unless stated otherwise, that semigroups have 0. Here is the basic definition.

Definition 2.1. A semigroup S (not necessarily finite) is said to be a graph semigroup if (1) S has enough idempotents, i.e., there is a (finite) set of idempotents E = {el, e 2 , . . . , en} such that (a) eiej = 0 for i # j , and (b) for every s E S , s # 0, there are unique idempotents ,e and e, of E such that ,es = s = se,.

(2) There is a finite irreducible (minimal) set G = (91,9 2 , . . . ,gk} of non-zero elements such that G S \ E and G generates the radical radS of S: radS = S \ E = (G) U (0). An immediate consequence is the following lemma.

.

100

Lemma 2.1. (1) If kes = s then es = 0 for all e E E , e # ,e. (2) If ses = s then se = 0 for all e E E, e # e,. (3) For any s,t E S such that se # te,

SSn t S = (0). We will use the notation SSUtS = ss U t S

in case the condition (3) is satisfied. Hence, for a graph semigroup S = S(E,G) with E = {el, e2,. . . ,en}, we have

S

= elS

U e2S U . . . U ens.

A graph semigroup carries the following data: a finite semigroup Eu{O} of “orthogonal” idempotents (eiej = bijei, with the Kronecker symbol & j ) , and a finite E-E-set G = (91, g2,. .. ,gk} such that, for each g E G, there are unique e,e’ E E with eg = g = gel.

Let as call the data (E,G) a scheme.

Example 2.1. A scheme is easily represented by a directed graph. For instance, E = {e1,e2}, G = {g1,g2} with gle = el,egl = e2,g2e= e2,eg2= el corresponds to the graph

The multiplication tables below define two obviously non-isomorphic graph semigroups over the same scheme: el e2 91 Q2

101

and

el ez 91 g2 a

a

a

For clarity, zero products are shown as blank space in multiplication tables. Rows and columns corresponding to 0 (and 1, if applicable) are not shown. The following concept is a natural generalization of the notion of a free semigroup.

Definition 2.2. A graph semigroup F = F ( E ,G ) is said to be a free graph semigroup if, for every 0 # a E r a d F , there is a unique presentation

a = gjzgj2 * . . g j ,

with

gj,

E G for 1 5 q

I p.

Of course, in a free graph semigroup, both E and G are uniquely determined. We may say that F is a free (or hereditary) over the scheme

( E ,GI. The following statement provides raison d’ztre for free graph semigroups.

Theorem 2.1. Given graph semigroup S = S ( E ,G ) , there is a canonical homomorphism cf, : F = F ( E , G )-+

S = S(E,G)

of the free graph semigroup F over ( E ,G ) onto S , fixing the elements of E and G. Thus S 2 F ( E , G ) / R ,where R is the congruence o n F defined by a

R N

b

i f and only if @ ( a )= @(b).

The proof of Theorem 2.1 is straightforward. Observe that the set R 0) is an ideal of F . Moreover, for all e E E and g E G , R = e and g a implies a = g. The following is a natural example of a graph semigroup. Let Q = ( Q O , 91) be a finite quiver, i.e., a finite oriented graph with the set QO of vertices and the set Q1 of arrows. The set of all (oriented) paths in Q together with adjoint zero forms a semigroup: product of two paths is their

-

I = { a E FJa R e a implies a

N

-

102

concatenation if the first path ends in the same vertex where the second one begins; otherwise the product is zero. Vertices of Q are considered to be paths of length zero. Putting E = Qo and G = Q1, we observe that the semigroup of paths is a free graph semigroup over the scheme ( E ,G). Let us point out that this semigroup will be finite if and only if the quiver Q has no oriented cycles. One can see immediately that every factor semigroup of a path semigroup is a graph semigroup. In fact, since every free graph semigroup F = F ( E ,G) can be easily identified with a path semigroup over Q = ( E ,G), Theorem 2.1 allows us to formulate the following statement.

Theorem 2.2. Every graph semigroup is isomorphic to a factor semigroup of a path semigroup over a suitable quiver. 3. Representations of semigroups Although the concept of an S-set (or, equivalently, an S-act) may be in certain situations useful, it is natural, as in the case of groups, to deal with a more appropriate concept of representations of semigroups in terms of matrices over a field.

Example 3.1. To illustrate a case where the concept of an S-set is useful, consider the Kronecker semigroup S = { e l , e2,g1,92,0) given by the multiplication table

$= el e2 91

92

Q2

Q2

Equivalently, S is the free graph ( { e l , e 2 L { Q l , Q 2 Hgiven by

semigroup over the

scheme

Here, all “discrete” indecomposable representations are given by S-sets. For each n 2 1, the set

103

with the S-action given by

xi . el = 21,

xi . e2 = w ,

xi *g1= Yi,

xi .g2 = YZ+l,

y3. . el = w ,

Y j . e2 = Y j ,

Y j .g1 = w ,

Yj

'

Q2 = W

is an indecomposable S-set. Together with the "transposed" S-sets

X n = { 21

1

. xn, xn+1, YI

~ 2* , *

7

7

YZ,

..

. l

Yn 7 W }

where g2 now acts by xi+l 'g2 = yi they exhaust all discrete indecomposable representations3 of S. Recall that we use here the term S-set in the following sense.

Definition 3.1. Given a semigroup S , a set X is an S-set if (1) X has a "distinguished" element w (so that X = X " U { w } ) ; (2) there is a map X x S -+ X, (2,s) H x . s such that ( x . s ) . t = z . ( s t ) , x . 1 = x and w . s = w for all s , t E S . Thus, an S-set is a semigroup homomorphism Q :S

4

Transf,(X)

=

{all maps X

-+ X

king w )

that maps 1 into the identity map on X and 0 into Q(0) : X Usually, (x)Q(s) is conveniently recorded as x s.

-

4

{w}.

Such semigroup homomorphisms are in one-to-one correspondence with semigroup homomorphisms Q :S

4

Lin(VK, X").

Here, Lin(VK,X") denotes the set of all linear transformations of the K space VK (for a chosen field K ) which map each element of the basis X " of XK into an element of X " or into the zero vector of VK. Thus there is a natural extension of the concept of an S-set X to a concept of an S-module over a field in the following sense.

Definition 3.2. Given a semigroup S and a field K , an S-module (or S-representation) over K is a semigroup homomorphism Q :S

4

Lin(XK)

of S into the multiplicative semigroup of the K-algebra of all linear transformations of a vector space XK over K .

104

Again, we will often write 2 . s (z E XK,s E S ) in lieu of (z)@(s). Those representations that correspond to S-sets will be called monomial. Indeed, fixing the appropriate basis of XK, the entries of the matrices representing the elements of S will be 0 and 1; moreover, there will be at most one non-zero entry in each row. In what follows, we shall consider only finite-dimensional representations, i.e., those with dimXK < 03.

Example 3.2. Let us now consider the free graph semigroup 5' = { e l , e2, e3, e4,g1, g2, g3,O)

over the scheme e2

That is, the multiplication in S is determined by elgle4

= 91,

e~gze4= 9 2 ,

e3g3e4 = g3.

Consider the indecomposable representation :S following action of S on the ordered basis B1 = K-space XI:

4

XI defined by the

( Z ~ , X ~ , Q , Z of ~ )

. el

= (21 10,010),

B1.gl

a1 ' e 2

= (0,2210,0),

8 1 .g2 = (0,547 O,O>,

,131

(070, z3,o>, = (O,O, 0 , ~ ) .

a1 . e3 = ,131

. e4

the

= (24,0,0,0),

131 . 9 3 =

(070, 2 4 , o>,

Clearly @ I is monomial. On the other hand, the indecomposable rep: S + X2 whose S-action on the ordered basis ,132 = resentation ( q , 2 2 , 5 3 , 2 4 , 2 5 ) of the K-space XZis defined by &.el

= (~1,0,0,0,0),

BZ * e2 = ( 0 , O,O, ~ ~ O), . e3 = (O,O, 23,0, 01,

.

,132 e4

= (O,O,

&*g1= B2

'

92 =

,132 * Q3 =

(~4,0,0,0,0), (O, z5i O, Oi O ) ,

(o,o, 5 4 -I-%,o, 01,

0,24,55),

is not monomial. Recall that the semigroup algebra A = KS of a semigroup S over a field K is the K-space of all iinite support functions f from the set of all nonzero elements of S to K with the convolution multiplication: if ( s ) f l = a,

105

and ( t )f 2 = Pt, then ( r ) (f1 . f 2 ) = Cst=r a,,&. It is easy to formulate the close connection to the representations of S.

Theorem 3.1. There is a one-to-one correspondence between the S modules over K and the A-modules, where A = KS is the semigroup algebra. Now if S is finite, A = KS is a finite-dimensional associative K-algebra, that is, a finite-dimensional vector space over K with a structure of a monoid, such that both left and right multiplications by elements of A are linear transformations of the K-space A. Consequently, if the field K is algebraically closed or finite, KS is a factor algebra of a path algebra over a quiver. We may now formulate the main theorem.

Theorem 3.2. For any finite semigroup S and a perfect field K , there exists a free graph semigroup F = F(E,G) and an ideal I in K F such that the K-algebras KS and KFII are Morita equivalent, i.e., their (abelian) categories of finite-dimensional representations are equivalent. If the ideal I is generated by monomials in G, then there is a graph algebra T = T(E,G) so that K S and KT are Morita equivalent. 4. Illustrations

In this final section, we are going to provide some illustrations of Theorem 3.2. In describing the representations of given semigroups, we shall exploit heavily the theory of representations of finite-dimensional associative algebras'.

Example 4.1. The symmetric inverse semigroup S = {0,1, e , f , a , a-l, b} of all partial one-to-one maps of a 2-element set has the following multiplication table:

Let K

= IF2

be the Galois field of two elements. Then

A = KS E Matzxz(K) @ K[z]/(z2).

106

Here,

+

+ + +

+

b). and K[z]/(z2)= K1+ K%= K(1+ e f) + K(1+ e f a u-l Denote by T the graph semigroup T = T ( E , G ) , where 3 = {el,e2}, G = {g}, with multiplication given by e2gez = g and g2 = 0. The graph of T is

and the multiplication table follows.

Writing B = KT, we have an equivalence of categories mod-A e mod-B. Thus the semigroup S has (up to isomorphism) just 3 indecomposable representations over F2.

Example 4.2. The symmetric inverse semigroup S of all partial one-toone transformations of a %set has 33 non-zero elements. If K = IFz, the algebra A = K S has the following structure: A

Mat3x3(K)@ Mat3x3(K[z]/(z2))@ Matnx2(K) @ K[z]/(z2).

The graph semigroup T = T ( E ,G) over the scheme

ii has E = {el,e2,e3,e4}, G = {gl,g2} and multiplication e2g1e2 = 91, e4g2e4 = 92 and gigj = 0 for 1 5 i , j 5 2. Equivalently, the multi-

107

i::.:'.

plication table of T is

el

e2 e3

e4

91

g2

Q2

and we have again an equivalence of categories mod-KT 21 mod-A. Hence, the semigroup S has 6 indecomposable representations over IF:!.

Example 4.3. Let us consider now the 15-element inverse semigroup = { e , f , g , a , b, c , d , p , q , a - l , b-l, c-l, d - l , 0 , l ) given by the following multiplication table.

c d d

f e

g e

a d

b

f , e f

e g a b

a d c c

b c

b

b

-

e 9 U

b

~e e

e b b

e b b

C

C

d P

d P

4

d-1

b-

b-l

C-1

c-l

d-l d-1 d-1 d-1

In fact, K

= IF2

p

b-lb-' b-lf2-l d-l f p b-' e P P c b P b d e b-' e

Q d

C C

c

P

c-l d-l

4 d-l

g e d e e d d-1 d-1 4 4

d-l

b-' b-

q

c-l

c-l

c-l

c-l

C-1

4

A = K S . Then A Z MatsX3(K)@ Matz,z(K)

and

= Pz(l+ e + f

Matsx3 =

d d

4

d-1 .-'

Let K

p qa-'b-'c-' b-1 b-1

@ K.

+ g) and

[!I: i-li] ,

[

Matzx2 = e + f + P u - l + b-l

+ d-l

+ +

ea + gb +qd

1

*

108

Thus, putting i? = {el, e2, e3}, the semigroup algebra KT of the free graph semigroup T = T ( Z ,0) satisfies mod-A 21 mod-KT.

Example 4.4. The subsemigroup S = { e, f ,g, a , b, c, d, 0,1} of S is of more interest. Here, for K = Pz,the semigroup algebra A = K S is isomorphic to the semigroup algebra KT of the free graph semigroup T over the following scheme.

Thus A is a tame hereditary algebra, a direct sum of a copy of K and a path algebra of the Euclidean (extended Dynkin)' diagram &. Explicitly, E = {el = e f,e2 = e, e3 = e g, e4 = 1 e f g} and G = (91 = b, g2 = a b d, g3 = d } . There is an infinite number of indecomposable representations. In fact, there are precisely 3 (non-isomorphic) indecomposable representations of S of dimension 3h 1 and 3h 2 for every natural h 2 0. The number of indecomposable representations of S over IF2 of dimension 3h is given by the number a(h) of irreducible polynomials of degree h over Fz. Referring to Gauss2, a ( h )= Cqlh p.($)24, where p. is the Mobius function, and thus the number of indecomposable representations of S of dimension 3 , 6 , 9 , 1 2 , 1 5 , .. . ,30,. . . , 6 0 , .. . ,120,. . . ,150,. . . is given, respectively, by 2 , 1 , 2 , 3 , 6 ,. . . ,99, . . . ,52 377, . . . ,27 487 764474,. . . , 2 2 517 997 465 744, . . . For example, the unique indecomposable representation of S of dimension 6 is given by

+ +

+

+

+

+ + +

+

;

where Eij denote the elementary matrices ( 1 5 i , j 5 6).

109

ef g h a b c d s e e f e a a c c f a f e f c f a e c a f ggdghhbbdh hbhghd bgd h a c a e a f c e f a bbhbddgghd c c a c f f e e a f dgdbdhgbhd s bhg hd bgd s

Observe that the semigroup S is generated by two elements: e and s.

91

where B is the path algebra of el

T-,

e2 modulo the ideal

92

(92919291). Denoting by T the graph semigroup over the scheme

92

with multiplication determined by the relation g 2 g l g 2 g 1 = 0, we have an isomorphism of algebras KT E A; explicitly, the generators of KT can be mapped as follows:

Hence the semigroup S has 10 indecomposable representations.

110

(2) Let K be any field with characteristic has now the following structure:

A

S

# 2.

The algebra A = KS

C @ Matzx2(K) @ K ,

91

where C is the graph algebra over el

e2

defined by the rela-

92

tion g2g1 = 0. We provide an explicit isomorphism: el

++

1

+h

-

s- i(e

+ f + g + h ) + :(a + b + c + d ) ,

e2++ i ( e + f + g + h + a + b + c + d ) , g1He+ f -g-h+a-b+c-d, g2-e-

Matzxz

K

f+g-h+a+b-c-d,

=

K(-h

r

z(e+ f - a - c ) g-h-b+d

e- f +a-c a(g+h-b-d)

+ s).

1

’

Therefore, over a field of characteristic different from 2, the semigroup S has only 7 indecomposable representations.

Example 4.6. Let us conclude this exposition by presenting three free graph semigroups S F ,ST and SW. Let E = {el, e2, . . . , e6) and G = {91,92,...,95). (1) SF = SF(E, G) is given by

(2) S,

= ST(E, G)

has multiplication

(3) SW = S w ( E , G ) is defined by eigie6 = gi

for 1 5 i 5 5 .

111

Here SF is described by the Dynkin diagram lE6 e5

el

91

e2

----g2

l g 5 94

e6 c-- e4

93

e3

t-

and thus has 36 indecomposable representations, the largest one of dimension 11. ST is given by the Euclidean diagram 6 s

and is of tame (infinite) representation type. Finally, the semigroup algebra of SW is a hereditary algebra over the graph

that is neither Dynkin nor Euclidean, and thus is of wild representation type'. References 1. V. Dlab, C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. SOC. 173 (1976) 2. C . F. Gauss, Disquisitiones Arithmeticae; Untersuchungen uber hohere Arithmetik, Chelsea Publ. Co., 1989 3. L. Kronecker, Algebraische Reduction der Schaaren bilinearer Formen, Sitzungsber. Akad. Berlin, 1890, pp. 736-776

GENERAL NOTIONS OF INDEPENDENCE KAZIMIERZ GLAZEK Dedicated to the memory of Professor B.H. Neumann

ABSTRACT. In 1958 E. Marczewski introduced a general notion of independence, which contained as special cases the majority of independence notions used in various branches of mathematics. A non-empty set I of the carrier A of an algebra 2l = ( A ; P) is called M-independent if equality of two term operations f and g of the algebra 2i on any finite system of different elements of I implies f = g in A . There are several interesting results on this notion of independence. However the important scheme of M-independence is not wide enough to cover stochastic independence, independence in projective spaces and some others. This is why some notions weaker than the M-independence have been developed. The notion of independence with respect to a family Q of mappings (defined on subsets of A ) into A , Q-independence for short, is a common way of defining almost all known notions of independences. There exists an interesting Galois correspondence between families Q of mappings and families of Q-independent sets. In our article after a brief survey of these topics we will mainly concentrate on a few easily formulated and interesting results.

Introduction. Notions of independence appear very often in many branches of mathematics and play an important role in its applications (e.g. in computer science, optimization theory and combinatorics, see references at the end of this article). Examples of such notions are very known: linear independence (of vectors, points, numbers), linear independence in abelian groups (in the sense of T . Szele), algebraic independence in the theory of numbers (and, more generally, in the theory of field extensions), independence of polynomials (more generally - of continuous functions), logical independence of axioms,independence in lattice theory and in the theory of Boolean algebras (and its applications in measure theory), independence (of vertices or edges) in graph theory, independence with respect t o (some) closure operators, stochastic independence, independence in the theory of data bases, etc. They not only have similar names, but also some common features of several notions of independence and freedom have been observed. There has been a tendency t o find general schemes containing all (or almost all) those notions as special cascs. Such a scheme permits simplifications and unifications of proofs of various results lying even in rather distinct branches of mathematics, e.g., dependence relations in some topological investigations and in electrical networks. Such abstract notions of independence have so far been based either on Set Theory ( E.H. Moore, H. 1991 Mathematics Subject Classification. 08B20, 06A15, 05B35, 52A01, 52B40, 90C27,

Key words and phrases. independence, Marczewski independence, M-indepence, independece with respect to a family of mappings, Q-independence, homomorphisms, free algebra, subalgebra, closure space, matroid, equicardinality of bases, abstract dependece, abstract convexity, equational basis, Tarski interpolation theorem, v-algebra, w *-algebra, v**-algebra, separable variables algebra.

112

113

Whitney, B.L. van der Waerden and T. Nakasawa) or on the Universal Algebra (G. Birkhoff, Ph. Hall, E. Marczewski). Both these directions are interrelated. The set-theoretical scheme is using the notion of independence with respect of closure operator and leads t o notions of matroids and greedoids which are important in combinatorics and the theory of optimization. We do not have enough space here t o discuss all interesting aspects of the theory of independence and the theory of abstract dependence relations. For more information and references the reader is referred to the monographs and survey articles cited here (see references below). This survey article is devoted mainly t o the algebraic scheme and can be treated as a supplement t o the articles [86] and [47], and Chapter 5 of [58],where you can also find many interesting problems for further inverstigations. In 1958 E. Marczewski ([Sl]) introduced a general notion of independence (which we call M-independence) which contained as special cases the majority of independence notions used in various branches of mathematics (see, e.g., [86], [47]). Several authors have investigated this notion. We do not have enough space to report many interesting results (but the reader can find references t o them at the end of this short article). However M-independence is not a wide enough notion t o cover stochastic independence, independence in projective spaces and some others. This is why some notions weaker than M-independence have been developed. A more general scheme was proposed by E. Marczewski in the mid-1960s (see [87], [86]).The notion of independence with respect t o a family Q of mappings (defined on subsets of A ) into A , Q-independence for short, is a common way of defining almost all known notions of independences. There exists an interesting Galois correspondence between families Q of mappings and families of @independent sets (see [48], [49]). Another kind of Galois correspondence concerns relations between Q-independences in an algebra 2l and congruences of algebras of (n-ary) term operations (see [50]).

I. Closure spaces. Several notions of independence or freedom appeared in various branches of mathematics have a very similar feature, for example the following kinds of independence have some common properties: linear independence (of vectors, points; or numbers), independence in graph theory (of vertices or of edges), algebraic independence (of numbers or, more generally, in the theory of fields). For an abstract setting of this three notions, it is enough to use closure spaces of finite character and with a so-called exchange property. It leads to matroid theory (and t o dimension theory). Let (A,C)be a closure space, i.e. A is a non-empty set and C : 2 A 4 2* is a (generalized) closure operator (in the sense of E.H. Moore), i.e.

C is extensive : monotonic

X :

C(X),

X & Y =+ C ( X ) c C(Y),

idempotent : C ( C ( X ) )= C (X) , ( X ,Y A ) . More general closure spaces are also useful for abstract consideration of logical independence (e.g. of axioms) and independence of bases in equational logics (see, e.g., [149]). For these notions suitable closure spaces can be considered and a useful notion is that of independence with respect t o a closure operator, i.e. if (A;C)is

114

a closure space and X & A , then X is said t o be C-independent (X E C-ind, for short) if

(Va E X ) [a $ C(X \ { a } ) ] . There exist some other notions of independence that are not covered by the above general “set-theoretical” scheme, but are covered by the “general algebraic scheme” of E. Marczewski (for example, independence in Boolean algebras, independence of polynomials or, more generally, continuous functions; etc., see, e.g., [47], [83], [8G], [89], and [SG]). 1.2. The theory of matroids (and dimension theory). In this case we consider some additional properties of closure operators. A closure space is said t o be a matroid if the following conditions are fulfilled: (i) : A is finite, or, more generally, (i’): the operator C is of finite character, i.e. : X C A , a E C(X) + (3 finite Y 2 X) [a E C(Y)] (in this case, a closure space (A,C)is called algebraic), and (ii) : the exchange property (EP) holds: X C A , b $ C(X), b E C(X U { a } ) + a E C(X U {b}). The notion of matroid was introduced (independently, and in equivalent forms) by H. Whitney ([lG3]),B.L. van der Waerden ([159]),and T. Nakasawa ([97]) in the mid-1930s (see also [72]). For infinite matroids, see, e.g., [lll]and [112]. Matroids also appear implicitly (in 1926 and 1928) in the papers by 0. Bor6vka ( [l3]) and K. Menger ([91]). The algorithmic approach also leads to a more general structure, namely t o the notion of greedoid (see: [12], [31], [71]). The family J of all C-independent subsets of a matroid has the following properties: (1) 0 # J G 2*, (2) X E J, Y C X + Y E J (hereditarity), (3) finite X, Y E J, c a r d ( X ) = c a r d ( Y ) 1 + ( 3 a E X \ Y ) [(YLJ { a } ) E J].

+

The notion of matroid can also be defined as a pair ( A ;J) with properties(l), (2) and (3). A set X A is said to be a C-basis (or an irredundant basis, see [149]) if A = C(X) and, simultaneously, X is C-independent (i.e. X E C-ind). Every C-basis is a maximal C-independent set (and also a minimal generating set). From (EP) we have the equicardinality property of C-bases. These facts generalize well-known ones from the theory of linear spaces (proved independently by A.N. Whitehead and E. Steinitz between 1898 and 1910). To define of the notion of matroid we can use also a family of C-bases. There exist several ways t o define the notion of matroid, e.g. by using a closure operator C, families C-ind or C-bases, rank functions, etc. (see, e.g., [72], [73], [113], [120], and [160]). The theory of matroids has many applications, for example in combinatorics and some optimization problems (see, e.g., [l],[31],[34], [47] [G3], [75], [113], [116], and [lGO] - [162]). Recently the theory of oriented matroids has been developed and applied (for example: to algebraic and differential topology; see, e.g., [5], [Ill, ~ 9 1 1421, , and [431). One possible way to introduce of the notion of matroid and the more general notion of infinite matroid is by using an abstract dependence relation; see [97], [125]

115

and [159]. Several kinds of general dependece relations have been used in algebraic logic, dimension theory and information sciences; see, e.g., [3], [4], [7], [S], [20] 1251, [331 1401, [41], [65], [67l, [931, [loo], [lo31 - [log], [114], [115], [122]- [123], [132] - [134], and [137]. Dependency structures (in the sense of W.W. Armstrong) and functional dependecies play an important role in the theory of relational databases (see the above references).

1.2. The theory of abstract convexity (or anti-matroids). In this case a closure operator C is again of finite character and C has a so-called anti-exchange property (AEP): X C A; a,b E A, a # b, b $ C ( X ) , b E C ( X u { a } )+ a $ C ( X u { b } ) . Then such a closure space ( A , C ) is said to be anti-matroid. The theory of antimatroids is a useful tool for investigating abstract Convexity; see the following survey aricles: [26], [30], and [64]. Let e z t ( X ) be the set of all extremal elements of a set X , i.e. a E e z t ( X ) if and only if X \ { a } is a C-closed set. Then (AEP) is equivalent to the so-called Krein-Milman Theorem: C ( X )= C(ezt(X)). For any anti-matroid, there exists only one C-basis. 1.3.. Equational Logic. In this case, closure operators of rank 2 (or of rank n ) are mostly used (see [149];we use a slightly different terminology here in accordance with the book [IS]). A closure operator C is of rank 2 iff for all X 5 A we have: C ( X ) = C 2 ( X )u Cz(Cz(X))u . . . , where

CZ(X) =

(J

C({a,b}).

{a,b)C_X

More generally, an operator C is of rank n if C(X) = C,(X) u C,(C,(X)) u . . . , where C,(X) = U { C ( Y ) I Y G X , c a r d ( Y ) 5 n } . Let BC be the set of all C-bases (irredundant bases) of a closure space (A; C) and VBc = {K I K = card(B) for some B E Bc}. The following theorem has very important applications: Tarski Interpolation Theorem (see [150], [149]). Let (A;C) be a closure space with a closure operator of rank n 2 2 and k , l E N n VBc with k < 1; { k + l , ..., l - l } n V B c = @ . Then k - 1 I n - 1 . In the case of n = 2, VBc is a convex subset of R?, i.e. ifk,m~R?nVU~andk
-

116

partially proved by A. Tarski, R. McKenzie, J. Ng) the following cases are (only) possible:

VUA = {0}, [k,rn], [ k , w ] ,{ w } , or 0, where 0 < k 5 rn < w . All these cases can be realized for equational logics of groupoids (a result obtained by J. Ng). In 1938 A. Tarski observed that the class of all abelian groups is onebased. It is well known that one identity is enough t o define the class of all groups as special groupoids (where the considered binary operation is the left-hand division) and that every finitely based class of groups is one-based (see, e.g., [62], [102]). For more general cosiderations, see also [45]. Theorem. (Tarski & Green, see [149] and [59]). If A is a finitely based equational theory of groups or rings, then either VUA = [l,w] or VUA = [2,w]. For lattices, R. Padmanabhan has proved that if A is a finitely based equational theory of lattices, then: [2,w ] VUA. Moreover, 1 E VBA if and only if there exists a lattice which is not a model of A and there exists a lattice L with c a r d ( l ) 2 2, which is a model of A. A more general interpolaton theorem applicable in combinatorics was proved by S. Givant ([45]) and by G. McNulty and W. Taylor ([go]).

11. General algebras. Closure spaces (A;C) appear in a very natural way in the theory of (abstract) general algebras. Let U = (A;IF) be an algebra. Then we A by taking can consider a mapping of the power set 2A into itself defined for X the subalgebra generated by X , i.e. X I-+ ( X ) n . This closure operator ( ) a is of finite character, i.e. the closure space (A;( )a) is algebraic. It is worth mentioning that families ( )g-ind for arbitrary algebras 2 were characterized (independently) by R. McKenzie and K. Truol in the 1960s. We recall the following important result: Theorem (G. Birkhoff & 0. Frink [lo]; see also [130]) If ( A ; C )i s a n algebraic closure space, then there exists a n algebra U = (A,F) such that C ( X ) = ( X ) , f o r all X

5 A.

Therefore considerations of some kinds of independeces mentioned in previous sections we can be also interpreted from the general algebraic point of view. Moreover, we can also consider from this point of view some special kinds of independences, such as 0 0

0 0 0 0

independence of polynomials, and more generally: independence of continous functions (defined using a Jacobian); w-independence of function (by means of a Wronskian); set theoretical independence, and more generally: independence in Boolean aIgebras; linear independence in abelian groups, etc., which were not covered by the set-theoretical scheme mentioned before.

In 1958 E. Marczewski introduced a general notion of independence (called also algebraic independence or M-independence), which contained as special cases the majority of independence notions used in mathematics (by using suitable algebras). For this survey, we need to recall some definitions and notations.

117

For many algebraic investigations, the sets T(U)of all term operations of algebras U = (A; P) play very important roles. of all n-ary term operations ( n > 0) of an algebra U = (A;F) is The set T(n)(U) the smallest set of operations such that 1" : lE(n) T(n)(U),where lE(n) is the set of all n-ary trivial operations (or projections, in another terminology), i.e. for e!") E IE(n),we have (n)

ei ( ~ 1 ,. ..,xi,. . . ,x,) = xi, 2" : i f f E (an m-ary fundamental operation of a),91,. . . ,gm E then f^(gl,.. . ,gm) E Here the composition of junctions: f^(gl,. . . ,gm) is defined by ' . rgm(z1r.. ( h , . ' . ,gm))(zl,. ,zn) = f(gl(z1,. . (for all xi E A , i = 1 , . . . ,n). Moreover, "(')(a) is the set of all (algebraic) constant operations (i.e. unary term operations whose values do not depend on their variable). Of course, their values are distinguished elements of A . Then: ' '

'

,%I,

'

.13,))

n=l

n=O

T(U)forms a clone (in the sense of Ph. Hall) and a so-called Menger system,while T(n)(21)forms a Menger algebra of dimension n (see: [92], [74], [129]). Moreover, = (Ti'(")(%); I@) is of the same type as U and it is free generated by the algebra e?), . . . ,&) in the variety 'HSP({U}). 11.1. M-independence. E. Marczewski observed at the end of 1950s that there are common features of linear independence of vectors and set-theoretical independence, and proposed a general scheme of independence called here M-independence. Recall that the notion of set-theoretical independence (or, more generally, independence in Boolean algebras, see, e.g., [6], [16], [17], [37], [46], [61], [70], [83], [94]) was introduced at the mid-1930s by G. Fichtenholz & L. Kantorovich ([37]) and also, independently, E. Marczewski himself, and this notion is very important in Measure Theory (see, e x . , [371, [611, 1781, [801, [96], and [1351). Define now the notion of M-independence (see [81], [84]). Let U = (A; F) be an algebra 0 # X 5 A. The set X is said t o be M-independent ( X E Ind(U; M), for short) if (a) : (Vn E N,n 5 card(X)) (Vf, g E T(")(U))(Val,. . . ,a, E X )

-

* f = g (in A)].

#

[ f ( a .~.,. ,an)= g(a1,. . . ,an)

This condition is equivalent t o each of the following (b) : (Vn E N , n 5 card(X)) (Vf,g E T(n)(21))(Vp: X + A ) ( V a l , . . . , a, E

X) [ f ( a l , . . . , a n ) = g(a1,... =+ f(p(al),.",P(an)) = g ( P ( a l ) , . . . , p ( a n ) ) l ; (c) : (VP E AX) (3E Hom((X)a,Q)) [?SIX =PI, where (X),is a subalgebra of U generated by X ; (d) : (X), is a K-free algebra IK-freely generated by X , where IK = {U} (or, by Birkhoff Theorem, IK = 'HSP{U},a variety generated by 8).

118

Basic properties of M-independence are the following:

(i) : M-independence is stronger than independence with respect t o the closure operator X ++ ( X ) m (X C A ) , i.e. I n d (U, M) C ( )!x-ind (ii) : (“hereditarity”) X E I n d (U, M ) , Y X Y E I n d (a,M ) (iii) : ( V X C A ) { (V’finite Y C X ) [Y E I n d ( Q , M ) ] + X E Id(%, M ) } (i.e. the family J = I n d ( U , M) is of finite character). Theorem (Swierczkowski [146]). Let subsets of A. T h e n

0 # J C 2A be a hereditary

( 3 3 2 A ) (3% = ( B ;G)) ( V f i n i t e X & B ) [ X E I d ( % ,M)

system of finite

+ X E J]

The following problems are still open: Problem 1 (E. Marczewski, 1958; see also Problem 52 in [58]). For which hereditary families J of finite subsets of a given set; A does there exist an algebra M = ( A ,IF) such that J is the family of all finite M-independent sets of the algebra U? Problem 2 (K. Glazek [47], P 1094). Describe I n d ( 2 , M) for U E K,where JK is a fixed variety of algebras (e.g. of groups, rings, lattices, semigroups, etc.) 11. 2. M-bases. Let U = ( A , F ) be an algebra, and X C_ A. The set X is said to be an M-basis if X is an M-independent set of generators (i.e. X E I n d ( U , M) and ( X ) , = A ) . There are several interesting results and open problems concerning M-bases. Theorem (Swierczkowski [144]). Let U = (A,IF), c a r d ( A ) > 2, and let the set A be M-independent. T h e n =

f o r any n (i.e. U is a trivial algebra).

Theorem (Marczewski & Urbanik [88]). Let A be a set with c a r d ( A ) = 2. T h e n there are only (up to t e r m equivalence) three algebras with the M-independent set A (the carrier of that algebras): 1) !?3* = ( A ; p , ) where p * ( z ,y , z ) is symmetric and p * ( z , z, y ) = y , 2) P* = ( A ; p * ) p * ( z , z , y ) = z and p* is symmetric, = z and P(z,Y,x)= P ( Y , z , z ) = Y . 3) !?3 = ( A ; p ) P(z,z:,Y)

The following problems seem to be interesting: Problem 3 (K. Glazek [47], P 1099). Give a set-theoretical description of the family of all M-bases of any based general algebra. In 1979 we defined so-called generalized matrices by using homomorphisms of algebras of the same type having M-bases. More generally, we can use so-called weak homomorphisms of non-indexed algebras with bases. For one algebra with an M-basis and its emdomorphisms, some partial results for the following problem were obtained in several papers by G. Ricci (see, e.g., [126] - [128]). Problem 4 (K. Glazek [47], P 1098). Work out the theory of generalized matrices. It is worth adding that some papers concerning so-called independence algebras are closely related t o this problem (see, e.g., [38], [39], [52], [53], [77], and [log]). Let B = B(M, M) be the set of all M-bases of U and

DB = { x : x = c a r d ( B ) for some B E B} (of course, for matroids, we have c a r d ( V B ) = 1). We recall now an interesting results:

119

Theorem (Marczewski & Ryll-Nardzewski, see [84]). If c a r d ( V B ) > 1 for an algebra U, then all M-bases of 8 are finite and their cardinal numbers form an infinite arithmetical progresion, i.e. V B = {no k m I V m E Pi,and f o r some no, k E M}. Such a situation was known for modules over non-commutative rings (see, e.g., [=I, [761,11251). Theorem (Swierczkowski [145];Goetz & Ryll-Nardzewski [51]). A n y arithmetical progression is the set of numbers of elements of all bases of a certain algebra. Theorem (Goetz & Ryll-Nardzewski [51]: see also [145]). If m,n E V B ( U , M ) , j = 1 , . . . ,n, m # n, then there exist fi E T(n)(8), i = 1 , . . . ,m, and gj E such that the following equalities hold: (*) : f i ( g l ( z 1 , . . . ,xm),. . . , g , ( z l , . . . ,zm))= xi,for all i = I , . . . ,m, (**) : g j ( f l ( x 1 , . . . ,x,), . . . , f m ( x l , . . . ,zn)) = zj, for all j = 1 , . . . ,n.

+

Varieties (of algebras) defined by equalities (*) and (**) (denoted by V,,,, if m is smaller than n, and frequently called Cantor varieties) or defined only by one series of equalities (*) and (**) have been considered by several authors (see, e.g., [2], [138] - [141] and [151]). F o r m = 1 and n = 2, algebras 8 E V1,2were considered by B. J6nsson and A.Tarski in.1661. In 1964 E. Marczewski raised the following question: Define S(U) = { n E MI L I E~ ~ ( ~ ' essentialy (8) n-ary). Does 1 < c a r d ( V B ( 8 ,M ) ) imply S ( 8 ) = M? In 1978 J. Dudek showed that the conjecture is true for algebras with one- and n-element ( n > 1) bases (see [ZS]). A positive answer t o Marczewski's question was given by A. Kisielewicz only in 1988 (see [69]). But there are still several interesting problems concerning algebras with c a r d ( V B ) # 1. For example, the following problems are still open. Problem 5 (K. Glazek, 1978). Do groupoids (or semigroups) having bases of different cardinality exist? Let a(%)be the arity of an algebra B = (A;F), i.e. a(%)= m i n { n E Nl " ( 8 )= T(A; Problem 6 (K. Urbanik, 1966). What is the arity a(%)for 8 with c a r d ( V B ( 8 ,M ) ) > 1 ? Problem 7 (K.Glazek, 1995). Give a common generalization of the 1) : Equicardinality Theorem of bases in matroid theory, 2) : Tarski Interpolation Theorem for closure systems of rank n, 3) : general algebraic theorem about arithmetical progressions of cardinalities of M-bases.

11.3. Near-linear properties of M-independence. For each algebra 8,we have I n d ( B , M ) C ( )a-ind. An algebra 8 is called a w**-algebra if I n d ( 8 , M ) = ( )a-ind. Theorem (Narkiewicz [99]). If !2l is v**-algebra and V B ( B ,M ) # 8, then c a r d ( V B ( 8 ,M ) ) = 1. Important subclasses of v**-algebras are the classes of v*-algebras (see [97]) and v-algebras (defined by E. Marczewski in [82]). Another interesting type of algebras are quasilinear algebras (or seperable variable algebras). M-independence in these

120

algebras has certain properties similar to the properties of linear independence in vector spaces. There are some deep results and representation theorems for the above-mentioned classes of algebras (see, e.g., [35], [36], [56], [58], [82], [98], [99], [153].- [156]). We recall some of them. An algebra U is called a v*-algebra if (i) : a E A \ (0)m + { a } E I n d ( U , M ) and (ii) : { a l , . . . ,an} E I n d ( B , M ) . For v*-algebras we can define a dimension. (Notation: dim(%)). It is worth mentioning that v*- algebras are very useful for algebraic logic cosidered by J. Baldwin, G.L. Cherlin, E. Hrushovski, A. Lachlan, S.Shelah, B.I. Zil’ber and others (see, e.g., [164] - [167]). An algebra 2l is called a v-algebra (or a Marczewski algebra) if E M, 1 I j I n) P f , g E T(~)(u),f # 9 ) [if the equality

(*I:

f ( X l , . . . , X j , Z j + l , , Z n )= g h , . . . , X j , . . . , X n ) depends on x j , then there exists an operation h E T ( n - l ) ( U ) such that (*) is equivalent to the equality

(**): ~j = h ( Z 1 , . . . , ~ j~ ,j + l , .. . ,x~)]. An algebra is called a separable variables algebra (an SV-algebra, for short) if (Vn E T) (Vk E N , k < n) ( V f , g E T(n)(U)) ( 3 f o E T ( ” ( U ) ) (3go E T+k)(U)) [the equation f ( ~ 1 ,. . ,x,) = g ( x 1 , . . . ,x,) is equivalent to the equation f O ( Z 1 , x 2 , . . . , Z k ) = gO(ZrC+l,... 7Zn)l. Of course, abelian groups are SV-algebras. Representation Theorem for v*-Algebras of Dimension 2 3: ( K. Urbanik [154]): (i) : If To(%)# 8 and T3(U) # T(3,1)(U)(2.e. there exists a ternary operation which is dependent on more than one variable), then there is a field K such that A is a linear space over K and, further, there exists a linear subspace An of A such that n

T(n)(U)= { f

I

f ( x l , . . . ,x n ) =

XkXk

+ a , Xi

E K , a E Ao}

k=l

(ii) : If To(%)= 0 and T3(U) # T(3.1)(U),then there is a field K such that A is linear space over K and, further, there exists a linear subspace A0 of A such that

T(n)(U)= { f I f n

(21,.

..,xn)= n

(iii) : If T3(U) = T ( 3 > 1 ) ( Uthen ) , there are a group 6 of transformations of the set A and a subset A0 5 A containing all fixed points of transformations (from 6 ) that are not the identity, and invariant under all transformations from 6 ,such that

121

Il’(n)(U) = {fl f ( X l 1 . . . ,z,) = g(zj), 1 5 j I n, g E 4 o r f(z1,. . . ,z,) = a, a E Ao} Corollary. A n y w*-algebra of d i m ( % ) 2 3 with T3(U)# T(311)is a w-algebra. The representation theorem for w*-algebras of dimension < 3 was stated by G. Gratzer ([56]) and K. Urbanik ([155]). Representation Theorem for Separable Variable Algebras (S. Fajtlowicz, K.Glazek & K. Urbanik [36]): Every separable variable algebra is t e r m equivalent to a quasi-linear algebra. An algebra U = ( A ,P) is said to be quasi-linear if the following conditions hold: (i) : the set A is a subset of an abelian group 6 = (G: +), (ii) : for any operation f E T(”)(U)( n = 1 , 2 , . . .) there exist unary operations f l , fz,.. . , f, on A (not necessarilly term ones) such that: n

f(Zl,~Z,...tZ,)

= Cfj(Zj), j=1

(where the summation is the group-operation in 6), (iii) : there exists an injective unary term operation q such that the binary operation r ( z ,y) = q(z) - q ( y ) is a term operation. Remark. By (iii), T ( Z , z) = 0, i.e. the zero-element of G is always an algebraic constant in U.

11. 4. Independence with respect to a family Q of mappings ( Q independence, for short) There are some notions of independence which are not special cases of Mindependence, such as: 0 linear independence in abelian groups, 8 independence with respect to a closure operator C (i.e. C-independence), 0 stochastic independence, 0 “independence-in-itself” defined by J. Schmidt (in 1962), 8 “weak independence” used by S. Swierczkowski (in 1964). For this reason a general notion of independence with respect a family of mappings was proposed by E. Marczewski in 1966 (and studied in [87] and [46]). It is a general enough notion t o cover the above-mentioned kinds of independences. We will use the following notation: Let 0 # X C A. Then: Q x AX = Mx = { p I p : X + A } ,

Q ( A ) = Q = U { Q x I X C_ A } , M ( A ) = A4 = U { A x I X A } . For an algebra U = ( A ,IF), a mapping p : X + A belongs t o H x ( U ) iff there exists a homomorphism p : ( X ) , + A such that plx = p . X is said to be Q-independent ( X E I n d ( U , Q ) ,for short) if Q x C Hx(W or, equivalently, (Vp E Q x ) (V finite n 5 c a r d ( X ) ) (Vf,g E T(,)(U)) (Val,.. . ,a, E X ) [f(a1,.. . I a,) = g(a1, ‘ ‘ . , an) f ( p ( a 1 ) , ...,p(a,)> = g(p(a1),’ ‘ ’ ,P(an))l.

*

122

Examples. (In the following examples we will use a terminology which differs from the original one) 1) : Q = M = u { A x I X A } ; M-independence (E. Marczewski: general algebraic independence, [Sl]), 2) : Q = S = u { (X)$I X & A } ; S-independence (J. Schmidt: independencein-itself, [131]), 3) : Q = So = u { X x I X A } ; So-independence or local independence (S. Swierczkowski: weak independence, [148]), 4) : Q = G = u { p I p E A X is diminishing, X A } ; G-independence (G. Gratzer: weak independence, [57]), where a mapping p is called diminishing if P f , g E "(')(U)) (Va E A ) [ f ( a )= d a ) f ( p ( a ) )= g ( ~ ( a ) ) l . For abelian groups, the notion of G-independence gives us well-known linear independence. Moreover, we consider R-independence (introduced in [46]): 5) : Q = R = u { p I p E A X injective, X A} and 6) : C-independence. Theorem ( [ 4 6 ] )Let ( A ,C ) be a closure space, with C of finite character. Then -+

( A , P ) )( 3 Q C M ( A ) (= U { A x I X C A } ) ) such that the family C-ind i s the same as the family I n d ( U , Q ) . 11.5. Properties and some Galois correspondence.The following list of properties (cf. [46])suggest considering Galois connections defined in a natural way. The obtained Galois correspondence (see [48], [49])seems t o be interesting and fruitful for further investigations of different kinds of independences and enables the use of so-called formal concept analysis for our research.

*

(i) : Q 1 C Q Z Ind(U,Q l ) 2 Ind(U,Qz); (ii) : I n d ( U , M ) C I n d ( U , Q ) ; (iii) : Let U = ( A ;P) be a fixed general algebra and I n d ( U , M ) J C 2 A . T h e n there exists a family Q of mappings such that I n d ( U , Q ) = J; (iv) : Let U = ( A ;IF) be a fixed general algebra. T h e n there ezists a family Q of mappings such that: ( )a-ind = I n d ( U , Q ) ; (v) : (VQ & M ( A ) ) (3greatest Q 2 Q ) such that: I n d ( U ,Q ) = I n d ( U , Q); (vi) : D(U) = { Q I Q C M ( A ) } ( f o r a n algebra U) i s a complete atomic Boolean algebra; (vii): Let U be a n algebra and Q' = ( M ( A )\ Q ) U M(U). Then, if the family I n d ( U , Q ) is hereditary, then family I n d ( U , Q') is offinite character. Now, we define a relation Q & 2 A x M in the following way: V 8 # X E 2A, V p E M , [ X Q P ( p $ A X ) or ( p E H ) and 0 ~ for p every p E M I . (viii) : W e have dually adjoint operators forming a Galois connection: M 2Q j ( Q ) = { X C A I (VP E Q ) [ X Q P ] } , 2A 2 J = { p E M 1 (VX E J) [.Yep]}. (ix): T w o Galois closure operators defined by:

+-+

6(J)

123

v(B)= j(Q(JI)) and A(Q) = Q(j(Q)) are additive. Now we recall some results obtained for linear spaces, abelian groups and, finally, w**-algebras. (x): Let U = (A;{Az py} I A, p E K ) be a n algebra that is t e r m equivalent to a linear space over a field K . T h e n we have: I n d ( U , M ) U ( ( 0 ) ) = I n d ( U , S ) = I n d ( U ,S O ) €3 Ind(U, M ) = ( )a-ind = Id(%, I ) X E Ind(U, G) ej ( X \ ( 0 ) E I n d ( Q , M ) )

+

{

Therefore, the following problem seems t o be interesting. Problem 8 (K.Glazek & F. Pastijn, 1980). Let U fulfil IXI. Is U term equivalent t o a linear space? (xi) : Let G be a torsion-free abelian group. Then: I n d ( G , S ) = Ind(G, S O )= I n d ( G , M ) u ( ( 0 ) ) X E Ind(C7, G ) @ ( X \ ( 0 ) ) E I n d ( G , M ) . (xii) : I n the v**-algebras, except one-element sets consisting of a n algebraic constant, the following properties are equivalent: (a) M-independence, (b) S-independence, ( c ) So-independence, (d) ( )%-independence.

REFERENCES [ 11 M. Aigner, Kombinatorik. Vol. 11: Matroide und Rznsversaltheorie, Springer-Verlag, Berlin 1976. [2] A.A. Akataev and D.M. Smirnov, Lattices of subvarieties of algebra varieties (Russian), Algebra i Logika 7 (1968), 5-25. [3] W.W. Armstrong, Dependency structures of database relationships, p. 580-583 in: “Information Processing ’74”, North-Holland, Amsterdam 1974. [4] W.W. Armstrong and C. Delobel, Decompositions and functional dependence in relations, ACM Trans. Database Systems 5 (1980), 404- 430.

[i]A.

Bachem and W. Kern, Linear Programming Duality. A n Introduction t o Oriented Matroids, Springer-Verlag, Berlin 1992. [6] B. Balcar and F. FranBk, Independent families in complete Boolean algebras, Trans. Amer. Math. SOC.274 (1982), 607-618. [7] J.T. Baldwin, First order theories of abstract dependence relations, Ann. Pure Appl. Logic 26 (1984), 215-244. [8] J.T. Baldwin, h n d a m e n t a l s of Stability Theory, Springer-Verlag, Berlin 1988. [9] G. Birkhoff, Lattice Theory (third ed.),Amer. Math. SOC.,Providence, RI, 1967.

[lo] G. Birkhoff and 0. Rink,

Representations of lattices by sets, Trans. Amer. Math. SOC.64

(1948), 291-316.

[ll] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler, Oriented Matroids, Cambridge Univ. Press, Cambridge 1993. [12] A. Bjorner and G. Ziegler, Introductions t o greedoids, p. 284-357 in: “Matroid Applications”, Cambridge Univ. Press, Cambridge 1992, [13] 0. Borhka, 0 jistem problemu minimalnim (Czech), Prace Moravsk6 Prirodovd. SpoleEnosti 3 (1926), 37-53. [14] N. Both, Independence, Stud. Cerc. Mat. 26 (1974), 491-494. 1151 N. Both, A synopis on independence, Mathematica (Cluj), 27 (50) (1985), 11-12 [16] F.M. Brown and S.Rudeanu, Consequences, consistency, and independence in Boolean algebras, Notre Dame J. Formal Logic 22 (1981), 45-62. [17] F.M. Brown and S. Rudeanu, Prime implications of dependency functions, Ann. Univ. Bucurqti Mat. 37 (1988), 16-26. 1181 S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York 1981.

124 (191 J.O. Ceideira and P. Barcia, Ordered matroids and regular independence systems, Discrete Math. 154 (1996), 255-261. [20] P.M. Cohn, Universal Algebra (second ed.), D. Reidel Publ. Co., Dordrecht 1981. [21] G. C z a l i , On dependencies in the relational model of data, J. Inform. Process..Cybernet. (EIK) 17 (1981), 103-112. [22] J. Demitrovics and Gy. Gyepesi, On the junctional dependencies and some generalizations of it, Acta Cybernetica 5 (1981), 295-305. [23] J. Demitrovics, G. Hencesey, L. Libkin and I. Muchnik, On interaction between closure operation and choice functions with applications to relational databases, Acta Cybernet. 10 (1992), no. 3, 129-139. [24] J. Demitrovics, L. Libkin and I. Muchnik, Functional dependencies and the semilattice of closed classes, Lecture Notes in Comput. Sci. 364 (1989), 136-147. [25] J. Demitrovics, L. Libkin and I. Muchnik, Functional dependencies in relational databases: a lattice point of view, Discrete Appl. Math.. 40 (1992), 155-185. [26] B.L. Dietrich, Matroids and antimatroids - a survey, Discrete Math. 78 (1989), 223-237. (271 J. Dudek, Remarks on algebras having two bases of different cardinalities, Colloq. Math. 22 (1971), 197-200. [28] 3. Dudek, On universal algebras having bases of different cardinalities, Colloq. Math. 42 (1979), 111-114. 1291 I. Duntsch and G. Gediga, Algebrazc aspects of attribute dependencies in information systems, Fund. Inform. 29 (1997), 119-133. [30] P.H. Edelman and R. Jamison, Theory of convex geometries, Geom. Dedicata 19 (1985), 247-274. [31] J. Edmonds, Matroids and greedy algorithm, Math. Programming 1 (1971), 127-136. [32] C.J. Everett, Vector spaces over rings, Bull. Amer. Math. SOC.48 (1942), 312-316. [33] R. Fagin and M.Y. Vardi, The theory of data dependencis - a n overview, Lecture Notes in Comput. Sci. 172 (1984), 1-22. [34] U. Faigle, Matroids in combinatorial optimization, p. 161-210 in: “Combinatorial Geometries”, Cambridge Univ. Press, Cambridge 1987. [35] S. Fajtlowics and K. GIazek, Independence in separable variable algebras, Colloq. Math. 17 (1967), 221-224. [36] S. Fajtlowicz, K. Glazek and K. Urbanik, Separable variable algebras, Colloq. Math. 15 (1966), 161-171. [37] G. Fichtenholz et L. Kantorovitch, Sur les ope‘mtion dans l’espace des fonctions borne‘es, Studia Math. 5 (1934), 69-98. [38] J. Fountain and A. Lewin, Products of idempotent endomorphisms of a n independence algebra of finite rank, Proc. Edinburgh Math. SOC.(Ser. 2) 35 (1992), 493-500, [39] J . Fountain and A. Lewin, Products of idempotent endomorphisms of a n independence algebra of infinite rank, Math. Proc. Cambridge Philos. SOC.114 (1993), 303-319. [40] F. G k s e g and H. Jiirgensen, Algebras with dimension, Algebra Univ. 430 1(1993), 422-446. (411 F. G h e g and H. Jiirgensen, Dependence in algebras, Fund. Inform. 25 (1996), 247-256. [42] I.M. Gelfand and R.D. MacPherson, A combinatorial fonnula f o r the Pontrjagin classes, Bull. Amer. Math. SOC.26 (1992), 304309. [43] I.M. Gelfand and G.L. Rybnikov, Algebraic and topological invariants of oriented matroids, Soviet Math. Doklady 40 (1990), 148-152. (441 I.M. Gelfand and V.V. Serganova, On the general definition of a matroid and a geedoid, Soviet Math. Doklady 33 (1987), 6-10. [45] S. Givant, Possible cardinalities of irredundant bases for closure systems, Discrete Math. 12 (1975), 201-204. [46] K. GIazek, Independence with respect to a family of mappings in abstract algebras, Dissertationes Math. 81 (1971), 1-55. [47] K. GIazek, Some old and new problems i n the independence theory, Colloq. Math. 42 (1979), 127-189. [48] K. Glazek and S. Niwczyk, A new perspective on Q-independence, p. 61-69 in the book: “General Algebra and Applications” (K. Denecke and H.-J. Vogel, eds.), Shaker-Verlag, Aachen 2000.

125 [49] K. Glazek and S. Niwczyk, Q-independence and wenk automorphisms as Galois connections, in the book: “Galois Connections” (K. Denecke, M. Ern6 and S. Wismath, eds.), Kluwer, Dordrecht, to appear (2003). [50] K. Glazek and S. Niwczyk, Q-independences and congmences of a n algebra of term operations, t o appear (2003). [51] A. Goetz and Cz. Ryll-Nardzewski, On bases of abstract algebras, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astr. Phys.) 8 (1960), 157-161. [52] G.M.S. Gomes and J.M. Howie, Idempotent endomorphisms of a n independence algebra of finite rank, Proc. Edinburgh Math. SOC.38 (1995), 107-116. [53] V. Gould, Independence algebras, Algebra Univ. 33 (1995), 294-318. [54] E. Graczyriska, EIS for nilpotent shifts of varieties, p. 116120 in: “General Algebra and Applications”, Heldermann Verlag, Berlin 1993. 1551 E. Graczyliska and F. Pastijn, Marczewski independence in Plonka sums, Math. Japonica 27 (1982), 49-61. [56] G. Gratzer, A theorem on doubly transitive permutation groups with application to universal algebras, Fund. Math. 53 (1963), 25-41. [57] G. Gratzer, On a new notion of independence in universal algebras, Colloq. Math. 17 (1967), 225-234. 1581 G. Gratzer, Universal Algebra (second ed.), Springer-Verlag, New York 1979., [59] T.C. Green and A. Tarski, The minimum cardinality of equational bases f o r varieties of groups and rings, Notices Amer. Math. SOC.17 (1970), 429-430. [60] M.H. Halldorsson, J. Kratochvil and J.A. Telle, Independent sets with domination constraints, Lecture Notes in Comput. Sci. 1443 (1998), 176-187. [61] F. Hausdorff, Uber zwei Siitze von G. Fichtenholz und L. Kantorovitch, Studia Math. 6 (1936), 18-19. [62] G. Higman and B.H. Neumann, Groups a s groupoids with one law, Publ. Math (Debrecen) 2 (1952), 215-221. [63] M. Iri, Applications of matroid theory, p. 158-201 in: “Mathematical Programming - The State of the Art”, Springer-Verlag, Berlin 1983. [64] R.E. Jamison-Waldner, A perspective on abstract convexity: classifying alignments by varieties, p. 113-150 in: “Convexity and Related Combinatorial Geometry”, Marcel Dekker, New York 1982. [65] J. Jarvinen, A representation of dependence spaces and some basic algorithms, Fund. Inform. 29 (1997), 369-382. [66] B. J6nsson and A. Tarski, On two properties of free algebras, Math. Scand. 9 (1961), 95-101. [67] H. Jiirgensen and S. Konstantinidis, Codes, p. 1-96 in: “Handbook of Formal Language Theory”, Springer-Verlag, Berlin 1996. [68] A. Kisielewicz, Marczewski’s problem on algebras with bases of different cardinalities - solutions and generalizations, Bull. Polish. Acad. Sci. Math. 36 (1988), 11-13. [69] A. Kisielewicz, On algebras with bases of different cardinalities, Fund. Math. 133 (1989), 147-154. [70] S. Koppelberg, Free constmctions, in: “Handbook of Boolean Algebras”, vol. 1, NorthHolland, Amsterdam 1989. [71] B. Korte, L. LovLz and R. Schrader, Greedoids, Springer-Verlag, Berlin 1991. 1721 J.P.S. Kung, A Source Book in Matroid Theory, Birkhauser-Verlag, Base1 1986. [73] J.P.S. Kung, Matroids, p. 157-185 in: “Handbook of Algebra”, vol. I, North-Holland, Amsterdam 1996. [74] H. Lausch and W. Nobauer, Algebra of Polynomials, Amsterdam 1973. [75] E.L. Lawler, Combinatorial Optymization: Networks and Matroids, Holt, Reinhart and Winston, New York 1975. [76] W.G. Leawitt, Modules wzthout znvanant baszs number, Proc. Amer. Math. SOC. 8 (1957), 322-328. [77] L.M. Lima, Nilpotent local automorphisms of a n independence algebra, Proc. Roy. Soc. Edinburgh (Sect. A) 124 (1992), 423-436. [78] E. (Szpilrajn-) Marczewski, Ensemles indipendants et mesures non siparables, C.R. Acad. Sci. Paris 207 (1938), 247-248. [79] E. Marczewski, Indipendance d’ensembles et prolongement de mesures (Riultats et problhmes), Colloq. Math. 1 (1948), 122-132.

I26 /SO] E. Marczewski, Ensembles indipendants et leurs applications d la the‘orie de la mesure, Fund. Math. 35 (1948), 13-28. [Sl] E. Marczewski, A general scheme of the notions of independence in mathematics, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astr. Phys.) 6 (1958), 731-738. [82] E. Marczewski, Independence in some abstract algebras, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astr. Phys.) 7 (1959), 611-616. [83] E. Marczewski, Independence in algebras of sets and Boolean algebras, Fund. Math. 48 (1960), 135-145. [84] E. Marczewski, Independence and homomorphisms in abstract algebras, Fund. Math. 50 (1961), 45-61. [85] E. Marczewski, Nombre d’kliments indkpendants et nombre d ’ilkments gknkmteurs dans les algebres abstraites finies, Ann. Mat. Pura Appl. (4) 59 (1962), 1-10 & 349. [86] E. Marczewski, Independence in abstract algebras. Results and problems, Colloq. Math. 14 (1966), 169-188. [87] E. Marczewski, Independence with respect to a family of mappings, Colloq. Math. 22 (1969), 17-21. [88] E. Marczewski and K. Urbanik, Abstract algebras in which all elements are independent, Colloq. Math. 9 (1962), 199-207. [89] F. MatdS, Stochastic independence, algebraic independence and abstract connectedness, Theoret. Comput. Sci. 134 (1994), 455-471. [go] G. McNulty and W . Taylor, Combinatory intevolation theorem, Discrete Math. 12 (1975), 193-200. [91] K. Menger, Bemerkungen zu Gmnlagenfragen. IV: Axiomatik der endlichen Mengen und der elementargeometrischen Verkniipfungsbeziehungen,Jber. Deutsch. Math.-Verein 37 (1928), 309-325. [92] K. Menger, The algebra offunctions: Past, present, future, Rend. Mat. Appl. 20 (1961), 409-430. [93] G. Metakides and A. Nerode, Recursion theory on fields and abstract dependence, J. Algebra 65 (1980), 36-59. [94] J.D. Monk, Independence in Boolean algebras, Period. Math. Hungar. 14 (1983), 269-308. 195) E.H. Moore, Introduction to a Form of General Analysis, New Haven,CT, 1906. [96] J. Mycielski, Algebraic independence and measure, Fund. Math. 61 (1968), 165-169. [97] T, Nakasawa, Zur Aziomatik der linearen Abhangikeit, I, I1 and 111, Sci. Rep. Tokyo Bunrika Daiku 2 (1935), 235-255; 3 (1936), 45-69; and 123-136. [98] W. Narkiewicz, Independence in certain class of abstract algebras, Fund. Math. 50 (1962), 333-340. [99] W. Narkiewicz, On a certain class of abstract algebras, and Corrections..., Fund. Math. 54 (1964), 115-124 and 58 (1966), 111. [loo] A.W. Nayor, O n decomposition theory - generalized dependence, IEEE Trans. Systems Man Cybernet. 11 (1981), 699-713. [loll B.H. Neumann, Special Topics in Algebra: Universal Algebra, Courant Institute of Math. Sci., New York 1962. [lo21 B.H. Neumann, On a problem of G. Gritzer, Publ. Math. (Debrecen) 15 (1967), 325-329. [lo31 V. Novik and M. Novotny, Relational structures and dependence spaces, Czechoslovak Math. J. 47 (122)(1997), 179-191. 11041 J. Novotny and M. Novotnf, On dependence in Wille’s context, Fund. Inform. 19 (1993), 343-354. [lo51 M. Novotnf and Z. Pawlak, Independence of attributes, Bull. Polish Acad. Sci. Math. 36 (1988), 459-465. [lo61 M. Novotny and Z. Pawlak, Algebraic theory of independence in information systems, Preprint no. 51 (50 pp.), Mat.Ustav, Cechosl. Akademie V&l 1989. [lo71 M. Novotny and Z. Pawlak, Algebraic theory of independence in information systems, Fund. Inform. 14 (1991), 454-476. [lo81 M. Novotny and Z. Pawlak, On a problem concerning dependence spaces, Fund. Inform. 16 (1992), 275-287. [lo91 A. Oliveira, Order-independence algebras, Algebra Universalis 39 (1998), 171-196. [110] P. Orlik, Introduction to Arrangements, Amer. Math. SOC.,Providence, RI, 1989. , (1111 J.G. Oxley, Infinite matroids, Proc. London Math. SOC.(3) 37 (1978), 259-272.

127 [I121 J.G. Oxley, Infinite matroids, p. 73-90, in: “Matroid Applications”, Cambridge Univ. Press, Cambridge 1992. [I131 J.G. Oxley, Matroid Theory,Oxford Univ. Press, Oxford 1992. [I141 Z. Pawlak, Rough Sets. Theoretical Aspects of Reasoning about Data, Kluwer, Dordrecht 1991. [I151 Z. Pawlak C.M. Rauszer, Dependency of attributes in information systems, Bull. Polish Acad. Sci. Math. 33 (1985), 551-559. [116] H. Perfect, Independence spaces and combinatorial problems, Proc. London Math. SOC.19 (1969), 17-30. [117] M.J. Piff, Properties of finite character of independence spaces, Mathematika 18 (1971), 201-208. [118] I. Purdea and N. Both, Qpes of relational independence in universal algebras, Studia Cerc. Mat. 40 (1988), 249-258. [119] I. Purdea and N. Both, Independence and duality, Mathematica (Cluj) 33 (56)(1991), 9-12. [120] R. von Randow, Introduction to the Theory of Matroids, Springer-Verlag, Berlin 1975. [121] C.M. Rauszer, Algebraic properties offunctional dependencies, Bull. Polish Acad. Sci. Math. 33 (1985), 561-569. [122] C.M. Rauszer, An equivalence between theory of functional dependencies and a fragment of intuitionistic logic, Bull. Polish Acad. Sci. Math. 33 (1985), 571-579. [123] C.M. Rauszer, Dependencies in relational databases. Algebraic and logical approach, Fund. Inform. 19 (1993), 235-274. [124] A. Recski, Matroid Theory and its Applications in Electric Network Theory, SpringerVerlag, Berlin 1989. [125] L. R&lei, Algebra, Vol. I, Leipzig 1959. Errata, Pure Math. Appl. (Ser. B) 3 (1992), [126] G. Ricci, Universal eigenvalue equations, 231-288 & 5 (1994), 241-243. [127] G. Ricci, A Whitehead Generator, Quadeni de Dipatimento di Maematica, vol. 86,Universit6 di Parma 1993. [128] G. Ricci, New characterizations of universal matrices show that neural networks cannot be made algebraic, Contributions to General Algebra 10 (1998), 268-291. [129] B.M. Schein and V.S. Trohimenko, Algebras of multiplace functions, Semigroup Forum 17 (1979), 1-64. [130] J. Schmidt, iiber die Rolle der transfiniten Schlussweisen in einer allgemeinen Idealtheorie, Math. Nachr. 7 (1952), 165-182. [131] J. Schmidt, Einige algebraische Aquivalente zum Auswahlmiom, Fund. Math. 50 (1962), 485-496. [132] M.J. Schroeder, Dependence systems with the operator-image exchange property, Discrete Math. 133 (1994), 237-248. [133] M.J. Schroeder, Stmcture and cardinality of the set of dependence systems on a given set, J. Combin. Math. Combin. Comput. 15 (1994), 181-196. [134] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, North-Holland, Amsterdam 1978. [135] R. Sikorski, Boolean Algebras, Springer-Verlag, Berlin 1964. [136] I. Singer, Abstract Convex Analysis, J. Wiley & Sons, New York 1997. [137] A. Skowron and C. Rauszer, The discernibility matrices and functions in information systems, p. 331-362 in: “Intelligent Decision Support. Handbook of Applications and Advances in the Rough Set Theory”, Kluwer Acad. Publ., Dordrecht 1992. (1381 D.M. Smirnov, Cantor algebras with one generator. I and I1 (Russian), Algebra i Logika 10 (1971), 61-75 and 658-667. [139] D.M. Smirnov, Bases and automorphisms of free Cantor algebras of finite rank (Russian), Algebra i Logika 13 (1973), 35-62. [140] D.M. Smirnov, O n the representability of Cantor varieties, Algebra i Logika 34 (1995), 464-471. [141] D.M. Smirnov, On the dimensions of Cantor and Post varieties (Russian), Algebra i Logika 35 (1996), 204209. [142] V.P. Soltan, Introduction to the Abstract Theory of Converity (Russian), “Shtiintsa”, Kishinev 1984.

+

128 [143] S. Swierczkowski, O n independent elements in finitely generated algebras, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astr. Phys.) 6 (1958), 749-752. [144] S. Swierczkowski, Algebras which are independently generated by every n elements, Fund. Math. 49 (1960), 93-104. [145] S. Swierczkowski, On isomorphic free algebras, Fund. Math. 50 (1961), 35-44. [146] S. Swierczkowski, A suficient condition for independence, Colloq. Math. 9 (1962), 39-42. [147] S. Swierczkowski, On two numerical constants associated with finite algebras, Ann. Mat. Pura Appl. (4) 62 (1963), 241-246. [148] S. Swierczkowski, Topologies i n free algebras, Proc. London Math. SOC.(3) 14 (1964), 5 6 6 576. (1491 A. Tarski, Equational logic and equational theories of algebras, p. 275-288 in: “Contribution to Mathematical Logic”, North-Holland, Amsterdam 1968. [150] A. Tarski, An interpolation theorem for irredundant bases of closure systems, Discrete Math. 12 (1975), 185-192. [151] W.F. Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), 263-303. [152] W.F. Taylor, Equational logic, Houston J. Math. 5 (1979), 1-83. [153] K. Urbanik, A representation theorem for Marczewski’s algebras, Fund. Math. 48 (1960), 147-167. [154] K. Urbanik, A representation theorem for v*-algebras, Fund. Math. 52 (1963), 291-317. [155] K. Urbanik, A representation theorem for two-dimensional v*-algebras, Fund. Math. 57 (1965), 215-236. [156] K. Urbanik, Linear independence in abstract algebras, Colloq. Math. 14 (1966), 233-255. [157] K. Urbanik, On some numerical constants associated with abstract algebras, Fund. Math. 59 (1966), 263-288. [158] K. Urbanik, On some numerical constants associated with abstract algebras. 11, Fund. Math. 62 (1968), 191-210. [159] B.L. van der W a r d e n , Moderne Algebra (Zweiten Auflage), Springer-Verlag, Berlin 1937. [160] D.J.A. Welsh, Matriod Theory, Academic Press, London 1976. 11611 D.J.A. Welsh, Matroids and combinatorial optimisation, p. 323-416 in: “Matroid Theory and Its Applications”, Liguori, Naples 1982. [162] D.J.A. Welsh, Matroidsr fundamental concepts, p. 481-526 in: Handbook of Combinatorics, vol. I, Elsevier. Amsterdam 1995. [163] H. Whitney, On the abstract properties of linear dependence, Amer. J . Math. 57 (1935), 509-533. [164] B.I. Zil’ber, Srongly minimal counably categorical theories (Russian), Sibirsk. Mat. Zh. 21 (1980), 98-112. [165] B.I. Zil’ber, Hereditary transitive groups and quasi- Urbanik structures (Russian), Trudy Inst. Mat. (Novosibirsk) 8 (1988),58-77 (English transl. p.165-186 in: .“Model Theory and Applications”, Amer. Math. SOC.,Providence, RI, 1999. [166] B.I. Zil’ber, Quasi-llrbanik structures (Russian), p. 50-67 in: “Model Theoretic Algebra” (Russian), Kazakh. Gos. Univ., Alma Ata 1989. [167] B.I. Zil’ber, Uncountably Categorical Theories, Amer. Math. SOC.,Providence, RI, 1993..

INSTITUTE OF MATHEMATICS, UNIVERSITY OF ZIELONA GORA, UL. PODG6RNA 50, P L 65-246 ZIELONAGORA,POLAND E-mail address: [email protected]

Finite groups with c-normal and f-hypercentral subgroups Guo, Wenbin* Department of Mathematics, Xuzhou Normal University Xuzhou, 221009, P.R. China E-mail: [email protected]

K. P. Shumt Science Faculty, The Chinese University of Hong Kong Shatin, Hong Kong, P.R. China (SAR) Email:[email protected]

Dedicated to the memory of Professor B H Neumann

Abstract We use local formations to study the finite groups in which some of its subgroups are c-normal and f-hypercentral. The pnilpotency and solvability of finite groups are studied. Some new results are obtained and some known results of pnilpotent groups and solvable groups are generalized.

1 Introduction I n the theory of groups, we usually use t h e properties of subgroups t o determine the structure

of t h e finite groups. N. Ito, in 1951 has already obtained that if for all o d d prime p, every subgroup of a group G of order p lies in t h e center of G then G is a p-nilpotent group [12]. I n 1996, Y.Wang [17] introduced the concept of c-normal subgroups. He called a subgroup H of a group G c-normal if there exists a normal subgroup K of G such that G = K H and 'Research of the first author is supported by a NSF grant of China and a Croucher Fellowship of Hong Kong. 'Research of the second author is partially supported by a UGC(HK) grant #2060187 (2002/2003) AMS Mathematics Subject Classification (2000): 20D10; 20D15 Keywords: c-normal subgroups; f-hypercenter; pnilpotent groups; solvable groups; local formations

129

130

H nK HG = CoreG(H),where HG is the largest normal subgroup of G contained in H . Then, he used the c-normality of the maximal subgroup to determine the properties of some finite groups. In fact, the c-normality of a subgroup has become an important tool in studying finite groups, for example, X.Guo and Shum used the c-normality of maximal subgroups and Sylow subgroup t o study the properties of a group G [lo]. Also, L. Miao, X. Chen and W. Guo have recently used the c-normality of primary subgroups t o determine the metanilpotence and p-length of a group [14]. In this paper, we shall use the concept of formation t o investigate some properties of finite groups. Recall that a class 3 of groups is called a formation if 3 is closed under homomorphic images and every group G has the smallest normal subgroup whose respective quotient is in 3. We then call such a subgroup the 3-residual of G , denoted by G3. In otherwords, G3 = n { N 9 GIGIN E 3}, Now, let F ( 9 ) be the set of all formations and P the set of all prime numbers. Then, a mapping f from P to F ( 9 ) is called a formation function. According t o Doerk and Hawkes [4], for a formation function f ,we denote L F ( f ) the set of all groups G such that GICG(GiIGi-1) E f ( p ) for every chief factor GilGi-1 of G and for every p E x(Gi/Gi-l), where x(Gi/Gi-l) is the set of prime divisors whose elements divide the order of GiIGi-1. Then we call a formation 3 local if there exists a formation function f such that 3 = L F ( f ) . In this case, f is called a local formation function or a screen of 3. The concept of formation was firstly introduced by W. Gaschiitz [6] in 1963 and has now become a powerful tool in studying group theory. we camm a formation 3 saturated if G E 3 whenever G/$(G) E 3, where $(G) is the Frattini subgroup of G. It is well known that a formation 3 is saturated if and only if F is local (cf. [7, Theorem 3.1.111 or [9]). A formation is said to be subgroup-closed, in brevity, s-closed if every subgroup of G belongs to 3 whenever G E 3 . We call a normal subgroup N of a group G f-hypercentralinGifforeveryG-chieffactorNi/Ni-lo f N , we haveGfCG(Ni/Ni-l) E f(p), for the formation function f and all p E n(Ni/Ni-1). It can be easily seen that if A , B are f-hypercentral normal subgroups of G , then their product AB is still a f-hypercentral normal subgroup of G. Thus, a finite group G always has a maximal f-hypercentral normal subgroup, denoted it by Z&(G).We call Z&(G) the f-hypercenter of G. It is known that the formation N of all nilpotent groups has a local formation function f such that f(p) = 1 for all prime numbers p . For the local formation function f of N, the f-hypercenter Z&(G) is also called the hypercenter of G and is usually denoted by Z,(G). We shall consider the f-hypercentral normal subgroups and the c-normal subgroups of a group G. By using the theory of formations, we shall determine the nilpotency, solvability and other properties of the group G with some f-hypercentral normal subgroups and c-normal subgroups. Some results obtained by Ito [12], Lam, Shum and Guo [13], Guo and Shum [lo], Wang [17], Miao, Chen and Guo [14] and others will be generalized or improved.

131 For notations, terminologies and definitions not given in this paper, the reader is referred to the monograph of W.B. Guo 171.

2

Preliminaries

We first state some terminologies which are useful in the paper. (1) A group G is called a pnilpotent if G has a normal Hall p'-subgroup, where p' is the complement set of a prime p in the set of all prime numbers.

(2) A group G is called pdecomposable if G = Gp x Gg, that is, G is the direct product of a Sylow pgroup Gp and a Hall p'-subgroup G@.

(3) Let 3 be a class of groups. Then a group G is called a 3-group if G E 3 (4)For a class 3 of groups, a group G is called a minimal non-3-group if G itself is not a F-group but all its proper subgroups are 3-groups. (5) A group G is called a Schmidt group if G itself is not nilpotent but all its proper subgroups are nilpotent. (6) A normal factor H / K of a group G is called the Frattini factor if H / K C d ( G / K ) , where 4 ( G / K )is the Frattini subgroup of G I K .

We cite here the following known results which are crucial in proving our theorems.

Lemma 2.1([8], Lemma 5). Let 3 be a s-closed local formation with a local formation Z L ( H ) , where Z L ( H ) is the ffunction f . If H is a subgroup of G, then H n ZL(G) hypercenter of the subgroup H .

Lemma 2.2(Wang[17]). Let G be a group. Then the following statements hold: (a) If H is a c-normal in G and H (b) Let K in G f K .

G and K

C_

H

5 K 5 G, then H is c-normal in K

C G. Then H

is c-normal in G if and only if H I K is c-normal

Lemma 2.3(Carter-Fischer-Hawkers [l]). Let G be a group and N a non-identity normal

132 subgroup of G . If 1 = N o 2 N1 2 ... 2 Nm = N

(1)

L1 2 . . .

(2)

1 =Lo

L, = N

are two subnormal series of N such that all its factors are G-chief factors, then there exists a 1-1 correspondence between the factors of (1) and (2) such that the corresponding factors are isomorphic, and they are both either Frattini factors or non-Frattini factors.

Lemma 2.4(Xu [18],Theorem 11.5.5). Let G be a group and p the minimal prime divisor of. the order of G. If G has a cyclic Sylow subgroup, then G is pnilpotent.

The following is the famous Feit-Thompson result for solvable groups.

Lemma 2.5 [5]. If the order of G is not divisible by 2, then G is a solvable group.

We now prove the lemmas for p-nilpotent groups and minimal non-p-nilpotent groups

Lemma 2.6. Let 3 be the set of all pnilpotent groups. Then 3 is a s-closed local formation.

Proof. It is easy t o see that 3 is a s-closed class of groups. We only need t o show that 3 is a local formation, that is, we need t o construct a local formation function for 3. For this purpose, we let f be a formation function such that f ( p ) = 1 and f(q) = 9 for g # p, where 9 is the formation of all finite groups. We now prove that 3 = L F ( f ) . Suppose that G E 3.Then, by ([7], Theorem 1.8.2), the pnilpotent group G has a normal series 1 = GQC G1 C . . . C Gt = G such that GilGi-1 is either a p'-group or a subgroup of Z(G/Gi-l), for every i E {1,2,. . . ,t } . Then, if p E r(Gi/Gi-l), we have G/CG(G,/Gi-l) = 1 6 f(p); and if G,/Gi-l is a p'-group, we have, obviously, G/CG(Gi/Gi-1) E 9 = f ( q ) for q # p. This means that G E L F ( f ) , and hence 3 C L F ( f ) . For the converse containment, we assume that L F ( f ) 3. Then, we can choose a group G E L F ( f ) \ 3 with minimal order. Because 3 is a formation, G has an unique minimal normal subgroup N such that GIN is p-nilpotent, that is, N = G3. Then, by definition of p-nilpotent group, we know that GIN = H / N x G,N/N, the semidirect product of HIN and

133 G p N / N ,where HIN is the normal Hall p’-subgroup of GIN and Gp is a Sylow subgroup of G. Since G E L F ( f ) ,by the definition of L F ( f ) ,either N is a p’-group or G / C G ( N )E f(p) = 1, when p E 7r(N).If G / C G ( N )= 1, then N C Z ( G ) , the center of G. This shows that N is an elementary abelian pgroup. By the well-known Schur-Zassenhaus theorem (see[7], Theorem 1.7.9), H has a Hall p’-subgroup H y and any two Hall p’-subgroups of H are conjugate in H . It is also clear that Hd is a Hall p’-subgroup of G . By using the Cunihin theorem on p-solvable groups (see[7], Theorem 1.7.6) and the usual Frattini argument (see[7, Theorem = N G ( G ~ , ) G ~= , NNG(G,,), where Gp’ is a Hall p’-subgroup 1.6.6]),we have G = N G ( G ~ ) H of G. Hence, it follows that G E 3, a contradiction. For the case that N is a &group, since HIN is a normal Hall p‘-group of G I N , we can immediately see that H = Gp’ a G. This leads t o G E 3, again a contradiction. Hence, L F ( f ) 3.This completes the proof.

Lemma 2.7. Let G be a minimal non-pnilpotent group. Then G is a Schmidt group and G3 is a Sylow subgroup of G , where 3 is the formation of all p-nilpotent groups.

Proof. By Cunihin [2], we know that G is a Schmidt group. We only need t o prove that G3 is a Sylow subgroup of G . Because G is a Schmidt group, by the well known structure of Schmidt group (see[7], Theorem 3.4.11), we know that G has a normal Sylow subgroup, say G, a G. Then GIG, S Gp E 3, where G p is a Sylow psubgroup of G . Hence G3 C G,. If G, # G3, then there exists a normal subgroup H of G such that G3 C H G, and G,/H is a chief factor of G . On the other hand, since G is a Schmidt group, G,/+(G,) is also a chief factor of G(see[7], Theorem 3.4.11). Since G,/H is a Sylow q-subgroup of G / H , we have +(G).But GIH E 3, we see that + ( G / H ) .It follows from Lemma 2.3 that H G,/H G E 3 since 3 is a saturated formation. This contradiction shows that G3 = G,. Hence, G3 is a Sylow subgroup of G .

5

For p-decomposition groups, we have the following results.

Lemma 2.8. Let 3 be the set of all pdecomposable groups. Then 3 is a s-closed local formation.

Proof. It is known that 3 is a s-closed formation. By using similar proof of Lemma 2.6, we can prove that 3 = L F ( f ) ,for the formation function f satisfying f(p) = 1 and f(q) = ,g, for all prime q # p , where is the formation of all p’-groups. We omit the details.

sp,

Lemma 2.9. Let 3 be the formation of all p-decomposable groups and G a minimal ’ is a Sylow subgroup of G. non-3-group. Then G is a Srhmidt group and G

134 Prooj The proof is the same as Lemma 2.7, and hence the proof is omitted

3

Nilpotent groups and solvable groups

We first prove a theorem for finite groups with f-hypercentral subgroups. Theorem 3.1. Let 3 = L F ( f ) be a s-closed local formation such that every minimal non-3-group is solvable. Then a group G is a 3-group if and only if all its subgroups with prime order or 4 lie in the f-hypercenter Z&(G) of G. Proof. The necessity is obvious. We only need to prove the sufficiency. Assume that the theorem is false and let G be a counterexample of minimal order. Let M be a proper subgroup of G. Since every subgroup of M with prime order or order 4 is a subgroup of G of prime order or order 4, by the given conditions of the theorem, we know that these subgroups are contained in the f-hypercenter Z&(G) of G. By Lemma 2.1, we know that Z&(G) n M Z & ( M ) .This shows that every proper subgroup of G satisfies the condition of our theorem. Then, by the choice of G, every proper subgroup of G is an 3-group. Thus, G is a minimal non-3-group. Now, by the well known Semenchuk theorem (see[l6] or [7, Theorem 3.4.2]), we know that G has the following properties:

c

1) G3 is a pgroup for some prime p ; 2) if p exceed 4.

> 2, then the exponent of G3 is p ; if p

= 2, then the exponent of G’ does not

c

Thus, by using the given conditions of our theorem, we can easily deduce that G3 Z&(G). ~ E f ( p ) . Consequently, Hence, for every G-chief factor Gi fGi-1 of G3, we have G / C G ( G fGi-1) G is an 3-group, a contradiction. Thus the theorem is proved.

The following corollaries follow immediately from our Theorem 3.1 and Lemmas 2.6-2.9.

Corollary 3.2. A group G is a nilpotemt group if and only if every element of G of prime order or of order 4 lies in the hypercenter Zm(G) of G.

Corollary 3.3. A group G is apnilpotent group if and only if every element of G of prime order or of order 4 lies in the f-hypercenter Z&(G) of G, where f is the formation function such that f(p) = 1 and f(g) = $j for all g # p.

135 Corollary 3.4. A group G is a pdecomposable group if and only if every element of G of prime order or of order 4 lies in the f-hypercenter Z&(G) of G, where f is the formation function such that f(p) = 1 and f ( q ) = Sfl for all q # p . By using Lemma 2.6 and Lemma 2.7, we can prove analogously as the sufficiency part of Theorem 3.1, the following theorem of Lam-Shum-Guo [13]. We hence omit the proof.

Theorem 3.5 [13]. (1) If p is an odd prime and every subgroup of G of order p lies in the hypercenter Z,(G), then G is p-nilpotent. (2) If p = 2 and all elemens of G of order 2 or 4 lie in the hypercenter Z,(G) of G, then

G is 2-nilpotent. In fact, Theorem 3.5 includes the well-known Ito’s theorem as its particular case because the center of a finite group G is always contained in the hypercenter of G. We now consider the finite groups with c-normal subgroups and f-hypercentral subgroups.

Theorem 3.6. Let 3 = L F ( f ) be a s-closed local formation such that every minimal non-3-group is solvable. If every cyclic subgroup of G of order 4 is c-normal in G and every element of G of prime order is contained in Z&(G),then G is a 3-group.

Proof Assume that the theorem is false and let G be a counterexample of smallest possible order. Since every cyclic subgroup of order 4 of each proper subgroup of G is c-normal in G, by Lemma 2.2, we know that every cyclic subgroup of order 4 of each proper subgroup of G is c-normal in this subgroup. On the other hand, since every element of prime order of each proper subgroup of G is an element of G of prime order, by our given conditions and Lemma 2.1, we can easily verify that every element of prime order in each proper subgroup of G is contained in the f-hypercenter of this subgroup. Therefore, every proper subgroup of G also satisfies the conditions of our theorem. Hence, every proper subgroup is now an 3-group by the choice of G. Then, by [7, Theorem 3.4.21, we know that G has the following properties: 1) G3 is a pgroup for some prime p ; 2) G3/4(G3) is a chief factor of G and the exponent of +(G3) is p;

3) If G3 is an abelian group, then it is an elementary abelian group; 4) If p > 2, then the exponent of G3 is p; if p = 2, then the exponent of G3 does not exceed 4.

136 Now assume that G3\4(G3) has an element x of order 4. Then, by the c-normality condition of < x >, there exists a normal subgroup K of G such that G =< x > K and <x > nK 5 <x > G . We now prove that < x >a G. In fact, if < x > n K =< x >, then < x >=< x > G a G. If < x > n K < < x >, then IG/KI = I < x > / < x > nKI > 1, and consequently, G # K . If K = 1, then clearly G =< x > is a 3-group, which is a contradition. Hence 1 < K < G. Now, because x E G3, we have

G =< x > K = G3K. Since every proper subgroup of G is a n 3-group, G / K Z G3/G3 n K E 3. It follows that G3 K , and hence G = G3K = K , again a contradiction. Therefore, < x > n K < < x > is impossible. Thus < x >a G. Since G3/q5(G3) is a chief factor of G and < x > q5(G3)/$(G3) a G/4(G3),we have G3 =< x > 4(G3) =< x > . Then, by the property 3) above, GT is an elementary abelian group. This contradiction shows that G3 has no element of order 4. Therefore the exponent of G3 must be a prime number. Now, by the conditions of our theorem, we have G3 C ZL(G) and thereby, G E 3, which is a contradition. The proof is completed.

The following corollaries can be easily deduced from our Theorem 3.6 and Lemmas 2.6-2.9.

Corollary 3.7. If any cyclic subgroup of G of order 4 is c-normal in G and if any minimal subgroup of G is contained in the hypercenter Zm(G)of G, then G is a nilpotent group. Corollary 3.8. If any cyclic subgroup of G of order 4 is c-normal in G and if any minimal subgroup of G is contained in the f-hypercenter ZL(G)ofG,where f is the formation function such that f(p) = 1 and f ( q ) = 9, the class of all groups, for all prime q # p , then G is a pnilpotent group. Corollary 3.9. If any cyclic subgroup of G of order 4 is c-normal in G and if every element of G of prime order is in the f-hypercenter Z L ( G )of G, where f is the formation function such that f(p) = 1 and f ( q ) = $&I, the classof all p'-groups, for q # p, then G is a pdecomposable group.

It is well known that the class 'U of all finite supersolvable groups is a s-close local formation and the formation has the formation function f such that f(p) = A ( p - l), the formation of all abelian groups with exponent dividing p - 1, and every minimal non-3-group is solvable

137 (cf.[3] and [7, Theorem 4.3.161). Therefore, by our Theorem 3.6, we can immediately deduce the following corollary.

Corollary 3.10. If any cyclic subgroup of G of order 4 is c-normal in G and if every element of G of prime order lies in the f-hypercenter of G , where f is the formation function such that f(p) = J l ( p - l), then G is a supersoluble group. The following theorem is t o consider the class 3 of all solvable groups. In this case, we can weaken the conditions of Theorem 3.5 in [lo]. In fact, we have the following theorem. Theorem 3.11. Let G be a finite group and P a Sylow 2-subgroup of G. Suppose that there exists a maximal subgroup Pi of P such that PI is c-normal in G. Then G is solvable.

Proof. Assume that IGI = 2u1p7 . .. p F , where p 2 , . . . ,ps are odd prime and cq,cq,. . . ,a, are natural numbers. If a1 = 1, then by the well known Feit-Thompson theorem (Lemma 2.5), we know G is solvable. Suppose that cq > 1. Since PI is c-normal in G, there exists a normal subgroup K of G such that G = P1K and P n K 5 ( P ~ ) G .

If (P1)G= 1, then Pi n K = 1 and IK(1= 2 p y . . . p F . By Lemma 2.4 and Lemma 2.5 wc see that K has a normal 2-complement K1 and K1 is solvable. It is clear that K1 is a normal subgroup of G. Note that K1 and G/K1 are both solvable. Hence G is solvable. Suppose that (P1)G # 1. We consider the factor group G / ( P l ) c .By Lemma 2.2, we see that P l / ( q ) Gis c-normal in G / ( P l ) c .Obviously, Pl/(Pi)Gis a maximal subgroup of P / ( P ~ ) G . Therefore, the factor group G / ( P l ) csatisfies the condition of the theorem. By induction, we can deduce that G / ( P l ) c is solvable. Hence, it follows that G is solvable. The proof is completed.

Corollary 3.12. Let G be a group and P a Sylow p-subgroup of G, where p is a prime

divisor of IG( and (IGl,p - 1) = 1. Suppose that there exists a maximal subgroup Pi of P such that PI is c-normal in G, then G is solvable. Proof. Suppose that p is an odd prime. Since (IGl,p - 1) = 1, G is a group of odd order. Then, by the well known Feit-Thompson theorem (Lemma 2.5), we know that G is solvable. I f p = 2, then the conclusion follows from Theorem 3.11. The following corollary is immediate from Corollary 3.12

138 Corollary 3.13. Let G be a finite group and P a Sylow psubgroup of G, where p is a minimal prime divisor of the order IGI of G. If there exists a maximal subgroup PI of P such that PI is c-normal in G, then G is solvable.

References [l] Carter R. W., Fischer B. and Hawkes T. O., Extreme classes of finite soluble groups, J.

Algebra, 9(1968), 285-313. [2] Cunihin S. A,, Subgroups of Finite Groups, Nauka i Tehanika, Minsk, 1964. [3] Doerk K., Minimal nicht uberauflosbare endliche Gruppen, Math. Z., 91(1966), 198-205. [4] Doerk K. and Hawkes, T. O., Finite Soluble Groups, Walter de Gruyter, Berlin-New York, 1992. [5] Feit W. and Thompson, J. G., Solvability of groups of odd order, Pacific J. Math. 13(1963),775-1029. [6] Gaschiitz W., Zur Theorie der endlichen auflosbaren Gruppen, Math. Z., 80(1963), 300305. [7] Guo Wenbin, The Theory of Classes of Groups, Science Press/ Kluwer Acadimic Publishers, Beijing-New York-Dordrecht-Boston-London, 2000.

[8] Guo Wenbin, The influence of minimal subgroups on the struture of finite groups, Southeast Asian Bull. Math., 22(1998), 287-290. [9] Guo Wenbin and K. P. Shum, On totally local formations of groups, Comm. Algebra, 30(2002) 2117-2131.

[lo] Guo Xiuyun and K. P. Shum, On c-normal subgroups of finite groups, Publ. Math. Debrecen, 58(2001), 85-92. [ll] Huppert B., Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967

[12] Ito N., Note on (LM)-group of finite order, Kodai Math. Seminar Report, 1951, 1-6 [13] Lam C.M., Shum K.P. and Guo Wenbin, A generalized Ito theorem onpnilpotent groups, Pure Math. Appl. 11(2000), 593-596. [14] Miao L., Chen X. and Guo Wenbin, Finite groups with c-normal subgroups, to appear in Southeast Asian Bull. Math. 25(2)(2001).

139 [15] Robinson D. J. S., A Course in the Theory of Groups, Springer-Verlag, New YorkHeidelberg-Berlin, 1982. [16] Semenchuk V. N., Minimal non-3-groups, Algebra i Logika, 18(3)(1979),349-382. [17] Wang Yanming, c-normality of groups and its properties, J. Algebra, 180(1996), 954-965.

[18] Xu Mingyao, A Introduction of Finite Groups, Science Press, Beijing, 1999.

Complemented minimal subgroups and the structure of finite groups Guo, Xiuyun" Department of Mathematics, Shanxi University, Taiyuan Shanxi 030006, People's Republic of China E-mail: gxyQmail.sxu,edu.cn

Dedicated to the memory of Professor B. 8. Neumann

Abstract

Complemented subgroups of a finite group have been recently studied because the properties of such subgroups provide useful information to the structure of finite groups. In this paper we give a brief survey on this topic under the assumption that some minimal subgroups are complemented in a special subgroup of a given group. Keywords: Complemented subgroups, Minimal subgroups, Saturated

formations, p-nilpotency, Supersolvable groups.

A subgroup H of G is said t o be complemented in G if there exists a subgroup K of G such that G = H K and H n K = 1. We also call the above subgroup K of G a complement of H in G. It is quit,e clear that t h e Let G be a finite group.

existence of complements for some families of subgroups of a finite group give a lot

of information about its structure, for instance, P. Hall in 1937 proved that a finite ~

~~

'The research of t,he author is supported by a research grant of Shanxi Province. PR China Mathematics Subject Classification (2000) 20D10, 20D20

140

141

group G is solvable if and only if every Sylow subgroup of G is complemented [I]. A new criteria for the solvability of finite groups were later obtained by Z. Arad and

M.B. Ward in 1982. They proved that a finite group is solvable if and only if every Sylow 2-subgroup and every Sylow 3-subgroup are complemented [2].In particular,

P. Hall [3] in 1937 proved that a finite group G is supersolvable with elementary abelian Sylow subgroups if and only if every subgroup of G is complemented in G. In a recent paper, A. Ballester-Bolinches and Xiuyun Guo[4] investigated the class of finite groups for which every minimal subgroup is complemented. In fact, they proved the following interesting result.

Theorem A. Every subgroup of a finite group G is complemented if and only if every minimal subgroup of G is Complemented. Therefore, a finite group G is supersolvable with elementary abelian Sylow subgroups if and only if every subgroup of G is complemented in G.

Theorem A illustrates that the existence of complements of all minimal subgroups of a finite group has strong influence on its structure. So we are lead to continue the investigation on the influence of the existence of complements of minimal subgroups on the structure of finite groups. For one hand, we drop the assumption that every minimal subgroup is Complemented. Then, we aim t o use a fewer complemented minimal subgroups t o determine the structure of the finite group. On the other hand, we drop the assumption that the minimal subgroups must be complemented in the whole group G, but instead, we only assume that the minimal subgroups are complemented in a subgroup of G. By using this idea, we first investigate the pnilpotency of a finite group under the assumption that every minimal subgroup of

P n @(G) is complemented in N G ( P ) ,where p is a prime dividing the order of the group G and P is a Sylow psubgroup of G.

Theorem ll5I.Let G be a group and p a prime number dividing the order of G. If every minimal subgroup of P n Op(G) is complemented in NG(P) and NG(P) is pnilpotent, then G is pnilpotent, where P is a Sylow psubgroup of G. In order to prove Theorem 1, we need to analyze the minimal counterexample.

142

We notice that if a finite group G is a minimal counterexample, then it is not difficult t o prove that the following fact (*) is true.

(*) P n OP(G) 5 Z(N,(P)), where Z ( N G ( P ) is ) the center of N G ( P ) Since G is not pnilpotent, G has a subgroup H such that pnilpotent group (that is,

H is a minimal nonH is not pnilpotent but every proper subgroup of H

is pnilpotent). By the structure of minimal non-nilpotent groups, we know that

H = Hp K Hq for a Sylow q-subgroup Hq and a Sylow psubgroup Hp in H ( q # p ) . Moreover, we may have H p 5

P n Op(G). Now, by the Frattini argument, we have

Furthermore, by the fact (*), we have that, N,(Hp)=

C,(Hp)and therefore H

=

H p x H,,which is a contradiction. This illustrates that the Theorem 1 is true. Remark 2. The assumption that N G ( P )is pnilpotent in Theorem 1 can not be removed. For instance, if we let G = A5, the alternating group of degree 5, then it is

N,(P) is a subgroup of G with order 10 for every Sylow 5-subgroup P of G. Hence every minimal subgroup of order 5 in P has a complement in N,(P) for Sylow 5-subgroup P of G. However, G = A5 is simple. easy t o see that

But if we assume that p is the smallest prime number dividing the order of G, then, the assumption that

NG(P)is pnilpotent in Theorem 1 can be removed. In NG(P)= G for

fact, if p is the smallest prime number dividing the order of G and a Sylow psubgroup

P of G, then, by using induction, we may have that the Hall

p'-subgroup of G is a q-group for some prime q and Op(G) is not a q-group. By induction and Burnside Theorem for pnilpotent, we can see that Op(G) is pnilpotent and therefore G is pnilpotent. So the remain case is

N,(P) < G. In this case, by

induction, we can have that, N,(P) is pnilpotent. Now by Theorem 1, we can have the following result.

Theorem 3L51. Let G be a group and p the smallest prime number dividing the order of G. If every minimal subgroup of

P n OP(G) is complemented in N G ( P ) ,

143

then G is pnilpotent, where

P is a Sylow psubgroup of G

By using Theorem 1 and Theorem 3, we obtain the following corollaries.

Corollary 4. Let N be a normal subgroup of a group G and p a prime number dividing the order of N . Also let

Np of the

F be

a saturated formation containing the class

all pnilpotent groups and GIN E 3. If N,(P) is pnilpotent and every

minimal subgroup of

P n OP(G)is complemented in Nc(P), then G E F , where P

is a Sylow psubgroup of N .

Corollary 5. Let, N be a normal subgroup of a group G and p be the smallest prime number dividing the order of G. Also let taining the class

Np of

F be

a saturated formation con-

the all pnilpot,ent groups and GIN €

subgroup of P n O q G ) is complemented in N G ( P ) ,then G E

F.If

every minimal

F , where P is a Sylow

psubgroup of N . Of course, there are some further applications of Theorem 1 and 3.

Theorem 6r51. Let

F be

a formation containing U , t,he class of supersolvable

groups. Let H be a normal subgroup of a group G such that G / H E Sylow subgroup

P

of H , every minimal subgroup of P

N G ( P ) ,then G is in

F,where GN is the nilpotent

n GN

F.If for every

is complemented in

residual of G.

Theorem 6 is true for any formation containing the class of supersolvable groups. In other words, we do not need to assume that ever, if the formation

F does not contain U

F is a saturated

formation. How-

(the class of supersolvable groups), then

Theorem 6 may not be true. For example, if we let, 3 be the saturat,ed formation of all nilpotent, groups, then the symmetric group of degree three is a counterexample. On the other hand, if we take the formation

F and the normal subgroup in The-

orem 6 special, we may get some interesting results. For example, if we let .F = U and let the normal subgroup be the nilpotent, residual GN of a finite group G , then, we have the following result,.

144

Corollary 7. If G is a finite group and, for every Sylow subgroup P of the nilpot,ent residual GN of G , every minimal subgroup of P is complemented in N,(P), then G is a supersolvable group. Furthermore, if G is assumed t,o be a solvable group, then t,he number of complemented minimal subgroups in Theorem 6 can be further reduced. In fact, we have the following theorem.

Theorem 8r5]. Let, F be a formation containing U , the class of supersolvable groups. Let H be a normal subgroup of a solvable group G such that GIH E F . If every minimal subgroup of the Fitting subgroup F(GN n H ) of GN n H has a complenient in G , t,hen G belongs to

F,where GN is the nilpotent, residual of G.

Since F(GN n H ) = GN n F ( H ) = (G”

n PI)x

(GN n Pz) x ... x (GN n Pk), we

know that, every minimal subgroup of F(GN n H ) in Theorem 8 is still a minimal subgroup of some subgroup GN n Pi, where P, is t,he Sylow pi-subgroup of F ( H ) for some prime p i . We also analyze the structure of finite groups in which every minimal subgroup of some maximal subgroups is complemented. For the sake of convenience, we let

BNS be the family of finite groups in which each minimal subgroup is complemented. In order to determine the structure of minimal non-BNS-groups, we first state the following result.

Theorem 9r6]. Let G be a group. If every maximal subgroup of G is in BNS and Ix(G)I 2 4, then G is in BNS. We now prove step-by-step the structure of minimal non-BNS-groups.

Theorem

lor6].Let G be a group.

If every maximal subgroup of G belongs to

BNS but G does not, then one of the following statements is true. (I) G = (alaP2 = l), where p is a prime.

145

(11) G = (a, 6 ) with

up =

bp = cp = 1,[a,b] = c, [a,c] = [b,c] = 1, where

P is an

odd prime. (111) G = P

K

Q is a minimal non-abelian group with order pqa and p 1((q - I), > 1 is a natural

where P E SylPG, Q E Syl,G, p and q are distinct primes and o number.

(IV) G is a minimal non-supersolvable group with order pqrp, pql(r

-

1) and

p l ( q - l), where G equals (c1, c2, ..., cp, a , b) satisfying the following relations cl'

=

czr

=

... = cpT = 1,cicj

=

cjci(i,j

=

1 , 2 , ...,p ) , aP

= bq =

1,b"

order of u(mod q ) is p , cia = ci+l, (i = 1 , 2 , ...,p

-

1,...,p; the order of v(mod

are distinct primes.

T)

is q , and p , q and

T

l ) , c p a = cl,cib = ci'

=

b"; the

,PP-i+l

,2 =

Conversely, every group G of type (I), (11), (111) and (IV) does not belong to

BNS but every maximal subgroup of G belongs to B N S . The following results are related to t,he BNS-groups

Theorem 11. Let G be a group. If for every Sylow subgroup P of G, N,(P) is a BNS-group, then G is a BNS-group.

Corollary 12. Let G be a group. If for every non-normal Sylow subgroup P of G, N,(P) is a BNS-group, then GUNS is nilpotent, where GUNSis the BNSresidual of G. Finally, we give a criterion for a finite group to be solvable. We show that a finite group G is solvable if there exists a maximal subgroup M of G such that every minimal subgroup of M is complemented in G. As a consequence, some conditions for a finite group G to be a supersolvable group are also given.

Theorem 13. Let G be a group and let M be a maximal subgroup of G. If every minimal subgroup of M has a complement in G, then G is solvable. Remark 14. Let M be a maximal subgroup of a group G. If every minimal subgroup of M has a complement in G, then every minimal subgroup of M has a complement in M and therefore, M is supersolvable and every Sylow sub-

146

group of M is elementary abelian. The following question now naturally arises. If

G has a supersolvable maximal subgroup M with elementary abelian Sylow subgroups, is G solvable? The answer to this question is negative, for instance, if we let G = PSL(2,7),then G contains a maximal subgroup M of order 21. It is clear that M is supersolvable and every Sylow subgroup of A4 is cyclic of prime order, however G is simple. This example illustrates that G is not solvable in this case. By using Theorem 13, we can give some conditions for a finite group to be a supersolvable group.

Corollary 15. Let M be a maximal subgroup of a group G with prime index. If every minimal subgroup of A 4 has a complement in G, then G is supersolvable.

Theorem 16. Let A 4 1 and A 4 2 be two maximal subgroups of a group G. If Ml and M2 are not, conjugates in G and every minimal subgroup of

M i(i = 1,2) has a

complement in G, then G is supersolvable.

[G

Corollary 17. Let, M I and M2 be maximal subgroups of a group G. If : MI] # [G : M2] and every minimal subgroup of Mi (i = 1 , 2 ) has a com-

plement in G, then G is supersolvable. We may also give a condition for a finite group to have a normal 7r-complement.

Theorem 18. Let G be a finite group and that every prime in

7r

7r

a nonempty set of primes such

n T ( G )is less than every prime in 7r’ n n(G). If H is a Hall

7r-subgroup of G and every minimal subgroup of H has a complement in G, then G has a normal 7r-complement and G is solvable. In closing this paper, we propose the following open question.

Question. Can we classify the finite groups in which every minimal psubgroup is complemented in N,(P) for a Sylow psubgroup P?

147

Acknowledgement The author would like to thank Professor

K.P.Shum for his valuable suggestions

and comments contributed t o this article.

References

[l]P. Hall, A characteristic property of solvable groups, J . L o n d o n Math. Soc., 12(1937), 198-200. [2] Z. Arad and M. B. Ward, New criteria for the solvability of finite groups, J . Algebra, 77(1982), 234-246. [3] P. Hall, Complemented groups, J . L o n d o n Math. Soc., 12(1937), 201-204. [4]A. Ballester-Bolinches and Xiuyun Guo, On complemented subgroups of finite groups, Arch. Math., 72(1999), 161-166.

[5] Xiuyun Guo and K. P. Shum, The influence of minimal subgroups of focal subgroups on the structure of finite groups. J. Pure Appl. Algebra, 169:l (2002),

43-50. [6] Xiuyun Guo and K. P. Shum, On complemented minimal subgroups in finite groups, t o appear in J . Group Theory.

FINITE TRANSLATION PLANES FROM THE COLLINEATION GROUPS POINT OF VIEW

CHAT YIN HO

DEPARTMENT O F MATHEMATICS, UNIVERSITY O F FLORIDA, 358 LITTLE HALL,P O Box 118105, GAINESVILLE, F L 32611-8105. [email protected]

IN MEMORY OF PROF. B.H. NEUMAN 1. Description of the problem. In the last century, Von Staudt might be the one who made the first reference to finite geometries. Projective spaces of dimension larger than three are Desarguesian (Fundamental Theorem of Projective Geometry). This distinguishes the role of projective planes. In the last four decades the emphasis in this area has been on finite planes. Planes in this article are of finite cardinalities. Collineation groups of projective planes, via difference sets of Singer groups, are related to cyclotomy in the Number Theory. This can be traced back to section VII of Gauss’s book: Disquisitiones Arithmeticae. In 1939 R. Baer initiated the study of planes by means of collineation groups. The last forty years saw great achievements in the Theory of Finite Groups, from the celebrated result of the Solvability of Groups of Odd Order of Feit-Thompson [I] to powerful techniques developed by Aschbacher and others for the Classification of Finite Simple Groups [4]. This tremendous work impacts greatly on the finite geometries. Among projective planes, translation planes have rich algebraic structures. The following is an algebraic description due to Andre (1954) A finite translation plane is a vector space of dimension 2d over a field of q elements, equipped with a spread. A spread is a set of qd 1, d-dimensional subspaces such that each non zero vector lies in exactly one of these subspaces. Each one of these subspaces is called a fiber. (We use the term fiber instead of component because of the term component has special meaning in the finite group theory.) The ( affine )points of the translation plane are the vectors of the vector space and the lines incident with the zero vector (the origin point) are the fibers. The other lines are the translates (i.e., cosets) of the fibers. The characteristic of the field is called the characteristic of the plane. The order of the plane is qd and the number d is called the Ostrom dimension. The case in which d = 1 merits some remarks. A translation plane with d = 1 is called a Desarguesian plane. It is a two dimensional vector space over a field F and the spread is the set of all one dimensional subspaces. This plane can be coordinated

+

Partially supported by NSA grant MDA904-02-1-0074

148

149

as {(x,y)lx,y E F } , and a fiber is either of the form {(x,y)ly = mx with m E F ) or ((0, y)ly 6 F } . This geometrical structure resembles the familiar Desarguesian affine plane over the reals. A similar coordinization for a translation plane in general can be arranged. In any geometrical structure the relationship between the geometry and the group of automorphisms is a basic topic to study. The group of automorphisms of a translation plane is called a collineation group. A natural question to ask is which group can be a collineation group. One of the main problem in this area is the following.

Main Problem. W h i c h n o n abelian finite simple groups can be collineation groups for a translation plane of odd order. iFrom now on we use the term a simple group to mean a non abelian simple group.

2. Collineation groups. The collineation group of a translation plane is a semi-direct product of the translation group and the translation complement. The translation group is a normal elementary abelian subgroup of order q2d. The translation complement is a group of semi-linear transformations. This shows that in order to understand a collineation group of a translation plane, one has to study the translation complement. We use the following lemma to show that the order of the translation complement is a divisor of n(n+l)(n-1)2 f , where f is the 1.c.m. of the orders of affine planar groups of the translation of order n. In fact we will prove the result in a more general setting for affine planes in 2.2 below.

2.1. Lemma. Let (H,O) be a finite group space and let f = l.c.m.{JH,I : x E O}. T h e n IHI divides 101f . Proof. This is [6,2.4].

We now apply this to affine planes. 2.2. Theorem. Let H be a collineation group of a n a f i n e plane of order n that fixes a n a f i n e point 0 . T h e n /HI divides n(n l ) ( n - 1)2f H , where f H = 1.c.m. of the orders of a f i n e planar subgroups of H .

+

Proof. Let O be the set of ordered quadrangles (0,A, B , C ) , where A , B are two different points at the line of infinity and C is an affine point not incident with the lines OA, OB. There are n 1 choices of A. The number of choices of B after A being chosen is n. There are n2 points outside the line of infinity. Among these points, n points are on O A , and n - 1 points are on OB different from 0. Thus the number of choices of C after A , B being chosen is n2 - n - ( n - 1). Hence = ( n + l ) n ( n- 1)2. Note that a subgroup fixing 0 ,A , B , C point-wisely is a planar subgroup.

+

150

Conversely an affine planar subgroup L of H always fixes 0. Let V ( L )be the affine subplane fixed point-wisely by L. Then L fixes all the points of intersections of the lines of V ( L )with the line of infinity. Let A # B be two of these points and C be any affine point not on O A or OB. Then ( 0 ,A , B , C} is an element in our 0 . The proof is now complete by 2.1.

Remark. This is best possible as can be seen in a Desarguesian plane. For a Desarguesian plane of order n, the number n(n 1 ) ( n - 1)’f~ equals to IrL(2,n ) / .

+

Application of 2.2 to translation planes is obvious. Note that in the case in which

n is a power of a prime p (e.g., a translation plane), 2.2 yields the following result which provides the existence of planar pelements. A collineation is called an elation (respectively, a planar element) if the set of fixed points is a fiber (respectively, a subplane). 2.3. Corollary. L e t H be collineation group an the translation complement of a translation plane of order n = p d , where p is a prime. If lHlp > n, t h e n H contains

nontrivial planar p-elements. For a translation plane of order n = p a , a planar collineation of order a power of p is called a Baer p-element if its set of fixed points is a Baer subplane, i.e., an affine subplane of order fi.We observe that the order of a group of elations with a common affine axis is at most n. On the other hand, the order of a group of Baer p-elements with a common Baer subplane is at most fi as this order must divide n - fi = fi(fi1).This provides one difference between groups of elations and groups of Baer p-elements. If a group in the translation complement is perfect, then it is a group of linear transformations. The group of all linear transformations in the translation complement is called the linear translation complement. Thus a simple collineation group in the translation complement of a translation plane is in a linear complement. We call a collineation (respectively, group) in the linear complement a linear collineation (respectively, group). Note that for q, a power of 2, L z ( q ) acts on the Desarguesian translation plane of order q. When q is an odd power of 2, &(q) acts on the Ott-Schafer plane of order q2 and Sz(q) acts on the Luneburg plane of order q 2 . The Lorimer-Rahilly plane and the Johnson-Walker plane (both of order 16) admit L2(7) E L3(2) as a collineation group. However, not one simple group has been shown to be a collineation group of a translation plane of odd order. In a survey article by M. Kallaher (1995, [lo]), he pointed out that three papers (Mason (1983); Foulser, Mason and Walker (1984); Fink and Kallaher (1987)) were written on the subject, all under the critical assumption that as a module the simple group acts irreducibly on the plane. The first two articles considered PSL(n,q ) , and the last article considered the sporadic simple groups. The last article also showed that one could extend the field to a bigger field such that the resulting module is absolutely irreducible if the plane is an irreducible module. In the process of extending the field, the spread becomes

151

a partial spread. It seems the study of partial spread is more appropriate in this setting. The key of the relationship between the geometry and the group is the action. In collineation groups, an important result of R. Baer states that an involution (i.e.., an element of order 2 ) is either a perspectivity or a Baer involution. 3. Some remarks on the Main Problem. Let us look at what could happen when the field bas infinite cardinality. Suppose the field F is a finite Galois extension of the rational numbers and consider the Desarguesian case, namely, d = 1. For any element c of the Galois group, define a collineation by sending (z,y) to (z", y"). A moment of thought shows that this is indeed a collineation over the rational numbers. Hence the Galois group acts on the translation plane. Thus any finite group which is a Galois group of a field over the rational numbers will be a collineation group. The recognition of which finite group can be a Galois group over the rational numbers is the Inverse Galois Theory, which is very active at the moment. J. G. Thompson has shown that the sporadic simple group "Monster" is such a Galois group. On the other hand the Galois group of a finite field over another finite field is always cyclic. This implies that any perfect collineation group in the translation complement is inside the linear complement. One way to tackle the Main Problem is to use Thompson's work on minimal simple groups. The smallest minimal simple group is As, the alternating group on five letters. However, so far no one has been able to eliminate As as a collineation group of a translation plane of odd order. For the moment, we decide to sort out other route. As indicated before there are simple groups acting on translation planes of even order and no known simple groups acting on translation planes of odd order. By the celebrated Feit-Thompson 's Odd Order Theorem, the order of every (non abelian) simple group is divisible by 4. ( Here we also use a result of Burnside.) Thus the prime 2 and subgroups of order 4 play special role in the study of simple groups. For collineation group of a translation plane we obtain the following observation, which shows how a subgroup of order 4 affects the characteristic of the plane. 3.1. Proposition [8]. Suppose a translation plane admits a linear collineation group H of order 4 such that the sets of fixed points of each non trivial element in H are equal. T h e n the order of the plane i s even or odd according t o H is elementary or cyclic.

4. Eigenspaces. The set of fixed points of a linear collineation (i.e., the eigenspace corresponding to the eigenvalue 1, ) carries a tremendous information. Eigenspaces are powerful tools for linear collineations that collineations of general projective planes lack. Consider the Desarguesian translation plane of order 8 as a vetor space V over the field GF(2). Thus the dimension of V is 6 over GF(2). Let c be a automorphism of order 3 in Gal(GF(8)/GF(2)).Then the map g , which sends (z, y) to (z", y") is a linear collineation (over GF(2)).Then V ( g ) := Cv(g)is a Desarguesian plane

152

of order 2 and [V,g ] is a direct sum of irreducible 2 dimensional modules of g. The 3 fibers X , Y,2 incident with V ( g )are each invariant under g . Let R E { X ,Y ,Z } . Then R = CR(g)@[R,g ] . Thus V = V(g)@[V, g] is also the canonical decomposition as g-modules. As d i m ~ p ~ 2 , [ V=, g4,] [V,g]cannot be linear (i.e., incident with a fiber) or planar (i.e., an affine subplane). On the other hand We prove [9]that an eigenspace is either linear or planar. For a diagonalizable linear collineation, if an eigenspace is planar, then all eigenspaces are planar with an common order m and m divides the order of the translation plane. We observe that an eigenspace is the sum of all isomorphic irreducible < g >modules of dimension 1 in the canonical decomposition of V as a < g >-module. The next two results are concerning with quotient spaces. In the rest of this section V will be a translation plane of odd characteristic p with a spread S. For W 5 V , let S ( W ) := { X E S : X n W > 0) and S, := { X n W : X E S ( W ) } .A fiber in S ( W ) is called incident with W . We now study a vector space with a partial spread. Let U be a vector space over a finite field of odd characteristic p and let u be a linear transformation of U . Denote C u ( a )by B. Let C be a set of mutually disjoint u invariant subspaces such that B is a translation plane with spread {X n B J X E C}, and U = X @ Y for two different elements X , Y E L. (In other words, C is a partial spread of the space U . ) Let = U / B and we use the bar convention. Note that ff is a u module and is u invariant for X E L.

u

4.1 Lemma.

x

If X # Y EC , then u=

and

xny=0 .

v

Remark. needs not be a translation plane as the images of the components of B might not cover the non zero vectors of ff.

x

v.

Proof. Let X # Y E L. Clearly @y = Suppose n y # 0. Then there is \ B such that x = y + b for some y E Y and b E B. Since x @ B , y @ B . Hence x - y = b = b" = x" y". This implies that x - xu = y" - y. The left hand side of this equality belongs to X but the right hand side belongs to Y as X " = X and Y" = Y . This forces x - xu = 0 = y" - y as X n Y = 0. Therefore x = xu. However this contradicts the fact that x $ B. The proof of 4.1 is now complete.

xEX

+

Corollary 4.2. Let T be a linear collineation of V . If W = C V ( T )as a Baer subplane of V , then V/W is a translation plane with a spread formed by the images of S ( W ) . Let H be a collineation group of V acting trivially o n W , then H is isomorphic t o a semi-direct product of a normal p-subgroup with a cyclic group OJ order prime t o p .

Proof. This first statement follows from 4.1 and a dimensional counting argument. For H , we observe that H induces a cyclic group of order prime to p on the translation plane V/W as it fixes each components of S ( W ) . The kernel of this action of H is an a normal p-subgroup. We now return to the space U

153

Proposition 4.3. If u has order a power of p , then D := Cu(u) is a translation plane with a spread n D(x E C } . In particular, the order of b is the same as

{w

C V(u).

Proof. Observe that fi # 0 as u has order a power of p . Let 3 E D. We claim that there is a E L E C such that = Z.This claim means that the image of {5?n OlX E C} covers the non zero vectors of B.This together with 4.2 show that n D1x E C} is a partition of 0. Since E B,b" = 6 b for some b E B. Let L E C such that b E L. Since B is a translation plane, there is M E C distinct with L. Hence U = L @ M . So S = a + p with a E L and p E M . Thus a + p + b = b b = 6" = a " + p " . This implies that 8" - p = CY - a" b. Since C Y , b E L , the right hand side of the last equality belongs to L. But the left hand side belongs to M . This forces the common element must be 0 as L n M = 0. Hence p = p" E B , which implies = 0. Therefore 5 = 5 = h as claimed. n D and To finish the proof, it suffices to prove that D = A @ B for A = B = L n_0,_ where X # Y are two different elements of C. Since X n Y = 0 by 4.2, A n B = 0. Let 0 # iii E O. By 4.2, iii = Z+y, with Z E and jj E 7. We will prove that 5 E A and Tj E B . jFrom Z+g = iii = iii" = 3"-+y", we see that Z = jj" - Tj. This element belongs to n 7 as rr" = X and -0 Y = y by 4.2. Therefore this element must be 0 as Ff-y = 0 by 4.2 again. This implies that 5 = Z" and y = 3". Therefore Z E A and 5 E B as desired. As the two translation planes Cu(u)and D have the spreads of equal sizes, so their order are the same. The proof is now complete.

s

{x

a

+

+

+

p

+

x

x

x

Remark. Let D be the complete preimage of D in U . Then the mapping, which sends u to [u,01, is a linear map from D to Cu(u) as [u,u, u] = 0. Thus [ D , a ]5 Cu(a). Thus the exact sequence: 0 + Cu(u) + D + [ D , a ] 0 yields that d i m D 5 2 d i m C v ( a ) . What we just prove in the present situation is that this linear map is onto. In other words, [D,o] = Cu(o).We now apply 4.3 to the translation plane V and obtain the following. 4.4 Theorem. Suppose r i s a collineation of p r i m e order p and C v ( r ) is a subplane of order m = p e . T h e n there i s a sequence of subspaces : 0 = Vo < Vl < Vz < . . . < V, = V such that V,/V,-, = C v p - l ( r ) is a translation plane of order m with a spread formed f r o m the images of the components in S ( C V ( r ) ) .I n particular, e divides d . (Recall that the order of V i s n = p d . ) Further, with respect t o this subspaces, V has a basis such that r is represented by

154

Remarks. 1. Note that the above result does not say much when p = 2 as the an involution has a minimal polynomial x 2 - 1. 2. In the Desarguesian translation plane of order n = Pp, the field automorphism z 4 z p induces a collineation (over G F ( p ) ) such that its set of fixed points is a subplane of order p . The matrix representing this collineation is the above matrix with e = 1. 3. When n = p3 instead of Pp in the above example, the order of r will be 3. Assume that ( 3 , p ) = 1. Then C := CV(T)is a subplane of order p again, but [V,T] CV/c(r) has dimension 4. It cannot be a plane of the same order as C. Note also that T does not induce a scalar multiplication on [V, T]. 4.5 Corollary. Suppose i is a n involution centralizing r (as in 4.4). T h e n i is a homology (respective, a f i n e ) if and only if the restriction of i on CV(T)is a

homology (respectively, afine). Proof. Let C

:= C ~ ( T The ).

subspaces V, are i-invariant. Working with V,+,/V, <

V,+2/y, we see that i induces the same action on each &+l/y.

If T acts on C as - I , then the last paragraph shows that i induces -I on V . This proves that i is a central homology if and only if it induces a central homology on C. Clearly if i is a homology, then it induces a homology on C . Now suppose that i induces an affine homology on C . The component A such that AflC being the axis of i is i-invariant. Note that A = A‘. Since AnC = CA(T) and i centralizes this subgroup of A, we obtained that i centralizes A by Thompson’s A x B Lemma (or just linear algebra here). Hence i fixes the points of the component A . So i is an affine homology as desired. 5. Translation Planes of Odd Order and Simple Collineation Groups. In this section we focus on a translation plane of odd order. Thus V is a 2ddimensional vector space over G F ( q ) with a spread S and q is a power of an odd prime p . Let G be a linear collineation group of V . An important consequence of a simple collineation group is the following. ( 1986 Ho [7], also see 1983 Fink and Kallaher [2]) 5.1. Theorem. If a translation plane of odd order n admits a simple collineation l(mod 4) and all involutions of G are Baer group G, then n = m2 with m involutions.

In the study of translation planes, two types of collineations: affine perspectivity

( the set of fixed points is a fiber of the spread ) and Baer elements (the set of fixed points is a Baer subplane ) attract most attention. These collineations have one common feature, namely, the dimension of the set of fixed points of any of these collineations is half the dimension of V . Several people have study the situation in which an element of order p is either an affine perspectivity or a Baer element. (for example: Bering, Ostrom, Foulser, Johnson, Schafer, Walker.) Note that the minimal polynomial of such an element is (z - 1)’. With the works of Thompson (1972 [13]) and (Ho 1979 [5]) on quadratic pairs we obtain the following sufficient condition for G to be not simple.

155

5.2. Theorem. If G contains a n element with minimal polynomial (x - l)’, then G is n o t simple. In a translation plane, the set of fixed points of a perspectivity or a Baer collineation is a subspace of half dimension of the ambiance vector space. It is not easy to distinguish these two objects. On the line of infinity, the set of intersection points with the fixed points of a perspectivity has the cardinality one or two. However the set of the intersection points with the fixed points of a Baer involution has cardinality m+ 1,where n. = m2 is the order of the plane. A moment of thought shows that two subplanes share common components have only the zero vector in common or they are equal. We could use permutation group theory t o study the action of the group on the points on the line of infinity. 6. Simple Groups and Perspectivities. Most known results on collineation groups in general require the existence of some perspectivities. In 1982, Reifart and Stroth determined simple irreducible ( no fixed points, lines or triangle) groups containing a perspectivity of a projective plane in two papers ([ll]and [12]). Thus it remains to study the situation in which a simple collineation group contains a perspectivity fixing a point and a line. This leads to the study of the Main Problem and it becomes the following . 6.1. Conjecture. No ”known” simple group could be a collineation group with a perspectivity of a translation plane of odd order. There are 17 families and 26 sporadic simple groups needed to be verified. A result in the article (Ho and Gonqalves 2000) [3] eliminates the alternating groups.

6.2. Theorem. G cannot isomorphic t o a n alternating group o n at least five letters. We intend to give an affirmative answer to this conjecture. A consequence of this is that the order of a translation plane admitting a “known” simple collineation group must be even. In the September 2002 Thompson Fest at Cambridge, M. Aschbacher (introduced by S. Smith) gave a talk on their 1300 pages preprint on Quasi Thin Groups. This might finally fill in the gap due to G. Mason and completes the classification of finite simple groups. Thus we might able to remove the word ”known” from the above statements.

7. Revisit the Main Problem. In 1917 H.F. Blichfelcht wrote that where purely geometrical methods are not available or less convenient than analytical methods, collineation groups are most easily studied through medium of linear groups. An observation of the dimensions of the sets of fixed points of affine perspectivities and Baer elements leads to study linear groups such that the dimension of the sets of fixed points is a constant on the set of non trivial elements of the centralizer of involutions. In 2003 (Ho), simple linear groups satisfying this condition has been

156

classified. An application of this to linear collineation groups shows that and La(7) are the only possible candidates. Note that this result gives an affirmative answer to the question that there is always an element in a simple collineation group of a translation plane of odd order such that its set of fixed points has dimension smaller than half the dimension of the underlying vector space. If no element of order 4 exists, then a Sylow 2-subgroup is elementary. A simple group with an elementary Sylow 2-subgroup has been classified. This leads us to focus on elements of order 4. Observing the translation planes, an element h of order 4 in a simple collineation group induces on the Baer subplane of fixed points of h2 either the identity or an involution. The case in which the action is either the identity or the negative of the identity has now been worked out via the medium of linear groups by determining the structure of a Sylow 2-subgroup. Thus we are able to apply Brauer-Wall-Suzuki Theorem or Alperin-Brauer-Gorenstein Theorem according to a Sylow 2-subgroup is a dihedral or semidihedral group. The possibility &(7) in the paper before now becomes the family L ~ ( u )with , u 5 7 (mod 8). In 1996 Ch. Hering constructed a translation plane of order 91 that admits a collineation group of order 128 such that each involution is Baer. At the Thompson Fest, Hering informs us that he believes that there might be simple collineation group for translation planes of odd order. This rekindles the importance of constructions of translation planes. In other words, find a spread for an even dimensional vector space using the group as a guide. For example, could L 2 ( 7 )be a collineation group for a translation plane of odd order? The module structure has been determined. On the other hand, how to use this algebraic information to affect the geometrical structure of the plane needs further study.

REFERENCES 1. W. Feit and J . G. Thompson, Solvability ofgroups of odd order, Pacific J. of Math. 13 (1963), 755-1029. 2. J.B. Fink and M. Kallaher, simple groups acting on translation planes, Rend. Del. Circ. Mat. Pal. (2) 32 (1983), 76-99. 3. A. Gon~alvesand C.Y. Ho, Alternating groups as collineation groups, J . of Algebra 225 (ZOOO), 581-601. 4. D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups, Number 1,2,3,4, Mathematical Surveys and Monographs, AMS. 5. C.Y. Ho, O n the quadratic pairs, 3 . of Algebra 42 (1976), 338-358. 6. C.Y. Ho, Finite projective planes that admit a strongly irreducible collineation group, Canadian J . of Math. 37 (1985), 579-611. 7. C.Y. Ho, Involutory collineations offinite planes, Math. 2. 193 (1986), 235-240. 8. C.Y. Ho, Order 4 elements of linear and collineation groups, J . of Algebra, t o appear. 9. C.Y. Ho, Eigenspaces of linear collineations, prepreint. 10. M. Kallaher, Handbook of incidence geometry, North Holland, Amsterdam, 1995, pp. 139-192. 11. A. Reifart and G. Stroth, O n finite groups containing perspectivities, Geometriae Dedicata 13 (1982), 7-46. 12. G. Stroth, O n chevalley groups acting on projective planes, J. of Algebra 77 (1982), 360-381. 13. J.G. Thompson, Quadratic pairs, preprint.

A POLYHEDRAL DESCRIPTION OF 3-MANIFOLDS ANN CHI KIM AND YANGKOK KIM Dedicated to the memory of B e r n h a d H. Ne um ann

1. INTRODUCTION

The cyclically presented groups comprise a rich source of groups, which are interesting from a topological point of views. The connection between cyclically presented groups and cyclic branched coverings of knots and links was studied, in particular, in [l],[2], [3] and [4].In [2], the study of the Fibonacci groups started with the construction of a family of %manifolds whose fundamental groups are isomorphic to the Fibonacci groups. For this purpose, they identified pairs of faces on the boundary of an oriented polyhedron which consists of n 3-gons(triangles) in the northern hemisphere, n 3-gons in the southern hemisphere, 2n 3-gons in the equatorial zone. In this paper we consider polyhedra whose boundaries are tessellated by m-gons and s-gons for integers m ( 2 3 ) , s ( 2 3). We then construct an infinite family of %manifolds M (m, s, n) and show that M (m, s, n) are n-fold cyclic branched coverings of the 2-bridge knot (m - 1)

+ A.

2.

POLYHEDRA AND %MANIFOLDS

Theorem 1. For integers m(> 3 ) , s ( > 3 ) not both even, there is a 3-dimensional manifold whose related polyhedron consists of m-gons and s-gons in the northern and southern hemisphere respectively. Proof. We consider a tessellation on the boundary of 3-ball, which can be regarded as a polyhedron 'P(m, s, n), consisting of n m-gons Fi in the northern hemisphere, n s-gons Ki in the southern hemisphere, n m-gons Fi and n s-gons ?7i in the equatorial zone. Then P ( m , s , n ) has 4n faces, n(m+s)+2nedges, n ( m i s ) - 2 n + 2 vertices(see Figure 1). We now consider two cases, m = 2k 1,s = 2 l + 1 and m = 2k 1,s = 21 separately. For m = 2k 1,s = 2 4 the oriented edges can be labelled as shown in Figure 2. Then the oriented edges fall into 2 n classes: x i , yi, i = 1,.. . ,n. Moreover the boundary cycle of the m-gons Fi and Fiis Y i X i + 2 k x & k - l . . . ~ 4 ~ 3 ~ and x 2the~ ; ~ ~ ) ' y ithe indices are boundary cycle of the s-gons Ki and 77, is x i + 2 ~ + 2 ( y i ~ ~ ~where taken mod n. Note that the set of all the faces splits into pairs of faces with the - same sequences of oriented boundary edges. We now identify Fi, Fi+n-2 and Ki, Ki for i = 1,.. . ,n, respectively such that the corresponding oriented edges on polygons carrying the same label are identified. The resulting complex M ( m , s, n) has one vertex, 2n

+

+

+

2000 Mathematics Subject Classafication. Primary 57M12, 57M25; Secondary 57M50, 57N10.

157

1 58

FIGURE 1. P(m,s, n)

Yi

..

"=i+Zk+l

E

"

Yi+l

x;

. 0

4'

.

.

Zi+zk+z

FIGURE 2 1-cells, 2n 2-cells and one 3-cell. Then we have a closed connected orientable 3manifold M ( m ,s, n ) since its Euler characteristic equals zero. For the case m = 2k+ 1,s = 2l+ 1,we can apply the similar argument. The only difference are the edge orientation and labelling, and face identification as follow; x ~ )the ~ boundary cycle The boundary cycle of the m-gons Fi and Fi is Y ~ ( x ~ ~ ,and is si(yi;',yi)' with the indices taken mod n(see Figure 3) of the s-gons Ki and and for the face identification, we identify Fi, Fi and Ki, Ei for i = 1,. . . ,n.

zi

I We note that P ( m , s, n) has a unique vertex and so the relators of the fundamental group of M ( m ,s , n ) can be obtained by running around the boundaries of

159

0

0

.

FIGURE 3 the 2n 2-cells of P ( m , s, n). Hence the fundamental group of M ( 2 k + 1 , 2 l + 1,n) is

(xi,.. . , z n , y i , . . . , Similarly, for M (2k (21,.

I

~ n9

1

e

i ( ~ 2 ~=~1,i si(yzqlyi) ) ~ = 1, indices mod n).

+ 1,2C,n), we have

. .,zn, ~ 1 ,. ., . yn I Yixi+2kxF:zk-1.

. .54531x2x;1 = 1, ~i+ze+2(yiy~+~) -1 e yi = 1, indices mod n).

Theorem 2. M ( m , s, n) are n-fold cyclic branched coverings of the 2-bridge knot 1 (m-1)+=. Proof. We note that the cylindrical symmetry of the polyhedron P(m, s, n) induces an orientation preserving homeomorphism g on M ( m , s, n) of order n. The set Fix(g) consists of the points of the polar diameter N S . Let G, be the cyclic group of homeomorphisms of M ( m , s, n) generated by g. Of course, G, has order n and Fzx(gk) = Fzx(g) for each Ic = 1 , 2 , ' . . , n - 1. The quotient space M ( m , s, n)/G, is homeomorphic to M ( m , s, 1) and the canonical quotient map p : M(m,s,n)/G, -+ M ( m , s , l ) is an n-fold branched cyclic covering, whose branching set is the 1-subcomplex of M ( m , s, 1) composed of N S . Indeed the polyhedron P(m,s, 1) defines in a natural way a decomposition of M ( m ,s,1) into handles which is represented in a Heegaard diagram H(m, s, 1) of genus 2 as shown in Figure 4 where the dotted line is the axis N S and lies below the diagram, inside the ball whose boundary is being identified along the discs pairs A, 2 and B , B. We draw a Jordan curve C1 around the meridian disk A . We note that the curve separates a pair of identified discs such that the number of points along their boundaries is greater than the number of intersections of the curve with the Heegaard diagram. We simply dig a ball along the curve and glue it back along the pair of disks determined by Jordan curve Cl. We can cancel 2-handle connecting 1 and m with a bhandle to get deformed graph as shown in Figure 5.a. We then

160

FIGURE 4. H ( m ,s, 1) do the same work with a Jordan curve Cz around A. We do this kinds of work times around the meridian disk B m23 times around the meridian disk A and to get a deformed graph as shown in Figure 5.b. By eliminating 1-handle B with 2-handle connecting CY and E , we have the graph shown in Figure 6.a. Finally the cancellation of 1-handle A and connection of two points N and S gives S3 and the as a branched set. For even integers m, s, we can apply 2-bridge knot (m - 1) the similar argument.

+

b.

a.

FIGURE 5

I

161

a.

b.

FIGURE 6 Corollary 3. The fundamental group of M(2k cyclic presentation with n generators:

+ 1 , 2 l + 1,n ) has the following

7rl(M(2k+1 , 2 l + 1 , n ) ) = (y1,...,y,/nly i((yG1lyi)e(yF-llyi)e)k= 1,indices modn) Proof. Note that the fundamental group of presentation: m ( M ( 2 k + l , 2!+1,

n ) )= (21,. . . ,z,

~1

( M ( 2 k +1 , 2 l + 1,n ) )has the following

y1,. . . ,gn I

R2i+l

= 1,

R2i =

1, indices mod n ) .

where = y i ( ~ 2 ~ xand i ) Rzi ~ = We simply isolate R2i = from the presentation with zi = (yt:lyi+l)e for i = 1,.. . ,n. Then we have the following cyclic presentation: m ( M ( 2 k + 1,2[+

L n ) )= ( y l , . ..,ynlyi((yi~l~yi+l)e(yt-lyi+l)e)k, indices modn).

We can get a similar presentation for M ( 2 k For odd integers m = 2k the homology groups;

+ 1 , 2 l , n).

+ 1, s = 2 l + 1, we have the following presentation for

Corollary 4. Hl(M(2k

.{ ZN ZN

+ 1 , 2 l + 1,n))E G(2k + 1 , 2 l + 1,n)ab

x Z ( ~ ~ Z + ~ ZN ) N=, . f (n’). g(n’) x ZN7 N = dn)

f o r n = 272’ f o r n = 2n’

+1

Here f ( n ) g, ( n ) are the generalized Fibonacci-Lucas numbers defined by the equation

h(n

+ 2 ) = 12h(n)+ k h ( n + l),

and the initial values f(O) = 0,f (1) = k , and g(O) = 2, g ( l ) = k 0

162

REFERENCES [l] P. Bandieri, A. C. Kim and M. Mulazzani, On the cyclic coverings of the knot 52, Proceedings of the Edinburgh Mathematical Society 42 (1999), 575-581. [2] H. Helling, A. C. Kim and J. L. Mennicke, A geometric study of Fibonacci groups, Journal of Lie theory 8 (1998), 1-23. [3] G. S. Kim, Y. Kim and A. Yu. Vesnin, The Knot 52 and cyclically-presented groups, J. Korean Math. SOC.35 (1998), no. 4, 961-980. [4] Y. Kim, About some infinite family of 2-bridge knots and 3-manifolds, Internat. J. Math. & Math. Sci. 24 (1999), no. 2, 95-108. [5] A. Mednykh and A. Vesnin, On the fibonacci groups, the Turk’s head links and hyperbolic 3manifolds, Groups-Korea’94(A.C. Kim and D.L.Johnson, eds), de Gruyter, 1995, Proceedings of the 3rd international conference on theory of groups held at Pusan National University, Pusan, August 18-25 (1994), 231-239. [6] D. Rolfsen, Knots and Links, Math. Lect. Series 7, Berkeley, Publish or Perish Inc., Berkeley, Calif. 1976 [7] H. Seifert and W. Threlfall, A textbook of topology, Academic Press, Inc., New York, 1980.

UNIVERSITY, KUNMING 650091, P.R. CHINA, (A.C. Kim) MATHEMATICAL INSTITUTE, YUNNAN DOLSAN MATHEMATICAL INSTITUTE, DONGRAEKU, SAJIK 1-DONG 19-46(1/2) PUSAN 607-814, KOREA E-mail address: ackimapusan. ac .kr CURRENT ADDRESS:

(Y. Kim) DEPARTMENT OF MATHEMATICS, DONGEUI UNIVERSITY, PUSAN614-714, KOREA E-mail address, Y. Kim: ykkim0dongeui. ac .k r

Theorems on admissible subsets of a semigroup Li Gang" Department of Mathematics, Zhongshan University Guangzhou, China and Department of Mathematics, Shandong Normal University Jinan 250014, China Email: ligngBsohu.com

K.P. Shumt Faculty of Science, The Chinese University of Hang Kong Hang Kong, China (SAR) Email: kpshum0math.cuhk.edu.hk

P.Y. Zhu Department of Mathematics, Qinghai Normd University Xining 810008, China Dedicated to the memory of Professor B H Neumann Abstract

In this paper, we classify the types of semigroups whose finite subsets are left admissible subsets. We dso prove that the lattice of group congruences defined on a regular semigroup S is isomorphic to the lattice of the full admissible subsemigroups of S. AMS Mathematics Subject Classification (2000): 20M10 Keywords: Admissible sets; Generalized Rees congruences. 'The first author is partially supported by a NNSF grant (No.10071068),China. +Thesecond author is partially supported by a UGC(HK) grant #2160187 (2002/03).

163

164

1

Introduction

According to Clifford and Preston [I], a subset A of a semigroup S is said to be admissible if for any a, b E S1,aAb n A # 0 implies that aAb C A. Naturally, we call a subset A of S a left admissible subset of S if for all a E S1, aA n A # 0 implies that aA A. Dually, we can define the right admissible subset of S . However, a subset A which is both left admissible and right admissible in a semigroup S is not necessarily an admissible subset of S, for example, if we let A be a subgroup of a group S , then A is both left and right admissible but A is not admissible unless A is a normal subgroup of S . Although the concept of admissible subset of a semigroup has been introduced for some times, there is no significant theorems in the literature concerning the admissible subsets of a semigroup S . In this paper, we will investigate the properties of such subsets and classify certain types of semigroups by considering its admissible subsets. We prove that the lattice of group congruences on a regular semigroup S is isomorphic to the lattice of the full admissible subsemigroups of S. Then the contents of Feigenbaum [2] and Hall [3] on regular semigroup congruences are enriched. In addition, a characterization theorem of left admissible subsets in an infinite semigroup is also obtained. In order to describe the admissible subsets of a semigroup S , we give the following basic definitions and terminologies.

Definition 1.1 The elements a , b of a semigroup S is said to be linked by an elementary @-transitionon S if a = zpy and b = zqy for some z, y E S', where ( p ,q ) E 6 or ( q , p ) E 6'. Definition 1.2 A congruence 6fl is said to be a tight linked congruence generated by an elementary 6-transition 6 on S if 611 is defined by ( a ,b) E OH if and only if either a = b or for some positive integer n, there is a sequence a = z1 + z2 3 . . . --t z, = b whose elements are linked by an elementary 6'-transition 6. (see 141) Shyr [7] has defined the following congruence on a semigroup S with respect to a given subset A of S.

Definition 1.3 Let A be a subset of the semigroup S . Then PA = { ( a ,b ) E S x Sl(Vz,y E Sl) zay E A # xby E A } is a congruence on S. We call PA the principle congruence on S determined by A. Let C(S) be the set of all congruences on a semigroup S. Then, as suggested by Y.Q. Guo, we call the following congruence

the generalized Rees congruence on S determined by the subset A of S. In fact, one can easily verify that A is an union of some pA-classes and if A is an ideal of S then P A is precisely

165 the usual Rees congruence on S determined by the ideal A of S. This is why we call P A the generalized Rees congruence on S determined by A . It is easy to see that A is an admissible subset of S if and only if PAJA= A x A. In particular, if A is a left admissible subset of S and za E A for some x E S and a E A then xb E A for any b E A. In extending the concept of admissible subset, we give the following definition.

Definition 1.4 Let A be a collection of subsets Ai(i E I ) of a semigroup S . Then A is called an admissible aggregate of Ai if ( V i , j E I) A (VU,b E

Sl)aAib n Aj # B + aAib & Aj.

The following concepts are also useful.

Definition 1.5 Let Ba(a E X ) be some subsets of a semigroup S and A = is called an admissible hull of &(a E X ) if for any a , b E A, there exist such that a E B,,,b E B,,, and Bai nB,i+l # 0 for all i = 0 , 1 , . . . , n - 1.

U B,. Then A

,EX cyo, a l , . .

. ,a,

E

X

Definition 1.6 Let X be a set containing a semigroup S. Let E be a mapping from X onto S such that [Is = Is, the identity transformation of [ on S . Then X forms an ideal extension of S under the following multiplication (Va, b E X ) a

ob

= (a[)(b().

4.

We naturally call the above semigroup (X, 0) an inflation of S , denoted by X = [ X ,S ;

Definition 1.7 A subsemigroup B of a semigroup S is called a full subsemigroup if E ( S ) g B , where E ( S ) is the set of all idempotents of S. For other concepts, notations and terminologies not given in this paper, the reader is referred to Howie [4].

2

Generalized Rees congruences on semigroups

Lemma 2.1 Let P A be a generalized Rees congruence determined b y a given subset A of a semigroup S , where A is the union of some P A - classes A, of S , i E I . Then P A = P A ; .

v

iEI

166

Proof. Since A =

U Ai and Ai is a pA-class

of S , we can easily verify that A = { A ; I i E I }

iEI

is an admissible aggregate of Ai. Define B = {(z,y) E S x SI x = y or (3i E I ) z , y E Ai}.

Then, we can extend 8 to a tight linked congruence @ on S generated by 8. Clearly, Bt P A , by the definition of en. On the other hand, if a E Ai and b@a, then either a = b or there exists a sequence u = 21 4 22 -+ ... -+ 2, = b (1) for some positive integer n, such that the elements in the sequence are linked by an elementary &transition linking a to b. Because A is an admissible aggregate of Ai(i E I ) , we see that for any a 6 Ai, we always have b E Ai. This implies that @ ) A = P A J A = PAJA, and consequently, P A C Of. Thereby, we have P A = On. Now, if we let ( a , b) E P A , then by the sequence (l),there exist A1, A2, . . . ,An-l E A such that a = zi PA^ 22 PA^ . . . PA,-^ zn = b. This leads to (a,b) E

V

and

iEI

V

Conversely, if ( a ,b) E

. . . ,zn-l E S such that

PA;,

SO P A

C V

iEI

PA;.

then there exist a positive integer n and a sequence x1,x2,

iEI

a

PA1

P A 2 2 2 PA3

...

PA,-1 X n - 1 P A ,

b,

that is, we have a -+ 2 1 + 2 2 -+ . . . -+ z,-1 -+ b, for some positive integer n. This means that a is linked to b by an elementary &transition and hence (a,b) E P A . Hence, we have shown 0 that P A = V P A ; , as required. iEI

Remark 2.2 (1) In the above lemma, if A itself is an admissible subset of S , then we can see immediately that P A = O i , where e A = {(x,y) E x SI x = y or x , y E A}.

s

(2) If A is an ideal of S , then A itself must be an admissible subset of S with

PA

= 6.4

Lemma 2.3 Let A be a n admissible subset of a semigroup S . T h e n the following statements hold. (1) P A i s the smallest congruence o n S such that A forms a congruence class. PI, where I = SIAS'. In particular, every element in S\I, p~-class.

(2) P A

i f exists, forms a

167 (3)

PA

is an idempotent pure congruence on S if and only if, for any x , y E S', zAy n E ( S ) # 0 + zAy

c E(S)

(4) A is a subsemigroup of S if and only if there exist some x , y E A

such that xy E A.

(5) If S is a regular semigroup then A is a subsemigroup of S if and only if A contains an idempotent. Proof. Statements (1) and (2) are trivial. We hence omit the proof, To prove (3), we let P A be an idempotent pure congruence on S . Suppose that 2, y E S' such that m y E E ( S ) for some a E A. Then, by Remark 2.2 (1) above, we have apAb for any b E A and so xaypAxby. This leads to xby E E ( S ) because P A is an idempotent pure congruence on S. Thus, zAy C E ( S ) . In the converse part of (3), we assume that x A y n E ( S ) # 0 + zAy C E ( S ) for any x , y E S1. Suppose that apAe for some e E E ( S ) . Then a can be linked to e by an elementary O~-transition. By using our assumption, we can see immediately that a E E ( S ) . Now, we prove (4). The necessity part of (4) is trivial. To prove the converse, we let some x , y 6 A such that z y E A. Since A is an admissible subset of S and zy E A, we have Ay n A # 0 and so Ay A. This shows that ay E A for all a E A. Again, by considering A as an admissible subset of S , we have aA A. Thus ab E A for all b E A. This shows that A is indeed a subsemigroup of S.

c

To prove (5), we first notice that the sufficiency of (5) follows directly from (4). In the direct part of (5), we assume that A is a subsemigroup of the regular semigroup S. Then, for any a E A, we have apA = A, and apA is an idempotent of S / ~ because A A is a subsemigroup of S . Now, by using the well known Lallement's lemma in semigroups, we know that there is 0 e E E ( S ) such that epA = apA. This implies that e E A. The proof is completed. Lemma 2.4 Let A , B be non-trivial admissible subsets of a semigroup S ( that is [ A /> 1 and IBI > 1). Then P A = p~ if and only if A is an admissible hull of some uBv (u,v E S') and B is an admissible hull of some sAt (s,t E S').

Proof. We first assume that P A = p ~ Then, . for a E A , we have A = a p = ~ apB. Let p~ = 6'1. Because the set A is a non-trivial admissible subset of S , I = SIBS1. Then p~ apa = A is not a singleton. This leads to a E I = SIBS1. In other words, A is contained in an union of some uBv(u,v E S'). Also, if u B v n A # 0 for some u,v E S1, then there is a b E B such that ubv E A. Hence, it follows from bpBc for any c E B , we have ubv p~ ucv and so ubv P A urn. This leads to ucv E A because A is an admissible subset of S. Consequently, uBv A and thus A = UUBW (u,v E S'). To show that A is an admissible hull, we let a, b E A. Since (a,b) E P A = p~ = O k , we have either a = b or for some positive integer n, there is a sequence of elements a = zo + z1 + . . . + z, = b in S linked by the elementary @a-transition from a to b. Hence, there exist u i , q E S',i = 1,2,. . . , n such that u E u0Bv0,b E u,Bv,

168 and zi-1 E ui-1Bvi-1 nuiBvi, i = 1 , 2 , . . . ,n. This shows that A is indeed an admissible hull of some uBw(u,v E S'). Similarly, we can show that B is an admissible hull of some s A t ( s ,t E 5''). Conversely, we assume that A is an admissible hull of some uBw(u,v E S') and B is an admissible hull of some s A t ( s ,t E S'). Then, for any a , b E A , we have a E uoBwo,b E u,Bv, with uiBvi ui+lBvi+l # 0, for ui,wi E S1,i = 0, 1,. . . ,n. This means that a is linked to b by an elementary BB- transition and so, we have apeb by definition. This leads to 0.4 C PB and 0 hence P A G p ~ Similarly, . we can prove that p~ C P A . Hence P A = p e r as required.

n

We now turn to the regular semigroups.

Lemma 2.5 Let C(S) and GC(S) be respectively the sets of congruences and group congruences on the regular semigroup S . Then, for p E C(S), there exists a subset A of S such that T r p A . I n particular, if p E BC(S), then p is a generalized Rees Kerp = KeTpA,TTp congruence on S , determined b y a full admissible subset B of S .

Proof. The proof is straightforward by taking A = K e r p . We omit the details.

0

We now establish a theorem on the lattice of group congruences defined on regular semigroups. This result enriches the content of Hall in [3].

Theorem 2.6 Let K be the set of all full admissible subsemigroups of a regular semigroup S. Then K = ( K ,C, n,V ) forms a lattice isomorphic to the lattice of group congruences on S. Also, (VA1,Az E K ) A1 V A z = {x E SI(3y E A2) Z P A , ~ } . Proof. It is routine to check that ( K ,C,n,V ) is a lattice, where, for A, B E K , A V B is the smallest full admissible subsemigroup of S containing both A and B . Also, by Remark 2.2 (1) and Lemma 2.3 (I), we can observe that for any A, B E K , A B if and only if pa p ~ , where P A and p~ arc the generalized Rees congruences on S determined by the subsets A and B , respectively. By Lemma 2.5, we can easily see that the mapping

4 : ( K ,c,n, v) -+ BC(S),A H p A is a lattice isomorphism. Now, let Ai, Az E K . Then A1 and A2 are full subsemigroups of S , we have p ~p~~~ E , BC(S). If ~7 is the smallest group congruence on S , then ~ A ~ / U , ~ AE ~C(S/g) / U so that ( p A l / o ) 0 ( p ~ , / f l )= ( P A , / C J )0 ( P A , / O ) . This leads to P A , 0 P A , = PA, 0 p ~ Hence, ~ . we have ~ ( A ~ v A = ~P A) , V P A , = P A , 0 PA^. For any e 6 E ( S ) ,by Lemma 2.3 (l), we have

A1 V Az = ~ P ( A ~ V A = , ) PA, V PA^) = 4 p A l 0 PA^) = {x E SI ( 3E~S ) ~ P A , Y P A ~ ~ ) = {z E SI (3y E Az) x p ~ , ~ (} since epAz = A2) This completes the proof.

169

3

Semigroups whose finite subsets are left admissible

Throughout this section, we assume that the semigroup S is always a semigroup whose finite subsets are left admissible. We now classify this kind of semigroups.

Lemma 3.1 For any a E S , we always have a2 = e E E ( S ) and ae = ea = a or ae = ea = e.

Proof. Clearly a E { a , a 2 } and by our assumption, { a , a 2 } is a left admissible subset of S. Then we have a3 E { a , a 2 } . If a3 = a , then a2 = e E E ( S ) so that ae = ea = a. On the other 0 hand, if a3 = a’, then we still have a2 = e E E ( S ) ,but ae = ea = e. Lemma 3.2 If IE(S)l = 1, then the S is a group with two elements or S as a square nilpotent semigroup. Proof. Without loss of generality, we assume that (SI > 1 and that E ( S ) = {e}. If a, b E S\E(S) such that ae = ea = a and be = eb = e, then by ea = a E { a ,b}, we obtain that e = eb E { a , b}, a contradiction. Hence, by Lemma 3.1, we have either ae = ea = a(Va E S ) or ae = ea = e(Va E S ) . In the first case, we claim that S is a group with two elements. In fact, if there exist a , b E S\E(S) with a # b, then ab E { a , b, e} by a2 = e E { a , b, e}. If ab = b, then e = b2 = ab2 = ae = a , a contradiction. If ab = a , then e = b, again a contradiction. If ab = e, then a = ae = a2b = eb = b, also a contradiction. This shows that [SI = 2. In the second case, S is a square nilpotent semigroup. This is because if a , b E S\E(S), then ab E { a , b, e} by a2 = e E { a , b, e}. Since { a , b, e} is a left admissible subset of S , we see that ab = a implies that a = ab2 = ae = e , a contradiction. Again, if ab = b, then e = b, a contradiction. This shows that ab = e, and thereby S forms a square nilpotent semigroup, 0 with e as its zero element. In what follows, we assume that IE(S)I > 1.

Lemma 3.3 One of the following statements holds an the semigroup S . (1) E ( S ) i s a left zero subsemigroup of S. (2) E ( S ) i s a right zero subsemigroup of

s

(3) E ( S ) i s a semilattice { a l p }of a right zero semigroup E, and a left zero semigroup Eg of S with (Y > p.

Proof. If e , f E E ( S ) , then by e2 = e E {e,f}, we see that e f E { e , f } and so E ( S ) is a subband of S. Thus, E ( S )is a semilattice Y of the rectangular bands E,(a E Y). If a, P,-y E Y

170 with a > p > y, then there exist x , E E,,xp E Ep and x7 E E, with x , > xp > x7 so that xp = xpx, E {x,,x7} by xpxr = x7 E {x,,xy}, a contradiction. Hence there does not exist a,p, y E Y with a > p > y. From this fact, it is routine to verify that IYI 5 2. If JYI = 1, then E ( S ) is a rectangular band. We now show that either (i) or (ii) holds. In fact, if (i) does not hold, then for any f , g E E ( S ) , there exists x E E ( S ) such that z # f and f z = x. Hence f = f g f E {x,gf} by fx E { x , g f } . Thereby, we have g f = f . This shows that (ii) holds.

{(.,a}

If IYI = 2, then E ( S ) is a a semilattice of rectangular bands E , and Ep with a > p. Now we show that E, is a right zero semigroup and Ep is a left zero semigroup. Assume without loss of generality that e,g E E, with e # g. Then, by a > /?,we have f E Ep with e > f. Hence by ef = f E { g , f}, we have eg E {g, f}, and so eg = g. This leads to E, is a right zero semigroup. If Ep is not a left zero semigroup, then we have x E EDsuch that x # f and f x = x . Hence by f x E ( x , e } , we have f = f e E { x , e } , a contradiction. Therefore Ep is 0 a left zero semigroup. Summing up the above facts, we see that (iii) holds. It is worth to remark here that for any e E E ( S ) ,we can define S, by

S, = { a E SI a’ = e } . Then, it is clear that S =

IJ

S,.

eEE(S)

Lemma 3.4 If E ( S ) is a left zero subsemigroup of S , then S is an inflation of the semigroup E(S). Proof. Assume that E ( S ) is a left zero semigroup. If there exists a E S,\{e} such that ae = ea = a for some e E E ( S ) , then, by taking f E E ( S ) with f # e and noting that ea = a E { a , f } , we have e = e f E { a , f } , which is a contradiction. Hence, for any e E E ( S ) , we have (Va E S), ae = ea = e. Now, let e E E ( S ) . If a, b E S,\{e}, then by ae = e E { e ,b}, we have ab E { e , b}. Also, if ab = b, then b = a2b = eb = e, a contradiction. Thus, we have ab = e so that S, becomes a square nilpotent subsemigroup. For any f E E ( S ) with f # e, we let c E Sf. Then ec E ( e , c } by e2 E { e , c } and so ec = e. If a E Se\{e}, then ac E {e,c} by ae E {e,c} and so ac = e . Summing up the above results, we know that (Va E S, A Vb E Sf) ab = e = e f ,

and so, if we define a mapping ( from S onto E ( S ) by ( :S

+ E ( S ) ,a H e

( if a2 = e ) ,

Then, we can check that S = [ S , E ( S ) , ( is ] indeed an inflation of E ( S ) .

Lemma 3.5 If E ( S ) is a right zero subsemigroup of S , then S = E ( S ) .

171

Proof. For any e E E ( S ) ,we let f E E ( S ) with f # e. Then, it is easy to verify that Se is a subsemigroup of S. If a E Se\{e}, we have a f E {a,e, f } by a2 = e E {a,e, f} and so af = f . Hence e = a2 E { a , f } by a f E { a ,f}, a contradiction. Thus S, = { e } and hence S = E ( S ) , as required. 0

Lemma 3.6 If E ( S ) is a semilattice {cy,p} of a right zero semigroup E, and a left zero semigroup Ep of S with cy > p, then S is a semilattice { c Y , ~ }of E, and an inflation Sp = [ X ,Ep,J] of Ep, and moreover, for any a E E,, b E So, we have ab = b, ba = b<. of a right zero semigroup E, and a left zero Proof. Assume that E ( S ) is a semilattice {a,@} semigroup Ea of S with a > p. Then, define S, and Sp by

=u eEE,

respectively. Form S = S,

U Sp.

We now divide the proof into several steps.

[Step 11 We first prove that for any a E S,, b E Sp, we have ab = b. If a E S,, b E S j for e E E, and f E Ep, then by e2 = e E {e, b}, we have eb E {e, b}. If eb = e, then e = eb2 = ef E Ep, a contradiction, and so eb = b. Also, by a’ = e E {e, a, b}, we have ab E {e, a, b}. If ab = a, then e = a’ = a2b2 = e f E Ep, a contradiction. Similarly, by eb = 6, we see that ab = e is also impossible. Hence ab = b.

[Step 21 We claim that S, = E,. In fact, if there exists a 6 S,\E,, we have a2 E { a ,b} by ab = b E { a , b}, a contradiction.

then, for some b E Sp,

[Step 31 We now show that for any f E Ep and e E E,, we have f e = f . In fact, by { e , f } , we immediately see that f e E {e, f}, and hence f e = f .

f2 =f E

[Step 41 We prove here that for any f E Ep, b E Sf and e E E,, we have be = f. In fact, if b = f , then by Step 3, we have be = f . If b # f , then by b2 E {b,e,f}, we have be E {b,e, f}. Again, we can show that if be E {b, e}, then f = b2 E {b, e } , a contradiction. Thus be = f . [Step 51 We show that for any f E Ep and a E S j , we have af = f a = f . In fact, if a = f , then af = f a = f. If a # f and if f a = a, then for some e E E,, we have f = f e E { a ,e } by f a = a E { a ,e}, a contradiction. Hence, by Lemma 3.1, we have f a = af = f . [Step 61 We show that for any f E Ep and a, b E S j , we have ab = f. In fact, if either a = f or b = f , then by Step 5, we have ab = f. If a # f and b # f , we know that a’ = f E {f,a, 6) implies that ab E {f,a, b}. Observe again that if ab E { a , b} then f = a2 6 {a, b}, a contradiction. Hence ab = f . [Step 71 Finally, we claim that for any f , g E Ep, if a E Sf and b E S,, then ab = f. Assume that a E S f ,b E S, for f , g E ED. If f = g , then, by Step 6, we see that ab = f. If f # g, then f b E { f , b } by f 2 E { f , b } . We note that if f b = 6, then f = f g = f b 2 = b2 = 9 , a contradiction. Hence f b = f . If a # f , then ab E { a , b, f} by a2 = f 6 {arb,f } . Also if ab E { a , b}, then f = a2 E {a, b}, a contradiction. Therefore ab = f.

172

Summing up all the above facts, we see that S is a semilattice { c u , ~of } En and Sp. If we define the mapping ( from X = Sp onto Ep by

t : X t E p , s f~ ( i f s 2 = f ) , then Sp = [X, Ep, (1 is an inflation of ED,and for any a E En, b E So, we have ab = b and 0 ba = b t , as required. By using the above lemmas, we now formulate the following theorem.

Theorem 3.7 T h e following statements are equivalent in a semigroup S: (1) S is a semigroup whose finite subsets are left admissible;

(2) S is one of the following types of semigroups: a group with two elements; a right zero semigroup; a n inflation of a left zero semigroup; a semilattice {a,/?}of a right zero semigroup E, and a n inflation Sp = [So,Ep,(] of a left zero semigroup Ep with a > ,B, and f o r any a E En and b E Sp, we have ab = b, ba = b t ; (3) S is a semigroup whose subsets are left admissible

Proof. The proofs of (2) details.

+ (3) and

(3)

3

(1) are straightforward and hence we omit the

(1) 3 (2). Note that square nilpotent semigroups are inflations of left zero semigroups. Thus the proof follows directly from Lemma 3.1 to Lemma 3.6.

Lemma 3.8 Let S be a semigroup. T h e n all infinite subsets of S are left admissible i f and only i f [SI < cc o r all finite subsets of S are left admissible. Proof. ==+). Assume that all infinite subsets of S are left admissible. If JSJ= 00, then we let A be a finite subset of S and x A n A # 0, for some z E S1.If zy @ A for some y E A, then B = S\{zy} is an infinite subset of S and z B B # 0. This leads to zB C B , and hence zy E B , a contradiction. Thus A is left admissible.

n

t) If. IS1 < 00, we can of course assume that infinite subsets of S are left admissible. If all finite subsets of S are left admissible, then by Theorem3.7, we can easily see that all infinite 0 subsets of S are left admissible.

By using Theorem 3.7 and Lemma 3.8, we formulate the following theorem for finite left admissible subsets of an infinite semigroup S.

Theorem 3.9 The following statements are equivalent in a n infinite semigroup S : (1) All finite subsets of S are left admissible;

173

(2)

s i s o n e of the following types of semigroups:

a right zero semigroup; a n inflation of a left zero semigroup; a semilattice {a,b} of a right zero semigroup E, and an anfiation Sp = [Sp,Ep,<]o f a left zero semigroup Ep with a > p, and f o r a n y a E E, and b E Sp, we have ab = b, ba = b<;

(3) S i s a semigroup whose subsets are left admissible;

(4) All

infinite subsets of S are left admissible.

By Theorem 3.7 and its dual, we obtain the following corollary. Corollary 3.10 T h e following statements are equivalent in a semigroup S: (1) All finite subsets of S are both left and right admissible;

(2) S i s o n e of the following types of semigroups: a group with t w o elements; a left zero semigroup; a right zero semigroup; a square nilpotent semigroup; a semilattice with two elements; (3) all subsets of S are admissible;

(4)

all finite subsets of S are admissible.

Acknowledgement: The authors would like to thank Professor Y.Q. Guo for his valuable comments and suggestions contributed to this paper.

References [I] A. H. Clifford and G. B. Preston. The algebraic theory of semigroups, Math. Survey, no. 7, Amer., Math. SOC.,Providence, R. I., Vol. I, 1961, Vol. 11, 1967.

[2] R. Feigenbaum. Regular semigroup congruences, Semigroup forum 17 (1979), 373-377.

[3] T. E. Hall. On the lattice of congruences on a regular semigroup, Bull. Austral. Math. SOC.1 (1969), 231-235.

[4] J. M. Howie. Fundamental to semigroup theory, Clarendon Press, Oxford, 1995. [5] W. D. Munn. Decompositions of the congruence lattice of a semigroup, Proc. Edinburgh Math. SOC.23 (1980), 193-198. [6] M. Petrich. Inverse semigroups, John Wiley & Sons, Inc., 1984. [7] H. J. Shyr. Free monoid and language, Published by Institute of Applied Math., National Chung-Hsing University, Taichung, Taiwan, 1991.

GENERALIZED SCHREIER VARIETY AND A CRITERION FOR NON-EXISTENCE OF COVERING GROUPS *

MOHAMMAD REZA R. MOGHADDAM Faculty of Mathematical Science Ferdowsi University 0.f Mashhad, Iran

ALI REZA SALEMKAR Department of Mathematics Sistan Baloochistan University, Zahedan, Iran t

Dedicated to the memory of Professor B H Neumann In this paper we introduce the generalized schreier variety u and show that every A ---t H --t G --t 1 is a stem group possesses a u-covering group. If 1 extension of the group G, with respect t o the variety u then similar t o Yamazaki (1964) for the variety of abelian groups, we show that H is a homomorphic image of a u-covering group of G. Also it is proved that, up t o isomorphism, the ucovering groups of a given group have the same marginal factor groups, which is a wide generalization of Read (1977 ). Finally, we present a criterion showing that a group may not have any covering groups with respect t o such a variety.

1. Preliminaries

Let F, be the free group freely generated by the countable set { q , 2 2 , ...}. Let V be a variety of groups defined by a subset V of F,. It is assumed that the reader is familiar with the notions of the verbal subgroup, V ( G ) , and the marginal subgroup, V * ( G ) ,associated with a variety of groups U and a given group G. We refer the reader to H. Neumann [17] for more information on the varieties of groups. Let N be a normal subgroup of a given group G, then we define [NV*G] to be the normal subgroup of G generated by the elements of the following set:

"AMS Subject Classification (2000) : Primary 20E34-20F14, Secondary 20E10 t Keywords : Bear-invariant, u-covering group, generalized Schreier variety, verbal and marginal subgroups

174

175

It is easily checked that [NV*G]is the smallest normal subgroup T of G contained in N, such that N/T V * ( G / T ) . Let 1 -+R F --+ G 1 be a free presentation of a group G, where F is a free group and R = kern. Then the Baer - invariant of G with respect to the variety V ,denoted by V M ( G ) ,is defined to be

- -

V M ( G )=

R nV ( F ) [RV*F]

'

One may check that the Baer-invariant of the group G is always abelian and independent of the choice of the free presentation of G (see [5]).In particular, if V is the variety of abelian or nilpotent groups of class at most c ( c 2 l),then the Baer-invariant of G will be ( R n F ' ) / [ RF, ] ,which is the Schur -multiplicator of G , or (Rn-yc+l(F))/[R,c F ] ( F repeated c times), respectively (see [6] , [7] for more details). We recall that an exact sequence 1 --+ A G* G 1 is called a V - stem extension with respect to the variety of groups V ,when A C: V ( G * ) n V * ( G * If ) . in addition A E V M ( G ) ,then the above extension is called a V - s t e m covei'. In this case G* is said to be a covering group of G, with respect to the variety V (or simply we say V-covering group of G). Now we define a series of a group G with respect to a given variety U , as follows:

- --

G = Vo(G)2 Vi(G)= V ( G )2 Vz(G)2 . . . 2 Vn(G)2 . . . , where Vn(G) = [Vn-l(G)V*G]for all n > 0. We call such a series the lower it/-marginal series of G with respect to the variety V. Similarly, we define the corresponding upper V-marginal series of G in the following way: 1 = V;(G)

c V;(G) = V * ( G )C . . . C V,"(G)G . . . ,

where for n > 0, V,*(G)will be define by

V,*(G) V L ( G ) -'*(V*

G (G)).

n-1

r+j

(GI G v, (GI v, (GI j 2 0. Clearly V,(G) . . = 1 if and only if V,"(G)= G, for all i 2 0. Now, by the above discussion we define a group G to be V-nilpotent if Vn(G)= 1, for some positive integer n. Also G is said to be V-nilpotent of class exactly c if Vc(G')= 1, for the least positive integer c (see [l]). In particular, if V is the variety of abelian groups then we obtain the notion

Using induction on i, one may check that V , * ( r )= 7 for, all

of nilpotent groups.

176

Finally, a group G is said to be V-perfect, if G = V ( G ) . In 1904, I. Schur [19]introduced the notion of covering groups for a given group, with respect to the variety of abelian groups. He also showed that every finite group has at least one covering group. E. W. Read [18] in 1977, obtained some interesting results on the concept of covering groups. In [ll] and [13] we also studied this concept with respect to arbitrary variety of groups V and assuming their existence we obtained some of their properties. However, it is of interest to know that with respect to which variety, a group enjoys a varietal covering group. In the next section, using a result of [5]we introduce the generalized Schreier variety and we will show that every group enjoys at least one covering group, with respect to such a variety. We also present some of the properties of varietal covering groups, which are a wide generalization of Read [18] and Yamazaki [21]. In final section we present some kind of varieties of groups, for which any group dose not enjoy any varietal covering groups. For more information on the concept of Bear-invariants and covering groups one may refer to [lo, 12, 141. 2. Varietal Covering Groups

C. R. Leedham-Green and S. McKay in [5] showed that any variety of groups V satisfies the following conditions: (i) subgroups of V-free groups are V-splitting; and (ii) subgroups of the V-marginal subgroup of a given group are normal, if and only if the variety V is the variety of all groups, the variety of abelian groups, or the variety of abelian groups of exponent dividing m, A, say, where m is a square free positive integer. Now we call V to be generalized Schreier variety, if it satisfies the above equivalent conditions. In this section we study the varietal covering groups of given group, with respect to the generalized Schreier variety. In 1904, I. Schur [19] introduced and proved the existence of covering groups for a finite group and then M. R. Jones [2o]generalized it to arbitrary groups, with respect to the variety of abelian groups. In [8] we have also shown that every group has an A,-covering group with respect to the variety of abelian groups of exponent dividing m, when m is a square free positive integer. One notes that in the variety of all groups every group may be considered as a covering group of itself, and so we have Theorem 2.1. I f U is the generalized Schreier variety, then eve? group enjoys a varietal covering group, with respect to such a variety.

177

Although, the existence of varietal covering groups is shown above but in the next section we will observe that, this is not true for some special varieties. In [9],we have shown that if G is a V-perfect group, then every group enjoys at least one covering group, with respect to arbitrary variety V and all such covering groups are mutually isomorphic. In other words, the existence of stem covers is shown above for the generalized Schreier variety V. Now in this section we show that every stem extension of a given group G is a homomorphic image of a stem cover of G , with respect to the variety V . To prove this result, we need the following lemmas.

-

Lemma 2.1. (Moghaddam and Salemkar [9]) Let V be a variety of groups de,fined by a set of laws V and G be a group with a free presentation 1 R F G 1. If S is a normal subgroup of F such that R nV ( F ) S --R X[RV'F] [RV'F] [RV'F] ' then G* = F / S is a V-covering group of G.

---

Lemma 2.2. (Moghaddam and Salemkar (81) Let V = A , be the variety of abelian groups of exponent dividing m, when m is a square free positive integer de.fined by a set of laws V , 1 R F 5G 1 be a free presentation of a n arbitrary group G and let 1 A H G 1 be a V-marginal extension of another group G. I f a : G G is an isomorphism, t h e n there exists a n epimorphism p : F / [ R V * F ] H such that the following diagram is commutative

- -- - -- ---f

l-1

F R --".G-1 [RV*F] [RV*F]

- - 1.~D1

10

A

H

la -

G-1

where ?i is the natural homomorphism induced by T and of P.

i s the restriction

Now we are able to prove the following theorem, which is a wide generalization of Yamazaki [21]for the variety of abelian groups.

-------

Theorem 2.2. Let 1 A H G of a group G. T h e n there is a n A,-covering a homomorphic image of G*.

1 be a n A,-stem extension group G* of G such that H is

-

Proof. Let 1 R F G 1 be a free presentation of the group G. By Lemma 2.2, there exists an epimorphism P : F / [ R V * F ] H (V= Am),such that the following diagram commutes

178

1--

1

--

R [RV*F]

-

-G-1 [RV*F]

1 P1

IP

A

H

+

-

II G-1

where P1 is the restriction of P. Put ker PI = ker P = T/[RV*F ] ,where T is a subgroup of R and T ( Rn V ( F ) )= R. Hence

R RV(F) F T T N T ( Rc1 V ( F ) ) N N-
where

T nV ( F ) [RV*F]

-

1,

is abelian. Hence

T [RV'F]

-=

where

T T --+ [RV*F] T nV ( F )

4 -

&%&.

S T nV ( F ) X[RV*F] [RV*F]'

Now we have

s n ( Rn v ( F ) ) = s n (T n v ( F ) ) = [RV*F] and

S ( R n V ( F ) )= S ( T ( R nV ( F ) )n V ( F ) )= ( R nV ( F ) ) ( S ( Tn V ( F ) ) ) = ( Rn V ( F ) ) T= R, which implies S R - R nV ( F ) X[ R V * F ] [RV'F] [ R V * F ]'

--

Thus by Lemma 2.1, F / S is a V-covering group of G. Moreover

F/S T/S which completes the proof. - N

-

F/[RV*F] 2 H, T/[RV*F]

Let G be a varietal perfect group with respect to the generalized Schreier variety V and

1-A-H-G-1,

(*)

be a V-stem extension of G. Now, the above theorem implies that if every epimorphism of an arbitrary V-stem extension of G onto the extension (*) is an isomorphism then (*) is necessarily a V-stem cover of G. In general, it is not true that any two covering groups of a given group are isomorphic (see [4]).However in the following theorem, we show that

179

up to isomorphism all V-covering groups of a given group with respect to the generalized Schreier variety V have the same marginal factor groups. This generalizes the work of Read [18], extensively.

Theorem 2.3. Let V be the generalized Schreier variety of groups and G1 and G2 be two V-covering groups of a given group G. Let

1-Ai-Gi-G-1,

i=1,2

be V-stem covers of G. Then

---

-- -

Proof. Let 1 R F G -+ 1 be a free presentation for the group G , and G* be a fixed covering group of G with respect to a given variety V. By the definition, there is an exact sequence 1 A G* G 1 such that A C V * (G * )n V ( G * )and A S Y M ( G ) . To prove the result it is sufficient to show that isomorphism class of groups V * ( G * ) / Aare uniquely determined by the presentation F I R E G. Using [9, Theorem 2.11 we may assume that G* Z F / S and A S R / S , for some normal subgroup S of F such that

-

S E'vJWG) x [RV*Fl'

j&

[RV*Fl' then clearly [LV*F]C [RV*F] S and hence L / S & V * ( F / S ) .Now, if xS E V * ( F / S )then for every v E V and = Put V*([RV*F]) F

,

f i , .. . fr

E

F , we have

U(fi,

. . . , f i a , . . . , f r ) U ( f l , . . . , fr)-'

So z [ R V* F ]E V*(&).

E

S n V ( F )= [ R V * F ] .

This implies that V * ( F / S )C L / S . Thus

V * ( F / S )= L / S . Hence V* G*

E

E

L/R. But the factor group

L / R is only determined by the free presentation F I R 2 G, and hence the result follows. C. R. Leedham-Green and S. McKay in [5] introduced the concept of Bear-invariant of a group with respect to two varieties of groups, in the following way. Let V and W be two varieties of groups defined by the set of laws V and W , respectively. Then for any group G in W , the generalized Bear - invariant of G with respect to two varieties is defined as following:

180

in which F I R % G. In this context we have introduced (see [15, 161) the generalized version of covering groups with respect to two varieties and obtained some of the generalized version of their properties. Now, one can easily observe that all the results of this section may be obtained, when W is taken to be arbitrary variety and V the generalized Schreier variety.

3. A Criterion for non-Existence Varietal Covering Groups In this section we show that a series of groups do not enjoy varietal covering groups with respect to a certain variety of groups(see Theorem 3.3). Let 24 and V be two varieties of groups defined by the set of laws U and V , respectively. Then the product U*V is the variety of all groups G such that U ( G )g V*(G).Note that U*U is indeed a variety (see [5]). The following theorem of N.S. Hekster [2], gives the relations of the marginal subgroup of the above product varieties (henceforth * -product), with respect to its components.

Theorem 3.1. Let U and V be varieties of groups and put W = U* V . Then for any group G the following hold. (a) V * ( G )5 w * ( G ) ; (b) W * ( G ) / V * ( G C ) U*(G/V*(G))C_ W * ( G / V * ( G ) ) . Although the *-product does not carry the associativity property (see [5, Example 2 on page 106 I), but the following lemma gives one way inclusion.

Lemma 3.1. (Hekster (21) Let U,V and W be varieties ofgroups. Then u * (v*w) g (U*U)*W. However, in the following theorem a sufficient condition is given, for which the *-product is associative.

Theorem 3.2. (Hekster (21) Let V and W be varieties of groups defined by the sets of laws V and W , respectively and S =V*W. Then f o r any group G, the following statements are equivalent: (a) * w S* G = V*(&); (b) [[NV*G]W*G] = [NS*G],for any normal subgroup N of G. Moreover, if (a) (and hence (b)) is satisfied, then f o r every variety U

U * ( V * W ) = (U*V)*W. Let U be the variety of groups defmed by the set of laws V, and let 7 be the variety of trivial groups .We define two series of the varieties of groups as follows : V O =Ub = 7 and for all n > 0,

181

U , = Vn-l* U and U i = U

* UiPl.

If G is a group then (V,)*(G) is the marginal subgroup of G with respect to the variety U , and V;(G) denotes the (n + 1)st term of the upper U marginal series of G with respect to the variety V. The following lemmas shorten the proof of our main theorem, considerably.

Lemma 3.2. Let U be a variety of gwups. Then for all n

i < 12,

v,

20

and 0 5

U,_i*Ui

Proof. Clearly U,*Uo =Urn =Uo*Um for each m 2 0. Now we proceed by induction on n. Hence using Lemma 3.1, for 0 i n we have

< <

( Y , - ~ * U i ) * V ~ V , - ~* (Vi*V) =V(,+,)-(,+1)*Y,+,. Clearly Vn+l =Vn+l*V0, which proves the assertion. Un+l =U,*VC

Lemma 3.3. Let V be a variety of groups. Then the following are equivalent: (a) Un+l =U,*Y, for all n >. 0. (b) Vn+, =Vm*U,, for all m,n >. 0. Proof. (a)+(b). We use induction on n, the cases n = 0 , l being obvious. Now Um+n+l CUm*Vn+l =Urn * (V,*V) C (V,*V,)*V =Yrn+,*V

(by Lemma 3.2) (by the assumption) (by Lemma 3.1) (by induction hypothesis)

=Vm+n+1. Hence the equality holds. (b)+ (a). It follows immediately by taking n = 1.

Lemma 3.4. Let U be a variety of groups. Then for all n 2 0, the following properties hold : (a) v* vn Svn+1 ; (b) v i C v n ; (c) (Vn)*(G)5 V:(G), for all groups G; In particular, if the equality holds then V, =Vi . proof. We proceed by induction on n in all cases. (a) The result being trivially true for n = 0. By Lemma 3.1 and the induction hypothesis, ws obtain

182

u*u,= u*(U,-1* u) (u*Y,_l)* u c u,*u=u,+1. (b) The result is obvious, for n = 0. Using induction hypothesis and part (a), we have UL+i= U

* U L C U * U n C Un+l.

(c) This part is also clear for n = 0. Now, Theorem 3.1 and the induction hypothesis follow that

which implies that (Vn+l)*(G)5 V,*,,(G). If the equality holds, then it follows that

So using Theorem 3.2 we obtain U* (Y,* U)=(U* Y,)* Y ,

for all n 2 0.

(*)

Now to prove U , =UL, we proceed by induction on n. The result is holds, for n = 0. By (*) and the induction hypothesis, it follows that

u;+,= u * u:,= u * u, = u * (un-l * u)= (u* u,-,) * u = u, * u = Un+1 This completes the proof of part (c). In [3], an example is constructed which shows that part (c) of the above lemma dose not hold, in general. If (Vn)*(G)= V,*(G) for all groups G and all n 2 0, then we call the variety U to be a marginal variety. Clearly, the variety of abelian groups is a marginal variety . If U is a marginal variety, then it is easily seen that for every group G and n 2 0, (V,) (G) = V, (G),in which (V,) (G)is the verbal subgroup of G with respect to the variety U , and Vn(G)is the (n+l)st term of the lower U-marginal series of G with respect to the variety U . The following main result of this section shows that, if U is a marginal variety of groups then a group G dose not necessarily have a U,-covering group. This result generalizes the work of C.R. Leedham-Green and S. McKay (1976), intensively .

Theorem 3.3. Let U be a marginal variety of groups and G be a V nilpotent group of class at most c 2 1, with non-trivial U,M(G), for n > c. Then G does not have any I/,-covering groups.

183

I n particular, i f G is a V-marginal group with non-trivial Baer-invariant with respect to the variety V,, then G does not have any covering groups with respect to V,, for all n 2 2.

-

Proof. We proceed by the way of contradiction. Let 1- A

G* - G -

1

be a V,-stem cover of the V-nilpotent group G of class at most c, where n > c . Then G * / A 2 G , A g V,(G*) nV,*(G*) , and A E V,M(G). Since G is V-nilpotent of class c, it implies that Vc(G)=< 1 > and hence Vc(G*)IA. Using the fact that A V,*(G*)we deduce that Vc+,(G+)is trivial. Thus we obtain

< 1 > = Vn+c(G*)= [Vn(G*)Vc*G*]2 [VC(,')Vc*C] = &,(G*). Now if n 2 2c then V,(G*) C I&(G*) = 1, and if n < 2c then by Lemma 3.4(c),

< 1 > = &(G*)

= [V,(G*)Vgc-,G*]

2 [V,(G*)V,*,-,G*] = V3c-n(G*).

Since n > c, we have 3c - n < 2c. Now continuing the above procedure it can be shown that V,+l(G*) =< 1 > and hence V,(G*) =< 1 >. Thus A C V,(G*) =< 1 >, V,M(G) which gives a contradiction to the non-triviality of V,M(G). Therefore G does not have any V,-covering groups, for all n > c. The following corollary is an immediate consequence of the above theorem. Corollary. Let Nc be the variety of nilpotent groups of class at most c, and G be a nilpotent group of class at most d 2 1 with non-trivial Baerinvariant AdC)(G)(c > d ) . Then G does not have any Nc-covering groups, for all c > d . In particular, if G is abelian and M(")(G)#< 1 >, for c 2 2 , then G has no Nc-covering groups.

References 1. W.K.H. Fung, 'Some theorems of Hall type', Arch. Math. 2 8 , 9-20 (1977). 2. N.S. Hekster, 'Varieties of groups and isologisms', J. Austral. Math. SOC.(Series A ) 46, 22-60 (1989). 3. J.A. Hulse and J.C. Lennox, 'Marginal series in groups', Proc. Royal SOC. Edinburgh, 76A,139-154 (1976). 4. G . Karpilovsky, 'The Schur multiplier', London Math. SOC.Monographs New Series 2, (1987).

184

5. C.R. Leedham-Green and S. McKay, ‘Baer-invariants, isologism, varietal laws and homology’, Acta Math., 137,99-150 (1976). 6. M.R.R. Moghaddam, ‘The Baer-invariant of a direct product’, Arch. Math., 33,504-511 (1979). 7. M.R.R. Moghaddam, ‘On the Schur-Baer property’, J. Austral. Math. SOC. (Series A ) , 31,43-61 (1981). 8. M.R.R. Moghaddam and A.R. Salemkar, ‘Characterization of covering and stem groups’, Comm. in Algebra, 27 ( l l ) , 5575-5586 (1999). 9. M.R.R. Moghaddam and A.R. Salemkar, ’Varietal isologism and covering groups’, Arch. Math. 75,8-15 (2000). 10. M.R.R. Moghaddam and A.R. Salemkar, ‘Some properties on isologism of groups’, J. Austral. Math. SOC.(Series A ) 68,1-9 (2000). 11. M.R.R. Moghaddam and A.R. Salemkar, ‘Schur-pair property and the structure of varietal covering groups’, Bull. Iranian Math. SOC.27 (2), 1-16 (2001). 12. M.R.R. Moghaddam, A.R. Salemkar and A. Gholami, ‘Some properties on isologism of groups and Baer-invariant’, Southeast Asian. Bull. Math. 24, 255261 (2000). 13. M.R.R. Moghaddam, A.R. Salemkar and A. Gholami, ‘Some properties on marginal extensions and the Baer-invariant of groups’, Vietnam Journal of Math. 29 ( l ) , 39-46 (2001). 14. M.R.R. Moghadda.m, A.R. Salemkar and M.M. Nasrabadi, ‘A remark on isologic extensions of groups’, to appear in Archzv der Math (Basel). 15. M.R.R. Moghaddam, A.R. Salemkar and M.R. Rismanchian, ‘Generalized covering groups’, Southeast Asian Bull. Math. 25,485-490 (2001). 16. M.R.R. Moghaddam, A.R. Salemkar and M. Taheri, ‘The Baer-invariants with respect to two varieties of groups’, Algebra Colloquium 8 (2), 145-151 (2001). 17. H. Neumann, ‘Varieties of groups’, Springer- Verlag, Berlin (1967). 18. E.W. Read, ‘On the centre of a representation group’, J . London Math. SOC. 16 (2), 43-50 (1977). 19. I. Schur, ‘Uber die darstellung der endlichen gruppen durch gebrochene lineare substitutionen’, J. Reine Angew. Math. 127,20-50 (1904). 20. J. Wiegold, ‘The Schur multiplier : An elementary approach’, Lecture Notes Series of London Math. SOC.71,137-154 (1982). 21. K. Yamazaki, ‘ Projective representations and ring extensions’, J. Fac. Sci. Univ. Tokyo, Sect. I, 10,147-195 (1964).

E-mail: [email protected]

This paper is dedicated to the memory of Professor B H Neumann

ISO-, GENO-, HYPER-MATHEMATICS AND THEIR ISODUALS CONSTRUCTED FROM OPEN PHYSICAL, CHEMICAL, AND BIOLOGICAL PROBLEMS Ruggero Maria Santilli Institute for Basic Research P. 0 . Box 1577, Palm Harbor, FL 34682, U S A . Ead d r es s [email protected],; Web Site http://www.i-b-r.org Abstract Pre-existing mathematical formulations are generally used for the treatment of new scientific problems. In this note we show that the construction of mathematical structures from open physical, chemical and biological problems leads to new intriguing mathematics of increasing complexity called iso-, geno- and hypermathematics for the treatment of matter in reversible, irreversible and multi-valued conditions, respectively, plus anti-isomorphic images called isodual mathematics for the treatment of antimatter. These novel mathematics are based on the lifting of the multiplicative unit of ordinary fields (with characteristic zero) from its traditional value $1 into: (1)invertible, Hermitean and singlevalued units for isomathematics; (2) invertible, non-Hermitean and single-valued units for genomathematics; and (3) invertible, non-Hermitean and multi-valued units for hypermathematics; with corresponding liftings of the conventional associative product and consequential lifting of all branches of mathematics admitting a (left and right) multiplicative unit. An anti-Hermitean conjugation applied to the totality of quantities and their operation of the preceding mathematics characterizes the isodual mathematics. Intriguingly, the emerging formulations preserve the abstract axioms of conventional mathematics (that based on the unit +l). As such, the new formulations result t o be new realizations of existing abstract mathematical axioms. The above mathematical advances are used for corresponding liftings of quantum mechanics into a new discipline known under the name of hadronic mechanics which has permitted basically novel advances in physics, chemistry and biology with numerous experimental verifications.

1. INTRODUCTION. Customarily, new problems in physics, chemistry, biology and other quantitative sciences are treated via pre-existing mathematics. Such a n approach is certainly valuable at the

185

186 initiation of new studies. However, with the advancement of scientific knowledge such an approach historically lead t,o serious limitations and controversies due to the insufficiency of the used mathematics for the problem at hand. The above occurrence is illustrated by the fact that the mathematics so effective for the study of planetary systems (Hamiltonian vector fields, symplectic geometry, etc.) resulted to be inadequate for the study of the atomic structure. In fact, the latter mandated the use of a new mathematics, that based on infinite dimensional Hilbert spaces over a field of complex numbers. Similar occurrences exist in contemporary science due to continued use for new scientific problems of pre-existing mathematics proved to be so effective in preceding scientific problems. This is the case for: (1) the lack of a classical formulation of antimatter, due to the inapplicability of conventional mathematics so effective for the classical treatment of matter; (2) the lack of quantitative studies of nonlocal-integral interactions as occurring in chemical valence bonds, due t o the inapplicability of conventional mathematics because of its strictly local-differential character; (3) the lack of representation of the irreversible and multi-valued nature of biological systems, due t o insufficiencies of both conventional mathematics and hypermathematics as currently formulated; and other cases. In this note we show that the construction of new mathematics from open scientific problems does indeed permit new, intriguing scientific horizons with far reaching implications in mathematics as well as quantitative science in general. As we shall see, the emerging new mathematics are based on progressive generalizations of the multiplicative unit +1 into everywhere invertible and sufficiently smooth, but otherwise arbitrary quantities (such as numbers, matrices or integro-differential operators) with corresponding generalizations of the associative product, thus implying corresponding generalizations of all branches of conventional mathematics (hereinafter defined as the mathematics based on the multiplicative left and right unit +1 over a field of characteristic zero). The reader should be aware that the literature in the topic of this note is rather vast because it encompasses numerous studies in pure mathematics, applications in various quantitative sciences, several experimental verifications as well as rapidly expanding new industrial applications. As a result, in this note we can only review the most fundamental aspects of the new formulations. A technical knowledge of the new advances can only be achieved via the study of the quoted literature. To avoid a prohibitive length, references have been restricted to contributions specifically based on the lifting of the unit with a compatible lifting of the product. Regrettably, we have t o defer to the specialized literature the treatment of numerous connections with other studies. References are grouped by main fields indicated with square brackets (e.g., [5]),while individual references are indicated with curved brackets (e.g., (201)). Except for monographs, proceedings and reprint volumes, the titles of the individual contributions are not provide to avoid a prohibitive length, as well as because of their lack of general availability in the physics literature without an extensive library search. The reader should be aware that the new mathematics and their applications are still in their initial stages and so much remains to be done. The author would be particularly grateful for the indication by interested colleagues of mathematical or or other references

in the origination of the new formulations which have escaped his knowledge, as well as for any comment.

2. CONSTRUCTION OF ISODUAL MATHEMATICS FROM CLASSICAL ANTIMATTER. One of the biggest scientific unbalances of the 20-th century has been the treatment of matter at all possible levels, from Newtonian to quantum mechanics, while antimatter was solely treated at the level of second quantization. In particular, the lack of a consistent classical treatment of antimatter left fundamental open problems, such as the inability to study whether a far away galaxy or quasar is made up of matter or of antimatter. It should be indicated that classical studies of antimatter simply cannot be done by merely reversing the sign of the charge because of inconsistencies due to the existence of only one quantization channel. In fact, the quantization of a classical particle with the reversed sign of the charge leads t o a particle (rather than a charge conjugated antiparticle) with the wrong sign of the charge. The origin of this scientific unbalance was not of physical nature, and was instead due to the lack of a mathematics suitable f o r the classical treatment of antimatter in such a way to be compatible with charge conjugation at the quantum level. In fact, charge conjugation is an anti-homomorphism. Therefore, a necessary condition for a mathematics to be applicable for the classical treatment of antimatter is that of being anti-homomorphic, or, better, anti-isomorphic to conventional mathematics. The absence of the needed mathematics is clearly seen by noting that classical treatments of antimatter require fields, functional analysis, differential calculus, topology, geometries, algebras, groups, etc. which are anti-isomorphic t o conventional formulations. The absence in the mathematics of the 20-th century of a formulation of numbers, trigonometric functions, derivatives, etc. which are anti-isomorphic to conventional expressions then mandated the construction of the needed new mathematics as requested by the physical reality here considered (rather than adapting physical reality to pre-existing mathematics). A novel mathematics verifying the above conditions was proposed by R. M. Santilli in Ref. (11)of 1985 and then developed in various works (see Refs.(l2, 14,15, 54,55,160,161) and [3]). The fundamental idea is the assumption of a negative-definite, left and right multiplicative unit, called isodual unit, and denoted I d , where I denotes the conventional positive-definite unit, I > 0, I d = -I < 0, (2.1) with corresponding reformulation of the conventional associative product A x B among generic quantities A, B (such as numbers, vector fields, operators, etc.) into the form

A x d B = A x (Id)-' x B ,

(2.2)

under which I d is the correct left and right multiplicative unit of the theory, A X ~ I=~I

V ~ A = A,

187

(2.3)

188

for all elements A of the considered set. More generally, isodual mathematics is given b y the image of a given mathematics admitting a left and right multiplicative unit under the following isodual map

A ( z ,...) + A d ( z d...) ,

= -At(-zt, ...).

(2.4)

when applied t o the totality of conventional quantities and their operations, with no exception of any type. In this note we cannot possibly review the entire isodual mathematics, and must restrict ourselves to an elementary review of only the foundations.

+,

DEFINITION 2.1: Let F = F ( a , x ) be a field of characteristic zero representing real numbers F = R(n, x), a = n, complex numbers F = C(c, x), a = c, or quaternionic numbers F = Q ( q , x), a = q, with conventional associative, distributive and commutative sum a b = c E F , associative and distributive product a x b = c E F , left and right additive unit 0, a 0 = 0 a = a E F , and left and right multiplicative unit I > 0, a x I = I x a = a,Va, b E F . The isodual fields (first introduced in Refs. (11,12)) are rings F d = Fd(ad,+d,x d ) with isodual numbers ad = -at, associative, distributive and commutative isodual sum ad +d bd = -(a + b)t = cd E F d , associative and distributive isodual product ad x d bd =ad x (Id)-' x bd = d E F d , additive isodual unit Od = 0 , ad +d Od = Od f d ad = ad, and isodual multiplicative unit I d = - I t , ad x d I d = I d x d ad = ad,Vad,bd E F d .

+,

+

+, +

+,

+

LEMMA 2.1 (12): Isodual fields are fields (namely, isodual field verify all axioms of a field with characteristic zero). The above lemma establishes the property (first identified in Refs. (11,12)) that the axioms of a field d o not require that the multiplicative unit be necessary positive-definite, because it can also be negative-definite. The proof of the following property is equally simple. LEMMA 2.2 (12): Fields (of characteristic zero) and their isodual images are antiisomorphic t o each other. Lemmas 2.1 and 2.2 illustrate the origin of the name "isodual mathematics." In fact, the needed mathematics must constitute a "dual" image of conventional mathematics, while the prefix "iso" is used in its Greek meaning of preserving the original axioms. It is evident that for real numbers we have nd = -12, while for complex numbers we have cd = (n1+ i x nz)d= -nl + i x n 2 = -?, with a similar formulation for quaternions.

-.

DEFINITION 2.2 (12): A quantity is called isoselfdual when it coincides with its isodual. = i. As It is easy t o verify that the imaginary unit is isoselfdual because id = we shall see, isoselfduality is a new symmetry with rather profound implications, e.g., in

189

cosmology. It is evident that, for consistency, all operations of numbers must be subjected to isodzlality. This implies: the isodual powers (ad))"d= ad x d ad x d ad... (n times with n an - ad, integer); the isodual square root ad("2)d = aJ('/2)dx d - 2 ; the isodual quotient adldbd = -(at/bt) = cd, bd x d cd = a d ; etc. An important property for the characterization of antimatter is that isodual fields have a negative-definzte norm, called isodual norm (12)

-&8,

ladld

=

latl x

zd = -(sat)"*

< 0,

(2.5)

where I. . ) denotes the conventional norm. For isodual real numbers nd we have the isodual norm = -In1 < 0 , the isodual norm for for isodual complex numbers (&Id = -(nf + n;)'/', etc. Recall that functional analysis is defined over a field. Therefore, the lifting of fields into isodual fields requires, for necessary condition of consistency, the formulation of the isodual cosdOd = functional analysis (54). We here merely recall that sindOd = -sin(-O), -cos(-6), with related basic property cosdZdBd+d sindZdOd = I d = -1; the isodual hyperbolic functions sinhdwd = - sinh(-w), coshd wd = - cosh(-w), with related basic property coshdZdw d -d sinhdZdwd = Id = -1; the isodual logarithm logdnd = -log( -n). Particularly important is the isodual eqonentiatzon which can be written edAd

=

I d + Ad/dl!d + Ad x d Ad/d2!d + ... = -eAt

,

(2.6)

Other properties of the isodual functional analysis can be easily derived by the interested reader (see also Refs. (14,21,22). It is little known that the differential and integral calculi are indeed dependent on the assumed basic unit. In fact, the lifting of I into I d and of F into F d implies the isodual dijjeerential calculus, first introduced in Ref. (14), which is characterized by the isodual differentials ddxd = dx with corresponding isodual derivatives adfd(xd))ld@xd= -%)/a(- 2 ) and other isodual properties interested readers can easily derive. Note that the differential is isoselfdual. Conventional vector and metric spaces are defined over a field. It is then evident that the isoduality of fields requires, for consistency, a corresponding isoduality of vector, metric and all other) spaces.

-af(

DEFINITION 2.3: Let S = S ( x ,g , R ) be an N-dimensional metric space with realvalued local coordinates z = {%'}, k = 1,2, ..., N , nowhere degenerate, sufficiently smooth, real-valued and symmetric metric g(z, ...) and related line element x2 = (dx g x z) x I = (xix gi3 x xj) x I E R. The isodual spaces, first introduced in Refs. (11,14), are vector spaces S d ( x dgd, , Rd) with isodual coordinates xd = -xt where t denotes transposed, isodual metric gd(zd,...) = -gt(-zt, ...), and isodual line element

190 The isodual Euclidean space E d ( x d6d, , R d )is a particular case of Sd when gt", = 6;. The isodual distance on Ed is negative definite and it is given by Dd = -D, where D is the conventional (positive-definite) distance on E. The isodual sphere on a 3-dimensional isodual space Ed is the perfect sphere with negative radius and expression rdZd= [(z$"+~ x;d +d xZd] x d I d = -r2 E Rd. The isodual Minkowskian, isodual Riemannian and isodual symplectic geometries can be defined accordingly (14,15). iFrom the above rudiments interested readers can construct the rest of the isodual mathematics, including: isodual topologies, isodual manifolds, etc. Particularly important for physical applications is the isodual Lie theory (first introduced in Ref. (11) (see also (14,22)), including isodual universal enveloping associative algebras, isodual Lie algebras, isodual Lie groups, isodual symmetries, and isodual representation theory, which we cannot review here for brevity. The main physical theories characterized by isodual mathematics can be outlined as follows. To resolve the scientific unbalance between matter and antimatter indicated earlier, the isodual mathematics has first permitted a Newtonian characterization of antimatter consistent with all available experimental data (14,22). Then, isodual mathematics has identified a new quantization channel (which is distinct from conventional symplectic quantization) leading t o an operator formulation which is equivalent t o charge conjugation (14,16,21). We first have the isodual Newton equations md X d

ddVd d d d d ddxd = F (t , X ,v),v"-; ddtd ddtd

~

with analytic representation via the isodual action functional (14) and the isodual Hamilton equations (14) (for the case when the Newtonian force F is representable with a potential) ddxd - d d H d d p d -__ ddHd ddtd ddpd ddtd - a d z d I)

Interested readers are encouraged t o verify that the above classical formulation does indeed provide a correct representation of all available experimental data on antimatter (22), such as the Coulomb attraction (repulsion) between charged matter and antimatter (charged antimatter and antimatter). The isoduality of the naive or symplectic quantization is elementary (14,16,21,55), leading in this way to the novel isodual quantum mechanics (14,21) defined on an isodual Hilbert space N d over the isodual field Cd with basic isodual Schroedinger and Heisenberg equations (for a Hermitean Hamiltonians H and observable A)

(H x

I$

=

>)d

id X d

-

< $1 x d H d = - < $1 x H

=

(Ex

I$

>)d

=

-

< $1

ddAd = [A,HId = -dHd x d Ad +d Ad x d H d = -[A, HI. ddtd

~

x

E, (2.10)

191

The equivalence of the above operator formulation with charge conjugation has been proved in Refs. (16,21), thus establishing the compatibility of the isodual theory of antimatter with available experimental data at the operator level too. Despite its simplicity, the physical, astrophysical and cosmological implications of isodual mathematics are rather deep. To begin, the isodual map (2.4) implies the change of the sign not only of the charge, but also of all other physical quantities of matter,including mass, energy, time, etc. For instance, the energy eigenvalue E of Eqs. (2.9) has negative values since it is positive in Eq. (9), yet computed on Rd. Note that the measurement of physical quantities with respect t o negative definite units resolves the traditional inconsistencies for negative mass and energy. In particular, the isodual theory of antimatter recovers the old hypothesis that antiparticles move backward in time (since they have a negative-definite time) by resolving the inherent violation of causality which lead t o its abandonment in the second half of the 20-th century. In fact, motion backward in time measured with respect to a negative unit of time is as causal as the conventional motion forward in time referred to a positive unit of tame. Note also that charge conjugation is an anti-homomorphic map, while isoduality is an anti-isomorphic map. Therefore, according to quantum mechanics, antiparticles exist in the same spacetime of particles, while, according to isodual quantum mechanics, antiparticles exist in the isodual spacetime which coexists with, yet it is independent from our spacetime. Most importantly, the isodual theory of antimatter mandates the existence of antigravity defined as a gravitational repulsion experienced b y antimatter in the field of matter and vice-versa (16,21,22), while resolving the historical objections against antigravity. As an illustration, the creation of an electron-positron pair is represented by an isoselfdual state on ‘Ft x Ed over C x C d ,whose eigenvalues can only be defined over the field of the observer (that is, R for a matte observer and Rd for an antimatter observer), as studied in detail in Refs. (21,60). Such an occurrence prevents configurations of correlated electronpositron systems which, in the case of antigravity without the isodual mathematics, could violate the principle of conservation of the energy (e.g., by obtained a blueshift without an applied force). Other objections against antigravity for antimatter-matter systems are resolved in similar ways. J. P. Mills has shown in Ref. (160) that the experimental verification of antigravity proposed in Ref. (17) can be conducted in a resolutory form with available technology. The proposed experiment essentially consists of a horizontal vacuum tube of about one meter in diameter and ten meters in length with internal collimators at one end and a scintillator at the other end. The release of photons through the collimators would establish on the scintillator the point of no gravity; the release of electrons would show at the scintillator a downward shift due t o gravity ; and the release of positrons would show an upward or downward shift on the scintillator depending on whether they arc experiencing antigravity or gravity, respectively. The experiment would be resolutory because, when the electrons and positrons have very small energy (of the order of electron Volts), the downward or upward shift of their impact on the scintillator would be visible

192

t o the naked eye. This low energy experiment has been ignored by experimentalists in favor of other high energy experiments due to the inconsistencies of the prediction of antigravity when treated with conventional mathematics. It is hoped that, in view of the resolution of these inconsistencies thanks t o isodual mathematics, experimentalists will reconsider their view and conduct indeed such a fundamental experiment. Above all, isodual mathematics has fulfilled the primary scope for which it was constructed, the initiation of quantitative classical studies as t o whether far away galaxies and quasars are made-up of matter or of antimatter. In fact, isodual mathematics predicts that antimatter emits a new photon, called the isodual ;photon (2l), which has experimentally detectable characteristics different than those of the ordinary photon emitted by matter, e.g., the isodual photon is repelled by the gravitational field of our Earth, and has a parity different than that of the ordinary photon. The experimental resolution on Earth whether light from a far away galaxy or quasar is made up of photons or of isodual photons would then resolve the open cosmological problem whether the universe is only made up of matter, or antimatter galaxies and quasars are equally present. An important application of the isodual theory of antimatter has been developed by J. Dunning Davies (161) who has developed the first (and only) known thermodynamics for antimatter stars. In this way, quantitative cosmological studies on the antimatter component of the universe are already under way. On historical grounds, we should note that the isodual mathematics and related theory of antimatter originated from an inspection of the celebrated Dirac equation (6). In fact, its basic unit is given by "i0 = Diag.(Iz,z, - I z X z ) , thus exhibiting the negative-definite unit I&, = - I z x z precisely for the antimatter component of the equation. Intriguingly, Dirac's gamma matrices turned out to be isoselfdual (Definition 2.2), thus implying a novel reformulation of the equation as representing the direct product of an electron and a positron (60). Regrettably, Dirac was unaware of the fact that a negative unit can indeed be the correct unit of an appropriate mathematics and, as a consequence, he developed the "hole theory" restricting the treatment of antiparticles t o the sole level of second quantization. Such a restriction has been lifted by isodual mathematics because it permits the formulation of antiparticles not only in second quantization, but also in in first quantization as well as at the classical level.

3. CONSTRUCTION OF ISOMATHEMATICS FROM NONLOCAL INTERACTIONS. Another large scientific unbalance of the 20-th century has been the adaptation of generally nonlocal-integral systems t o preexisting mathematics which is notoriously localdifferential, with consequential serious limitations despite attempts conducted over three quarter of a century, such as: the lack of numerically exact representations of chemical valence bonds in molecular structures; the equally historical inability t o achieve an exact representation of nuclear magnetic moments; and other unresolved problems. In all these cases we have the mutual overlapping/penetration of particles and/or their wavepackets at distances of the order of 10-13cm, which conditions are strictly nonlocal-integral (for theoretical and experimental studies on nonlocality, see, e.g., C. A. C. Dreismann

193 (220-223) and references quoted therein). Quantum mechanics and its underlying conventional mathematics permitted a numerically exact representation of all experimental data of the Hydrogen atom. By contrast, the same mathematics and related discipline have not permitted an equally exact representation of the experimental data of the Hydrogen molecule, since a historical 2% of molecular binding energy has been missed for about one century under the rigorous applicability of quantum axioms (thus excluding screenings of the Coulomb law which imply noncanonical/nonunitary transforms, thus exiting the class of equivalence of the original theory). Since the sole difference between one isolated Hydrogen atom and two atoms coupled into the Hydrogen molecule is given by the electron valence bonds, the above occurrence illustrates the exact validity of quantum mechanics and related mathematics when the systems can be effectively appropriate as point particles at sufficiently large mutual distances (as it is the case for the structure of the Hydrogen atom), while the same theory and related mathematics have a merely approximate character when systems contain interactions at short distances (as it is the case for the mutual overlapping of the wavepackets of valence electrons with antiparallel spins). The insufficiency is due t o the fact that conventional mathematics is 1ocal-di;fferential in its structure, thus solely permitting the representation of valence bonds as occurring between point particles interacting at a distance. This representation is evidently valid in first approximation because electrons have indeed a point charge. Nevertheless, such a local-differential representation is insufficient because of the lack of treatment of the mutual penetration of the electron wavepackets which cannot be consistently reduced t o a finite set of isolated points. It is then evident that a more adequate treatment of valence bonds in chemistry, as well as all nonlocal-integral interactions in general, requires a new mathematics which is partly local-differential (to represent conventional Coulomb interactions) and partly nonlocal-integral (to represent the overlapping of the wavepackets). Additional physical requirements establish that the needed mathematics must be nonlinear in the wawefunction, thus preventing the use of conventional quantum mechanics because nonlinear quantum theories violate the superposition principle with consequential inapplicability to composite systems (45,46). Moreover, the nonlocal-integral and nonlinear interactions of valence bonds are also nonpotential, i.e., not representable with an additive potential V in a Hamiltonian H = p 2 / 2 m + V . In fact, the interactions here considered are of contact and zero-range type, thus preventing their representation via action-at-a-distance potentials in a Hamiltonian. Finally, t o be meaningful in applications, the needed mathematics and related physical theories must be time-invariant, i.e., their mathematical avions and numerical predictions must remain the same under the action of the one-dimensional group representing time evolution. This latter requirement illustrates the difficulties of the task at hand. In fact, a necessary condition for a formulation t o be more general than a Hamiltoniari one is that of being noncanonical at the classical level and nonunitary at the operator level. Recent studies have established the following

194

THEOREM 3.1 (45,46): All formulations with classical noncanonical and operator nonunitary time evolutions do not leave time-invariant the basic units, thus lacking timeinvariant mathematical axioms and numerical predictions, and, therefore, haave no known mathematical or physical value. Recall that theories with a (Hermitean) Hamiltonian H = p 2 / 2 m + V = Ht defined on a Hilbert space 'H. over the field of complex number C possess a unitary time evolution U ( t ) x I$(&) >= I$@) >,U x Ut = U f x U = I , U ( t ) = e i X H x t ,which, as such, leaves invariant all numerical predictsions. For instance, if H(t,) x I$(to)>= 5eV x I$,(&,) >, the value 5 eV is preserved at all subsequent times, U ( t ) x H(t,) x I$(t,) >= (U x H x Ut) x (U x Ut)-l x (U x I$ >) = H (t) x I$(t) >= U ( t ) x (5eV x I$(&,) >) = 5eV x I$(t) > . Recall that the main features of the interactions herein considered is that of being nonlocal, nonlinear and nonpotential, thus representable with anything except the Hamiltonian (to avoid granting potential energy to contact interactions which have none and for other reasons). Therefore, a necessary condition for the representation of the interactions here considered is that their operator time evolution is nonunitary, i.e., U ( t ) x Iqh(to)>= I$(t) >,U x Ut f I , U ( t ) # e i X H X. tIt is then easy to see that, when the nonunitary theory is formulated via the mathematics of unitary theories, it does not admzt time-invariant numerical predictions, thus hawing no known physical meaning or value (46). To illustrate this important occurrence, supposed that the nonunitary theory is such that at the initial time we have H ( t o )x I$(&) >= 5eV x [$(to)> and that 15 seconds later (U x Ut)t=lSsec= 1/3. We then have U ( t )x H(t,) x I$(to)>= (U x H x Ut) x (U x Ut)-' x (U x I$ >) = U ( t ) x (5eV x I$(to) >), namely, H ( t ) x I$(t) >= (U x U t ) x ( s e V l $ ( t ) > ) = 15eV x I$(t) >. The lack of preservation in time of numerical predictions then implies consequences today known as "catastrophic physical inconsistencies," such as: the lack of applicability of the theory to measurements (because of the loss of invariant basic units); Lopez's Lemma (172,173) (loss of Hermiticity in time with consequential lack of acceptable observables); violation of causality and probability laws; etc. (45,46,171-175) The use of conventional mathematics for broader nonunitary theories leads to equally serious mathematical problems today known as "catastrophic mathematical inconsistencies." Suppose that a nonunitary theories is formulated at a given initial time t, on a metric space defined over the field of real numbers R with basic unit I. But, by their very definition, nonunitary time evolutions do not leave invariant the basic unit, I + U x I x Ut # I . Therefore, nonunitary time evolutions do not admit the original basic unit I at all times t > to, with consequential loss of the base field R. In turn, the loss of the base field implies the inability t o properly define the metric space acting on it, with consequential catastrophic collapse of the entire mathematical structure. Classical noncanonical theories are afflicted by similar catastrophic mathematical and physical inconsistencies (45,46,171-175). It should be recalled that, when facing non-Hamiltonian systems, a rather general tendency is that of transforming such systems into a form which is (at least locally) Hamiltonian via the use of Darboux's theorem of the symplectic geometry or the Lie-Koening

195 theorem of analytic mechanics (51). Unfortunately, these transformations cannot be used for the study of valence bonds because: (i) the system considered are nonlocal (thus implying the inapplicability a priori of the topologies needed for the quoted theorems); (ii) Darboux’s transformations are nonlinear, thus implying the impossibility of placing measuring apparata in the new Darboux’s coordinates; and (iii) in view of their nonlinearity, Darboux’s transformations cause the loss of the inertial character of reference frames with consequential loss of Galileo’s and Einstein’s relativities. In view of these shortcomings, the only physically acceptable representations are those occurring in the fixed coordinates of the observer, called direct representations (51). Only after such a representation is consistently achieved the transformation theory may have value. In summary, the representation of chemical bonds as well as other nonlocal interactions at short distances requires the abandonment of Hamiltonian theories which, in turn, imply the necessary use of theories whose time evolution is noncanonical at the classical level and nonunitary at the operator level. Still in turn, such noncanonical/nonunitary theories require the necessary construction of a new mathematics capable of resolving the catastrophic inconsistencies reviewed above. The new mathematics specifically constructed for the above needs is today known under the name of isomathematics, (where the prefix ”iso” also denotes the preservation of conventional axioms), as first proposed by R. M. Santilli in Ref. ( 2 3 ) of 1978 and subsequently studied by several mathematicians, theoreticians and experimentalists (see Refs. [2,4-111). The main idea is that, as indicated earlier, valence bonds include conventional localdifferential Coulomb interactions, plus nonlocal, nonlinear and nonpotential interactions due to wave-overlappings. The former interactions can be effectively represented with the conventional Hamiltonian, while the latter interactions can be represented via a generalization of the basic unit as a condition to achieve invariance (since the unit is the basic invariant of any theory). Therefore, the main assumption of isomathematics is the lifting of the conventional unit of current formulations, generally given by an N-dimensional unit matrix I = Diag. (1, 1, ..., 1) > 0 , into a quantity i called isounit which possesses all topological properties of I (such a positive-definiteness, same dimensionality, etc.), while having an arbitrary functional dependence on coordinates x, velocities v, wavefunctions $, their derivatives and any other needed variable (23,51),

a=$,

I = D i a g . ( l , l ,..., l ) > O + j ( z , v , $ , & $

,...) = l / ~ ( ~ , v , $ , d ,... , $) > O .

(3.1)

The conventional associative and distributive product A x B among generic quantities A, B (such as numbers, vector fields, operators, etc.) is jointly lifted into the more general form A x B -+ A ; ~ B = A x T x B, (3.2) which remains associative and distributive, thus being called isoproduct, under which the left and right character of I is preserved, i.e.,

IxA= A x I

=A

--t

i ; =~~

i ; f =A ,

(3.3).

196 for all elements A of the set considered. An example of isounits (hereon assumed t o be multiplicative) representing the interactions due t o wave-overlappings of valence bonds is given by

and illustrates the desired representation of nonlocal, nonlinear and nonpotential interactions. Another example of isounits is given by I = Diag.(nl,ni,n:), and illustrates the representation of the actual, extended, generally nonspherical and deformable shape of the particle considered, in this case, a spheroidal ellipsoid. Numerous additional examples of isounits exist in the literature [4-111. Note that the features represented by the isounits are strictly outside any representational capability by the Hamiltonian.

+,

DEFINITION 3.1: Let F = F ( a , X ) be a field as per Definition 2.1. The isofields, first introduced in Ref. (23) of 1978 (see Ref. (12) for a mathematical treatment) are i, k ) whose elements are the isonumbers ti a x i,with associative, rings F = p(&, distributive and commutative iso:um ti$bA= ( a b) x I = 2 E F, associative and distributive isoproduct & k b = ti x T x b = 2 E F, additive isounit 0 = 0, t i l o = O h = ti, and multiplicative isounit i = 1/T > 0, iikf = f k t i = &,V&,i E F, where is notAnecessarily an element of F. Isofields are called of the first (second) kind when I = 1/T > 0 is (is not,) an element of F .

+

LEMMA 3.1: Isofields of first and second kind are fields (namely, isofields verify all axioms of a field with characteristic zero). The above property establishes the fact (first identified in Ref. (12) that, by no means, the axioms of a field require that the multiplicative unit be the trivial unit +1, because it can be a negative-definite quantity as for the isodual mathematics, as well as an arbitrary positive-definite quantity, such as a matrix or an integrodifferential operator. Needless t o say, the liftings of the unit and of the product imply a corresponding lifting of all conventional operations of a field. In fact, we have the isopowers = &:ti2 ..., kti (n times) = an x i with particular case i” = i;the isosquare root till2 = all2 x i l l 2 ; the isoquotient 276 = (&/;) x i = ( a / b ) x i; the isonorm 161 = la/ x i, where la1 is the conventional norm; etc. Despite their simplicity, the above liftings imply a complete generalization of the conventional number theory particularly for the case of the first kind (in which f E F ) with implications for all aspects of the theory. As an illustration, the use of the isounit i = 1/3 implies that ” 2 multiplied by 3” = 18, while 4 becomes a prime number. An important contribution has been made by E. Trell (143) who has achieved a proof of Fermat’s Last Theorem via the use of isonumbers, thus achieving a proof which is sufficiently simple to be of Fermat’s time. . A comprehensive study of Santilli’s isonumber theory of both first and second kind has been conducted by C.-X. Jiang in monograph (68) with numerous novel developments and applications. Additional studies on isonumbers ^

^

197

have been done by N. Kamiya et al. (156) and others (see mathematical papers [lo] and proceedings [8]). The lifting of fields into isofields implies a corresponding lifting of functional analysis into a form known as isofunctional analysis studied by J. V. Kadeisvili (132-133), A. K. Aringazin et al. (144) and other authors. A review of isofunctional analysis up to 1995 with various developments has been provided by R. M. Santilli in monographs (54,55). We here merely recall the isofunctions f(i) = f ( x x I ) x I ; the isologarithm log,a = I x log,a, log,; = I , log,I = 0; the isoexponentiation *

^

^

The conventional differential calculus must also be lifted, for consistency, into the isodifferential calculus first identified by R. M. Santilli in memoir (14) of 1996, with isodifferential di = T x dP = T x d ( x x f) which, for the case when f does not depend on x, reduces t o dP = d x ; isoderivatives 8 j ( k ) / a k= f x [i3f(i)/di],Aandother similar properties. The indicated invariance of the differential under isotopy, dZ = d x , illustrates the reason why the isodifferential calculus has remained undetected since Newton's and Leibnitz's times. Particularly important for these notes are the isotopy of the Euclidean topology independently identified by G. T. Tsagas and D. S. Sourlas (139) and R. M. Santilli (14), as well as the isotopies of the Euclidean, Minkowskian, Riemannian and symplectic geometries first identified by Santilli in various works (see Refs. (14,15,29,54,55). We cannot possibly review here these advances for brevity. We merely mention the invariance of the fundamental symplectic f o r m under isotopies, w = dp A d r = o$Ad? = & due to the invariance of the differentials under isotopies as well as the fact that certain geometric reasons require the isounit of the variable p in the cotangent bundle (phase space) to be the inverse that of x (i.e., when I = l/T is the isounit for x, that for p is T = l / f ) . These invariances provide a reason why the isotopies of the symplectic geometry escaped identification for about one century. Despite their simplicity, these isotopies have vast implications, e.g., the identification of a broader quantization channel leading to a structural generalization of quantum mechanics known as hadronic mechanics, as outlined below. As well known, Lie's theory (4) is based on the conventional (left and right) unit I = Diag. (1, 1, ..., 1) of the universal enveloping associative algebra. The lifting I + I ( x ,...) implies the lifting of the entire Lie theory, first proposed by R. M. Santilli in Ref. (23) of 1978 and then studied in numerous works (see, e.g., memoir (14) and monographs (51,54,55)). The isotopies of Lie's theory are today known as the Lie-Santilli isotheory following studies by numerous mathematicians and physicists (see the monographs by D. S. Sourlas and Gr. Tsagas (64), J. V. Kadeisvili (66), R. M. Falcon Ganfornina and J. Nunez Valdez (67), proceedings [8], and contributions quoted therein). Let ( ( L ) be the universal enveloping associative algebra of an N-dimensional Lie algebra L with (Hermitean) generators X = (Xi), i = l , 2, ..., N, and corresponding Lie transformation group G over the reals R. The Lie-Santil!i isotheory is characterized by: (I) The universal enveloping isoassociative algebra ( with infinite-dimensional basis

198

characterizing the Poinca&Birkhoff- Witt-Santilli isotheorem

i : i,Xi, XiG X j , Xi A X j

GXh,

_ _i_5, j 5 IC;

(3.6)

where the ”hat” on the generators denotes their formulation on isospaces over isofields; (11) The Lie-Santilli isoalgebras

i

%

ti)- : [Xi;Xj]= Xi2Xj -@Xi

=c;xXk;

(3.7)

(111) The Lie-Santilli i s o t r a n s f o n a t i o n groups

G : $8)= ( ~ ~ ~ x ~ ~ ) ~ ) A (eixRxFxW ( o ) ~) ( A(O) ~ - ~( e - i ~X W X~? X x ~ )x> ~

(3.8)

where 8 E R are the isoparameters; the isorepresentation theory; etc. The non-triviality of the above liftings is expressed by the appearance of the isotopic element f ( z , ...)- at all levels (I), (11) and (111) of the isotheory. The arbitrary functional dependence of T ( z ,...) then implies the achievement of the desired main features of the isotheory which can be expressed by the following: LEMMA 3.2 (14): Lie-Santilli isoalgebras on conventional spaces over conventional fields are generally nonlocal, nonlinear and noncanonical, but they verify locality, linearity and canonicity when formulated on isospaces over isofields. To illustrate the Lie-Santilli isotheory in the operator case, consider the eigenvalue equation on 32 over C, H ( z , p , $ , ...) x I$ >= E x I$ >. This equation is nonlinear in the wavefunction, thus violating the superposition principle and preventing the study of composite nonlinear systems, as indicated earlier. However, under the factorization H ( z , p , $, ...) = H ’ ( z , p ) x f ‘ ( z , p ,$, ...), the above equation can be reformulated identically in the isotopic form H ( z , p , $ , ...) x Isi >= H ’ ( z , p ) x T ( s , p , $ , ...) x I$ >= H’xl$>= E x I$ >= Ex14 > whose reconstruction of linearity on isospaces over isofields (called isolinean’ty (14)) is evident and so is the verification of the isosuperposition principle with resulting applicability of isolinear theories for the study of composite nonlinear systems. Similar results occur for the reconstruction on isospace over isofields of locality (called isolocality) and canonicity (called zsocanonicity). A main role of the isotheory is then expressed by the following property: LEMMA 3.3 (29): Under the condition that i is positive-definite, isotopic algebras and groups are locally isomorphic t o the conventional algebras and groups, respectively. Stated in different terms, the Lie-Santilli isotheory was not constructed t o characterize new Lie algebras, because all Lie algebras over a field of characteristic zero are known. On the contrary, the Lie-Santilli isotheory has been built to characterize n e w realizations of known Lie algebras generally of nonlinear, nonlocal and noncanonical character as needed for a deeper representation of valence bonds or, more generally, systems with nonlinear, nonlocal and noncanonical interactions.

199

The mathematical implications of the Lie-Santilli isotheory are significant. For instance, Gr. Tsagas (142) has shown that all simple non-exceptional Lie algebras of dimension N can be unified into one single Lie-Santilli isotope of the same dimension, while studies for the inclusion iof exceptional algebras in this grand unification of Lie theory are under way In fact, the characterization of different simple Lie algebras, including the transition from compact t o noncompact Lie algebras, can be characterized by different realizations of the isounit while using a unique f o r m of generators and of structure constants (see the first examples for the SO(3) algebra in Ref. (23) of 1978 and numerous others in the quoted literature). The physical implications of the Lie-Santilli isotheory are equally significant. We here mention the reconstruction as exact at the isotopic level of Lie symmetries when believed to be broken under conventional treatment. In fact, R. M. Santilli has proved: the exact reconstruction of the rotational symmetry for all ellipsoidical deformations of the sphere (12); the exact SU(2)-isospin symmetry under electromagnetic interactions (28,33); the exact Lorentz symmetry under all (sufficiently smooth) signature-preserving deformations of the Minkowski metric (26); and the exact reconstruction of parity under weak interactions (55). R. Mignani (180) has studied the exact reconstruction of the SU(3) symmetry under various symmetry-breaking terms. In all these cases the reconstruction of the exact symmetry has been achieved by merely embedding all symmetry breaking terms in the isounit. The main physical theories characterized by isomathematics are given by: (i) The isonewtonian mechanics based on the equations (first achieved by R. M. Santilli in Ref. (14) of 1996 as the first application of the isodifferential calculus reached in the same memoir) ~

A

d.i,

mx, =mx dt

d[v x f ( t , x , u , .)I dt

~

~

x I= F=

dV x I, dX ~

--

(3.9)

where t^ E Rt, and j. E R,, .i, E &. To our best knowledge, the above equation constitutes the first structural generalization of Newton’s equation since Newton’s times permitting the representation of extended, nononspherical and deformable particles under local and nonlocal, linear and nonlinear and potential as well as nonpotential interactions. In fact, the assumption of the isounit I = Diag.(n?, ni, n$)r(t, 2 , u , ...) in the isoderivative ^d.i,/& = d^(ux f)/& permits the representation of the actual shape of the particle considered as well as all possible nonpotential forces, thus permitting the force in the r.h.s. of Eq. (3.9) to be of potential type. Note the necessity for the achievement of these results of the prior lifting of the unit, numbers, differential calculus), topology and the entire mathematics of Newton’s equations. As a simple illustration, the assumption of the isounit f = eYt,y E R, permits the direct representation of the equation with resistive force m x a = -y x v. Sufficiently smooth, but otherwise arbitrary nonpotential Newtonian forces can be represented via a suitable identification of the iosounit (see Ref. (14) for universality theorems and otehr examples).

200 (ii) The incorporation of all nonpotential forces in the isounit then permits the construction of the isohamiltonian mechanics (14), based on the action isofunctional here omitted for brevity and the isohamilton equations

whose primary property is that of formally coinciding with the conventional Hamilton equations. Eqs. (3.10) are directly universal f o r all suficiently smooth, closed-isolated, discrete systems with linear and nonlinear, local and nonlocal and potential was well nonpotential interactions, in the sense of representing all the systems of the class admitted (universality) in the fixed frame of the experimenter (direct universality). (iii) The formal identity of isohamilton equations (3.10) with the conventional equations permits a simple isotopy of symplectic quantization, by leading in this way t o the isotopic branch of hadronic mechanics defined over a isohilbert space (25) 'FI with isoinner product < $lklG > x i over the isofield C (see memoir (31) for a review), and characterized by the isoschroedinger equations (first derived in Refs. (25,179) with ordinary mathematics and first formulated vis the isodifferential calculus in Ref. (14), where A = 1)

and the isoheisenberg equations (first derived in Ref. (38 via conventional mathematics and first formu1;ated via the isodifferential calculus in Ref. (14))

A simple method has been identified in Refs. (14,31) for the construction of the entire isomathematics and its physical applications. It consists in: (i) representing all conventional interactions with a Hamiltonian H and all nonhamiltonian interactions and effects with the isounit i; (ii) identifying the latter interactions with a nonunitary transform U x U t = i # I ; and (iii) subjecting the totality of conventional mathematical and physical quantities and all their operations to said nonunitary transform,

20 1

<$I

I$ > + u x <$I x I$ > x u t = u x I$ > X(U x U t ) =< 7 j ( X ( ? j> x i , H X ( $> + U X ( H X ~ $>) = ( v x H x u + ) x ( u x u + ) - ' x ( u x I ~ > ) =HicllZ,>,etc.(3.13) < $1

x Ut x

x

(Ux ut)-' x

It should be indicated that not all Lie-Santilli isoalgebras can be constructed via inonunitary transforms-of conventional Lie algebras. As an illustration, t.he classification of all possible isotopic SU(2) algebras exhibits eigenvalues different then the conmventional ones, while only conventional eigenvalues are admitted under nonunmitary transforms (see Refs. (28,33) for brevity). The invariance of the basic mathematical axioms and numerical predictions under time as well as other transforms was first achieved in Refs. (14,31), and it can be proved on isospaces over isofields by reformulating any given, nonunitary transform in the isounitary form, and then showing that the basic isoaxioms are indeed invariant, i.e.,

w x Wt= i,w=w x T'l2,wx Wt = w x w t = wt;w i i'= 6 ' x i ; w t 4

=

= j,&B

(6'x T x A x T x r/irt) x (T x =

2x

(Wt x

T x W)-'

w;(A;B);fit

i,

=

T x (r/ir x f)-' x (6'x T x B x T B = al x T x B' = A ' ; B , etc.

x

x

4

=

x

r/irt) = (3.14)

The invariance is ensured by the numerically invariant values of the iscunit [ and- of the isotopic element T under nonunitary-isounitary transforms, namely, I + I' = I , T --t 9 = T , ;+ ;' = G ,etc. The resolution of the catastrophic inconsistencies of Theorem 3.1 is then consequential. The isodual isotopic branch of hadronic mechanics for the treatment of antimatter is given by the image of the theory under map (2.4) and its outline is here omitted for brevity. The contributions on hadronic mechanics and underlying isomathematics are too numerous t o be reviewed here in detail [4,6,7,8,9,10,11], Let us note that an inspection of Eqs. (3.10)-(3.12) reveals the emergence at both classical and operator levels of a new notion of bound states, called closed variationally nonselfadjoint (or nonpotential) systems (51,55). They essentially consist of systems of extended, nonspherical and deformable particles verifying all conventional total conservation laws (closure), yet admitting internal nonlinear, nonlocal, and nonpotential interactions. An illustration of classical nonpotential systems is given by the structure of Jupiter which, when considered as isolated from the rest of the universe, does indeed verify all conventional total conservation laws, yet its structure exhibits vortices with varying angular momenta and other effects solely due to contact nonpotential interactions (51). The expected operator counterpart of Jupiter is the structure of hadrons for which the name of "hadronic mechanics" was proposed (38). In fact, the interior of hadrons constitutes the densest medium ever measured in laboratory, within which the assumption of the exact validity of quantum mechanics should not be expected t o pass the test of time.

202 The application of hadronic mechanics t o hadrons has permitted: the achievement of an exact quark confinement, that with an identically null probability of tunnel effects for free quarks, simply assured by the incoherence of the conventional external and the isotopic internal Hilbert spaces (216,217); the identification of quark constituents with actual, physical, ordinary massive and stable particles, the proton and the electron, although obeying hadronic laws (thus being isorepresentations of the PoincarB-Santilli isosymmetry); the prediction of new clean energies currently under test which are based precisely on the capability of quark constituents t o be produced free (58); and other intriguing advances. Particularly significant is the achievement, permitted by isomathematics, of the exact symmetry (rather than covariance) for all possible Riemannian line elements, which symmetry resulted t o be isomorphic to the PoincarC symmetry and it is today called the Poincark-Santilli isosymmetry (26,29,137). In turn, the achievement of an exact invariance of gravitation has permitted the resolution of vexing controversies on gravitation (see, e.g., (55,158,220)). For instance, the proof of the existence of total conservation laws for gravitation theories on curved manifold and, above all, their compatibility with relativistic conservation laws, is reduced to the mere visual inspection that the generators of the PoincarC and the PoincarB-Santilli isotheory are identical (since these generators are the conserved quantities). Again, as it was the case for the belief in broken symmetries, controversies in gravitation were due to the use of insufficient mathematics, rather than physical shortcomings. The achievement of a universal invariance implied a new conception of gravitation based on: (1) the factorization of (nowhere degenerate and sufficiently smooth) locally Minkowskian Riemannian metrics g(x) into the form g(z) = T,,(z),v. where 7 = Diag.(1,1,1, -1) is the conventional Minkowski metric and F g r ( x )is a 4 x 4 positive-definite matrix (from the local Minkowskian character of g(x)); (2) the assumption of j g r ( z )= l/Tgrp)as the new basic unit of the theory with corresponding isoproduct A k B = A x TgTx B ; and (3) construction of the isominkowskian geometry by subjecting all aspects of the Minkowskian geometry to the noncanonical transform U x Ut = f g r ( x ) . This procedure yields the Santilli’s isominkowskian geometry (15) which: is isomorphic to the Minkowskian (rather than the Riemannian) geometry, preserves identically the Einstein-Hilbert (or any other) field equations, and have null curvature on isospace over said isofield. Thanks t o the elimination of curvature in isospace, the above isominkowskian formulation of gravity permitted: (A) a geometric unification of the special and general relativities which are solely differentiated by their basic unit (15); (B) an axiomatically consistent grand unification including gravitation and electroweak interactions (32,35) in which gravity is merely embedded in the unit of electroweak theories; (C) a reinterpretation of gravitational singularities as the zeros of the gravitational isounit, f g r ( x )= 0; (D) the extension of the preceding exterior isogravity to interior gravitational problems by applying the additional noncanonical transform U x Ut = Diag.(n:, n;,ni, where n4 is the index of refraction or, equivalently, a geometrization of the density of the interior

ni),

203 problem considered (thus resulting in a direct geometrization of locally varying speeds of light c = c,/n4 within physical media) and n:,n;,n: represents the shape of the body considered; and other advances. Note the teaching of isomathematics according to which a n axiomatically consistent grand-unified theory of gravity and electroweak interactions always exasted. It did creep in un-noticed for about a century because existing where nobody looked for, in the unit of electroweak theories (32,35). Still in turn, these novel gravitational horizons imply the new isoselfdual cosmology (34) which is based on a universal invariance under the isoselfdual symmetry S = P(3.1) x Pd(3.1) = Sd, where P(3.1) is the Poincar-Santilli isogroup for matter and Pd(3.1) is its isodual for antimatter. At the limit of equal matter and antimatter, isoselfdual cosmology predicts that the universe has identically null total time, total mass, total energy, etc., thus permitting mathematical models of creation without discontinuities such as that of the "big bang" theory. The experimental verifications of hadronic mechanics are equally numerous and have been achieved in particle physics, nuclear physics, chemistry, superconductivity and astrophysics [9]. We here mention that isomathematics and related hadronic mechanics achieved in full the objective for which they were constructed, namely, the first exactnumerical representation of all characteristics of the Hydrogen, water and other molecules (125,126, 59), while, as indicated earlier, quantum chemistry still misses 2% of the binding energies under the strict application of its axioms, with much greater deviations for electric and magnetic moments for which even the sign is at times incorrect. Hadronic mechanics also achieved the first exact-numerical representation of all nuclear magnetic moments (114), while quantum mechanics still misses about 1%of the magnetic moment of the deuteron with embarrassingly greater deviations for heavier nuclei despite seventy five years of attempts. Equally significant is the achievement of the first exact-numerical representation of the experimental data of the nonlocal Bose-Einstein correlation under the exact reconstruction of the Poincare' symmetry (112,113),the achievement of the first exact-numerical representation of the behavior of the meanlife of unstable particles with speed (110,111), and other experimental verifications. Perhaps the most significant implications of isomathematics and related hadronic mechanics and chemistry has been the theoretical prediction, experimental verification and industrial development of new clean non-hydrocarbon fuels (59) with a new non-valence chemical structure called magnecules (127-130). In closing with a historical note, the reader should be aware that hadronic mechanics is a completion of quantum mechanics essentially along the historical argument by Einstein, Podolsky and Rosen ( 5 ) . In particular, isotopies are a co?cret! and explicit realization of "hidden variables" because the modular isotopic action H 2 > and the conventional > coincide at the abstract, realization-free level. Particularly intriguing action H x are the Bell-Santilli isoinequalities since they admit indeed a classical image.Therefore, it appears that the vast studies on local realism conducted in the 20-th century are also afflicted by limitations caused by the use of insufficient mathematics. For all these aspects we refer the interested reader t o Ref. (33).

204

4. CONSTRUCTION OF GENOMATHEMATICS FROM IRREVERSIBLE SYSTEMS. As it is well known, systems are called irreversible when their images under time reversal, t + -t, are prohibited by causality and other laws, as it is the case for nuclear transmutations, chemical reactions and organisms growth. Systems are called reversible when their time reversal images are as causal as the original ones, as it is the case for planetary and atomic structures when considered isolated from the rest of the universe (see reprint volume (81) and vast literature quoted therein). Yet another large scientific unbalance of the 20-th century has been the treatment of irreversible systems via mathematical and physical formulations of reversible systems which are themselves reversible, resulting in serious limitations in virtually all branches of science. The problem was compounded by the fact that all formulations were essentially of Hamiltonian type, while all known Hamiltonians are reversible since all known potential interactions are reversible. This third scientific unbalance was dismissed by academic interests with unsubstantiated statements, such as ”irreversibility is a macroscopic occurrence which disappears when all bodies are reduced t o their elementary constituents.” The underlying belief is that mathematical and physical theories which are so effective for the study of one electron in reversible orbit around a proton are tacitly assumed to be equally effective for the study of the same electron when in irreversible motion in the core of a star with local nonconserwation of energy, angular momentum, etc. These academic beliefs have been disproved by the following: THEOREM 4.1 (224): A classical irreversible system cannot be consistently decomposed into a finite number of elementary constituents all in reversible conditions and, vice-versa, a finite collection of elementary constituents in reversible conditions cannot yield an irreversible macroscopic ensemble. It should be indicated that the above scientific unbalance existed only in the 20-th century because Newton’s equations (1) are generally irreversible since the force F ( t ,x , v ) can be decomposed into a component which is variatzonally selfadjoint (i.e. derivable from a potential), FSA = - d V / d x and the remaining component which is variationally nonselfadjoint (of contact type non representable with a potential), F = FSA + F N ~ A (48,51). It is evident that, since all known Fsa are reversible, in Newtonian mechanics irreversibility is set t o originate in the contact nonpotential forces F N ~ A . In a way fully aligned with Newton’s teaching, Lagrange (2) and Hamilton (3) formulated their celebrated analytic equations in terms of a function, today called the Lagrangian or the Hamiltonian, representing FSA, plus external terms representing precisely the contact nonpotential forces FNSA. Therefore, according to the teaching of Lagrange and Hamilton, irreversibility is characterized by the external terms representing contact zero-range interactions among extended particles. Unfortunately, Lagrange’s and Hamilton’s external terms were truncated at the beginning of the 20-th century, resulting in conventional Hamiltonian formulations which are fully reversible.

205 More recent studies (23,38) have shown that the true Lagrange’s and Hamilton’s equations (those with external terms) cannot be used in applications due t o a number of insufficiencies, such as: (1) the lack of invariant numerical predictions in accordance with Theorem 3.1 (due t o their evident noncanonical character); (2) the lack of characterization of any algebra by the brackets of the time evolution, let alone the loss of all Lie algebras (because the external terms violate the right distributive and scalar laws); (3) the lack of a topology suitable t o represent contact nonpotential interactions (since they can only occur among extended particles while the topology of Hamiltonian formulation is strictly local-differential, thus solely characterizing point particles); and other limitations. The only resolution of these problematic aspects known t o this author was the construction of a novel, structurally irreversible mathematics, namely, a mathematics whose basic axioms are not invariant under time reversal for all possible reversible Hamiltonians. Stated in different terms, the manifestly inconsistent reduction of irreversible macroscopic systems to elementary particles in reversible conditions was due, again, t o insuficiencies of the used mathematics. It should be noted that the isomathematics of the preceding section is also reversible in time (because the isounit is Hermitean), thus mandating the construction of a broader mathematics specifically suited t o represent irreversibility. The achievement of a structurally irreversible mathematics resulted t o be as rather long scientific journey due to the need of achieving znvariance under irreversible conditions. The first studies can be traced back to Ref. (8) of 1967 which presented the first known parametric deformation of Lie algebras with product ( A ,B ) = p x A x B-q x B x A , where p, q, and p & q are non-null parameters and A, B are Hermitean matrices. The studies continued with the first known presentation in Ref. (38) of the operator deformations of Lie algebra with product

(A;B) = A X P x B - B x Q x A =

=

( A XM x B - B X M x A ) + ( A x N x B f B x N x A ) , M + N

=

P,N - M

= Q,

(4.1)

where P, Q and P z t Q are nowhere singular matrices. On historical grounds, the above deformations were introduced in Refs. (8,38) as realizations of Albert’s Lie-admissible and Jordan-admissible products (7), namely, products whose antisymmetric and symmetric parts are Lie and Jordan, respectively. Note, however, that the Lie and Jordan algebras attached t o brackets ( A ; B ) are not conventional because of the broader isotopic nature [4]. The transition from the parameter t o the operator deformations of Lie algebras was mandatory because all time evolution which can be characterized by the former brackets are nonunitary. Therefore, the reader can easily verify that the application of a nonunitary transform t o the parametric deformations leads to the operator ones. Operator (rather than parametric) deformations (4.1) are promising for the representation of irreversibility because they are no longer totally anti-symmetric (as it is the case for Lie brackets) and, therefore, they can indeed represent nonconservation as needed in irreversible processes. Moreover, operator deformations (4.1) are universal in the sense of admitting as particular cases all possible algebras as currently known in mathematics

206 (characterized by a bilinear product), including Lie, Jordan, Kac-Moody, supersymmetric and all other possible algebras. Moreover, the joint Lie- and Jordan-admissibility is preserved by nonunitary transforms, as the reader can verify, thus confirming that brackets (AYB) characterize the most general possible algebras. Nevertheless, the formulation of the parametric and operator deformations via conventional mathematics is afflicted by the catastrophic mathematical and physical inconsistencies of Theorem 3.1 (because of the lack of invariance of the deformation parameters or, equivalently, of the product). While the physical and mathematical literature saw an explosion of contributions in Lie deformations during the past decades (generally without a quotation of their origination in Refs. (8,23,38) as well as the lack of quotation of their Lie-admissible content), by the mid 1980’s this author had abandoned their study according t o their original formulations (8,23,38) because of said catastrophic inconsistencies. A breakthrough occurred with the discovery, apparently done for the first time by R. M. Santilli in Ref. ( 1 2 ) of 1993, that the axioms of a field also hold when the ordinary product of numbers a x b is ordered to the right, a > b, or, separately, ordered to the left, a < b. In turn, such an order permitted the construction of two generalized units, called genounits to the right and to the left

I

=

D i a g . ( l , 1,..., 1 ) + i > ( t , z , v , $ , a z $ ,...) = l / F > ( t , z , v , $ , a , l L , ...) > 0 ,

with corresponding ordered genoproducts to the right and to the left

Ax B

+A

>B

=Ax

P-x B , A x B

+A


= A X<

F x B ,A > B = ( B< ~

) t (4.3) ,

under which the left and rights character of the genounits is preserved,

I ~ =AA ~ z A =

i> > A = A > i> = A , I X A =A

~ I = + A< i < A

=

A

< < i = A,

(4.4). for all (Hermitean elements A, B, of the considered set. Examples of genounits and genoproducts will be provided shortly. In this way, the ordering ”>” can describe motion forward in time while the ordering ” <” can describe motion backward in time, with interconnecting Hermitean (or transposed) conjugation. This approach permitted the embedding of irreversibility in the most fundamental quantities, the basic units and operations, thus assuring ab initio the construction of a structurally irreversible mathematics, today known as genomathematics, as summarized below. DEFINITION 4.1: Let F = F ( a , + , X ) be a field as per Definition 2.1. The forward genofields (first introduced in Ref. ( 1 2 ) of 1993) are rings F’ = k>(?, +>, >) with forward genonumbers a> = a x f > , associative, distributive and commutative for( u + b) x i> = ’t E F > , assqciative and distributive forward genosum ii>;>b> : ward genoproduct ii> > b> = ii > xT’ x b> = t> € F , additive forward genounit

207

(j>+>&> = &> E F > , and multiplicative forward genounit i> = =a ^ > E F’, V 2 , b> E F>, where i> is a complex-valued nonHermitean, or real-value non-symmetric, everywhere invertible quantity generally outside F. The backward genojields (ii>, P,&>I,e.g., = O,&>+>(j>

If+,;>

> i>

=

f>

1

> >;

LEMMA 4.1: Forward and backward genofields are fields with characteristic zero (namely, they verify all axioms of said fields). In Sect. 2 we pointed out that the conventional product ” 2 multiplied by 3” is not necessarily equal to 6 because, depending on the assumed unit and related product, it can be -6. In Section 3 we pointed out that the same product ” 2 multiplied by 3” is not necessarily equal t o +6 or -6, because it can also be equal to a n arbitrary number, or a matrix or an an integrodifferential operator. In this section we point out that ” 2 multiplied by 3” can be ordered t o the right or to the left, and yield different numerical results for different orderings, ” 2 > 3 # 2 < 3, all this by continuing to verify the axioms of a field per each order (12). Once the forward and backward fields have been identified, the various branches of genomathematics can be constructed via simple compatibility arguments, resulting in the genofunctional analysis, genodzfferential calculus, etc (14,54,55). Particularly intriguing are the genogeometries (loc. cit.) because they admit nonsymmetric metrics, such as the genoriemannian metrics g>(x) = T > ( x ) while , bypassing known inconsistencies since they are referred to the nonsymmetric genounit i> = l/T>. In this way, genogeometries are structurally irreversible and actually represent irreversibility with their most central geometric notion, the metric. Particularly important for this note is the lifting of Lie’s theory permitted by genomathematics, first identified by R. M. Santilli in Ref. (23) of 1978, and today knows as the Lie-Santilli genotheory [7,8],which is characterized by: (1) The forward and backward universal enveloping genoassociative algebra, i>,< with infinite-dimensional basis characterizing thePoincare‘-Birkhoff- Witt-Santilli genotheorem ,...,i < j < k ,

i,

i>:~,Xi,xi>xi,xi>xi>X~

where the “hat” on the generators denotes their fqrmulation on genospaces over genofields and their Hermiticity implies that X > =< X = X; (2) The Lie-Santilli genoalgebras characterized by the universal, jointly Lie-and Jordanadmissible brackets (4.1)

208

(3) The Lie-Santilli genotransformation groups

A ~

?i,

( )

=

4GR h)> A(0) < (<e-;2&x ) = ( e i X x X ? ’ X W )

A(O)

(e>

(e-iXWX
)> (4.7)

where 6’ E R> are the genoparameters; the genorepresentation theor;, etc. The mathematical implications of the Lie-Santilli genotheory are significant because of the admission as particular cFses of all possible algebras, as well as because, when computed on the genobimodule ‘t x [’ Lie-admissible algebras verify all Lie axioms, while deviations from Lie algebras emerge only in their projection on the bimodule <[ x <> of the conventional Lie theory. This is due to the fact that the computation-of the left action A < B = A x< T x B on ‘[ (that is, with respect t o the genounit < I = 1/ B on (>. In this way, thanks t o genomathematics, Lie algebras acquire a towering significance i n view of the possibility of reducing all known algebras to primitive Lie axioms. The physical implications of the Lie-Santilli genotheory ar equally significant. In fact, Noether’s theorem on the reduction of conservation laws to primitive Lie symmetries can be generalized to the reduction of, this time, nonconservation laws to primitive LieSantilli genosymmetries. As a matter of fact, this reduction was the very first motivation that suggested the construction of the genotheory in memoir (23) (see also monographs (49,5O)). The reader can then foresee similar liftings of all remaining physical aspects treated via Lie algebras. It is important for this note to outline the physical theories characterized by genomathematics. First, we have the genonewtonian mechanics with manifestly irreversible forward equations (14)

Note that the topology underlying Newton’s equations is the conventional, Euclidean, local-differential topology which, as such, can only represent point particles. By contrast, the topology underlying the isonewton equations is the S a n t i l l i - S o d a s -Tsagas genotopology (14,139) for the representation of extended, nonspherical and deformable particles via x , u , ...) where k = 1,2,3 isounits, e.g., of the diagonal type i> = Diag.(nT,n;,n$)r(t, represents the semiaxes of an ellipsoid (with more general nondiagonal realizations here omitted for simplicity), and represents contact interactions. Note also that in Newton’s equations the nonpotential forces are incorporated as part of the general applied force F, while in genonewton’s equations potential and nonpotential forces are separated, the latter being embedded in the forward genounit. As a result, irreversibility is represented in the isonewton equations with the genodifferential calculus, that is, with a structurally irreversible mathematics, as desired. We then have the genohamiltonian mechanics with forward equations (14,28,51,52,55)

ni,

=

d^>i

gi;

=

dt

& H --I >

~

8>p

dH ap

-

p I> = ~

m

IA> ,,

209

^>

Diag.(i,>, i?), H = P > p p / 2 m + v(x), (4.9) where we have: ignored the arrow of time for real-valued quantities for notational simplicity; used the alternative realization in which the genoderivative in time is conventional and the genoderivative in space is generalized; the genounit for x is I > = 1/T> while that for p is ?> because of geometric reasons connected t o genosymplectic geometry (14); consequently, the genoproduct in x is given by A B = A x T>x B while that in p is given by A > p B = A x I> x B. Note the direct universality of Eqs. (4.9) in representing all infinitely possible genonewton equations (4.8) (universality) directly in the fixed frame of the experimenter (direct universality). Note also that Eqs. (4.9) formally coincide with Hamilton’s equations with external terms. They are reformulated via genomathematics as the only known way t o admit a consistent algebra in the brackcts of the time evolution (38). Therefore, genohamilton equations (4.9) are indeed irreversible for all possible reversible Hamiltonians, as desired. The origin of irreversibility rests in the contact nonpotential forces according to Lagrange’s and Hamilton’s teaching. Note finally that the extension of Eqs. (4.9) t o include nontrivial genotimes implies a major broadening of the theory we cannot review for brevity (14,55). A simple lifting of the naive or symplectic quantization then yields the genotopjc brflnch of hadronic mechanics defined on ‘H>with forwared inner genoproducts < $1 > I$? x I > E C> and characterized by the forward genoscroedinger equations (first formulated in Refs. (42,179) via conventional mathematics and in Ref. (14) via genomathematics) ITot-PhaseSpace =

$;GI@

>= -? > &?I?,??

>= -i

x

?zi

x

ail$‘ >,i> >

>= I&‘

>,

(4.10)

and the genoheisenberg equations first formulated in Ref. (38) via conventional mathematics and in Ref. (14) via genomathematics dA

i x , = (A;H)= dt ~

~

.

A < H - H > A,

A

(4.11) (2i;pj)= i x 6; x I > , ( 2 y ) = ($&) = 0 , where time has no arrow, since Heisemberg’s equations are computed a t a fixed time. As it was the case for the isotopies, a simple method has been identified in Ref. (44) for the construction of genoformulations. It consists in subjecting the totality of quantities and all their operations to the following dual unitary transforms:

u x Ut # 1,w x Wt # 1,u x wt = i’, w x ui =< i = ( i > ) t ,

210

-i>

I a

a x b + ti’

+

ti’

> 6’

=

=

u x I x W ~ ,+
u x a x ~t

= U x (U x

= a x i>,a+< ti = w x a x Ut =<

i x a,

b) x W’ = (Ux a x Wt) x (Ux Wt)-’ x (Ux b x W t ) ,

u x (a/ax)x wt = i’ x (a/ax), >= u x (< $1 x 111, >) x W + , < $1 x 111, >+<< $1 > a/ax B’/S’O> +

H x 111, >--t H > >

=

I$’ >= (Ux H

x Wt) x

(Ux Wt)-’

x (V x 11, >),etc.(4.12)

The fundamental invariance of the formulations under the Lie-Santilli genotransformation group, (first achieved in Ref. (44) of 1997) can be illustrated by

u = fi x T”l2,w

i’ =

4

i”=

=

r/ii x p / 2 , u x Wt = fi > fit

U > f’ > W t = i’,A > B

(U x?’ x A xT’x fit) x (f”x Wt)-’ = A’.x

x?’

((jx T’ x W t ) - l x B

+U

= fit

2 x f”

= I/?’

> ( A> B) > Wt =

x (fix?’)-’ =

> 0 = i’

x ( f i x T’A xT’ x

x B‘ = 2

> B‘,

fi’) = (4.13)

from which all remaining invariances follow, thus resolving the catastrophic inconsistencies of Theorem 3.1. Note again the numerical invariance of the genounzt i’ + I” = i’ and >. of the genoproduct >+>I= As one can see, genomathematics and its related formulation do indeed achieve the objective for which they were built, namely, a n invariant representation of irreversibility at all levels, from Newton to second quantization. Such an objective is achieved via the following main rules: (i) identify the classical origin of irreversibility in contact nonpotential forces among extended particles, much along the historical teaching of Lagrange and Hamilton; (ii) represent said nonpotential forces via non-symmetric genounits and then construct a mathematics which is structurally irreversible for all reversible Hamiltonians in the sense indicated earlier; (iii) achieve an identical reformulation of Hamilton’s equations with external terms, Eqs. (4.9), with a consistent algebra in the brackets of the time evolution of Lie-admissible type according t o Albert; (iv) complement the latter with the underlying genosymplectic geometry, permitting the mapping of the preceding classical formulations into operator formulations preserving said Lie-admissible character; and (v) identify the origin of irreversibility in the most elementary layers of nature, such as elementary particles in their irreversible motion in the interior of stars. Note that a basic requirement t o achieve the latter objective is the nonconservation of the energy and other physical quantities which is readily verified by the genoheisenberg’s equations (4.11) due to the lack of totally anti-symmetric character of the Lie-admissible brackets for which i x d H / d t = ( H ; H ) = H x (T’ -< T)x H # 0. The isodual genoformulations for antimatter are readily constructed via conjugation (2.4). As such, their explicit construction is left to the interested reader. In closing with a historical comment, it is hoped that Lagrange’s and Hamilton’s legacy of representing irreversibility with the external terms in their analytic equations, if studied

21 1

seriously, can result in covering theories with momentous advances in mathematics and all quantitative sciences.

5. CONSTRUCTION OF HYPERMATHEMATICS FROM BIOLOGICAL SYSTEMS. In this author’s opinion, by far the biggest scientific unbalance of the 20-th century has been the treatment of biological systems (hereon denoting DNA, cells, organisms, etc.) via the mathematics developed for inanimate matter. The unbalance is due t o the fact that conventional mathematics and related formulations, such as quantum mechanics, are inapplicable for the treatment of biological systems for various reasons. To begin, biological events, such as the growth of an organism, are irreversible. Therefore, any treatment of biological systems via conventional reversible mathematics and related physical formulations cannot pass the test of time. Quantum mechanics is ideally suited for the treatment of the structure of the Hydrogen atom or of crystals. These systems are represented by quantum mechanics as being ageless. Recall also that quantum mechanics is unable t o treat deformations because of incompatibilities with basic formulations, such as the group of rotations. Therefore, the rzgorous application of quantum mechanics to biological structures implies that all organisms f r o m cells to humans are perfectly rigid and eternal. Even after achieving the invariant formulation of irreversibility outlined in the preceding section, it was easy to see that the underlying genomathematics remains insufficient for in depth studies of biological systems. Recent studies conducted by C. R. Illert (56) have pointed out that the shape of sea shells can certainly be represented via conventional mathematics, such as the Euclidean geometry. However, the latter is inapplicable for a representation of the growth in time of sea shells. Computer simulations have shown that the imposition to sea shell growth of conventional geometric axioms (e.g., those of the Euclidean or Riemannian geometries) implies t,he lack of proper growth, as expected because said geometries are strictly reversible, while the growth of sea shells is strictly irreversible. The same studies by C. R. Illert (loc. cit.) have indicated the need of a mathematics which is not only structurally irreversible, but also multi-dimensional. As an example, C. R. Illert achieved a satisfactory representation of sea shells via the doubling of the Euclidean reference axes, namely, a geometry which appears to be six-dimensional. A basic problem in accepting such a view is the lack of compatibility with our sensory perception. In fact, when holding sea shells in our hands, we can fully perceive their shape as well as their growth with our three Eustachian tubes. In particular, our senses are sufficiently sensitive t o perceive deviations from the Euclidean space, as well as for the possible presence of curvature. These occurrences pose a rather challenging problem, the achievement of a representation of the complexity of biological systems via the most general possible mathematics which is: (1) is structurally irreversible (as in the preceding section); (2) admits the deformation theory; (3) is invariant under the time evolution; (4) is multi-dimensional; and, last but not least, (5) is compatible with our sensory perception.

212

A search in the mathematical literature revealed that a mathematics verifying all the above five requirements did not exist and had t o be constructed from the main features of biological systems. As an example, hyperstructures in their current formulations (see Ref. (96)) lack a well defined left and right generalized unit even under their weak equalities, they are not structurally irreversible and, above all, they do not possess invariance properties necessary t o bypass Theorem 3.1. After numerous trials and errors, a yet broader mathematics verifying the above five conditions was identified by R. M. Santilli in Ref. (14) (see also Refs. (13,47) and monograph (57)); it is today known under the name of hypemathematics; and it is characterized by the following hypemnits here expressed for the lifting of the Euclidean unit

a,*, ...) = Diag.(i:,i,>,i;, = = Diag.[(i:,, iA,..., i:m), (i&,i$, ...)i&J, (i2,ig,..., = l / T > , I

= Diag.(l, 1 , l ) + i > ( t , z , v , * ,

I = D i a g . ( l , l ) ..') 1 ) + < i ( t , 2 , v , 7 j , a z $,...) = D i a g . ( < & , < i 2 , < & ) = = Diag.[(
ih),(
=

(
(5.1)

with corresponding ordered hyperproducts to the right and to the left as in Eqs. (4.3) and identifies as in Eqs. (4.4) expressed in terms of ordinary equalities and operations. S A further broadening is then permitted by weak operations. As one can see, the above mathematics is not 3m-dimensional, but rather it is 3dimensional and m-multi-valued. Such a feature permits the increase of the reference axes, e.g., for m = 2 we have the six axes used by C. R. Illert (loc cit.), while achieving compatibility with our sensory perception because at the abstract, realization-free level the hypermathematics characterized by hyperunit (5.1) is indeed 3-dimensional. The various branches of hypermathematics (hypernumbers, hyperspaces, hyperalgebras, etc.) can be constructed via mere compatibility arguments with hyperunit (5.1) (see monograph (57) for brevity. The isodual hypermathematics can be constructed via the use of map (2.4). The reader should be aware that the complexity of biological structures requires the use of hypermathematics as well as its isodual, e.g., for quantitative interpretations of bifurcations. In fact, the latter require the following four directzons in time: motion forward in future time i>; motion backward in past time d. The necessity of these four directions time 3 = 346.765. By comparison, in hypermathematics the product of two numbers yields, by conception, a set of values, e.g., 2 > 3 = (12.678,341.329,891.556, ...). Such a feature appears t o be necessary for the representation of biological systems because the association of two atoms in a DNA (mathematically representable with the hypermultiplication) can yeld a large variety of different organs. This feature serves t o indicate that the biological

213

world has a complexity simply beyond our imagination at this time, and that the study of biological problems, such as understanding the DNA code via numbers dating back to biblical times, is manifestly insufficient. We close this note with the view that mathematics and all quantitative sciences will never admit ”final formulations” because, no matter how advanced and valid a given formulation may appear to be, the discovery of structural generalizations is only a matter of time.

Acknowledgments The author would like to thank several colleagues for invaluable and inspiring, constructive and critical remarks, including Professors E. Trell, S. Johansen, P. A. Bjorkum, L. Horwitz, T. L. Gill, A. K. Aringazin, M. C. Duffy, K. P. Shum, F. Winterberg, and several others.

References [l] HISTORICAL REFERENCES: (1) I. Newton, Philosophiae Naturalis Principia Mathematica (1687), translated and reprinted by Cambridge Univ. Press. (1934). (2) J. L. Lagrange, Mechanique Analytique (1788), reprinted by Gauthier-Villars, Paris (1888). (3) W. R. Hamilton, O n a General Method in Dynamics (1834), reprinted in Hamilton’s Collected Works, Cambridge Univ. Press (1940). (4) s. Lie, Over e n Classe Geometriske Transformationer, English translation by E. Trell, Algebras Groups and Geometries 15, 395 (1998). (5) A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). (6) P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, fourth edition (1958). (7) A. A. Albert, Trans. Amer. Math. Soc. 64, 552 (1948). [2] BASIC MATHEMATICAL PAPERS: (8) R. M. Santilli, Nuovo Cimento 51, 570 (1967). (9) R. M. Santilli, Suppl. Nuovo Cimento 6, 1225 (1968). (10) R. M. Santilli, Hadronic J. 3, 440 (1979). (11) R. M. Santilli, Hadronic J. 8, 25 and 36 (1985). (12) R. M. Santilli Algebras, Groups and Geometries 10, 273 (1993). (13) R. M. Santilli and T. Vougiouklis, contributed paper in New Frontiers in Hyperstmctures, T., Vougiouklis, Editor, Hadronic Press, p. 1 (1996). (14) R. M. Santilli, Rendiconti Circolo Matematico di Palermo, Supplemento 42, 7 (1996). (15) R. M. Santilli, Intern. J. Modern Phys. D 7, 351 (1998). [3] ISODUAL FORMULATIONS: (16) R. M. Santilli, Comm. Theor. Phys. 3, 153 (1993). (17) R. M. Santilli, Hadronic J. 17, 257 (1994). (18) R. M. Santilli, Hadronic J. 17, 285 (1994). (19) R. M. Santilli, Communication of the JINR, Dubna, Russia,. No. E2-96-259 (1996). (20) R. M. Santilli, contributed paper in New Frontiers of Hadronic Mechanics, T.L.Gil1, ed.,

214

Hadronic Press (1996). (21) R. M. Santilli, Hyperfine Interactions, 109, 63 (1997). (22) R. M. Santilli, Intern. J. Modern Phys. A 14, 2205 (1999). [4] ISOTOPIC FORMULATIONS: (23) R.M.Santilli: Hadronic J. 1, 224 (1978). (24) R. M. Santilli, Phys. Rev. D 20, 555 (1979). (25) C.Myung and R.M.Santilli, Hadronic J. 5, 1277 (1982). (26) R. M. Santilli, Novo Cimento Lett. 37, 545 (1983). (27) R. M. Santilli, Hadronic J. 8, 25 and 36 (1985). (28) R. M. Santilli, JINR Rapid. Comm. 6, 24 (1993). (29) R. M. Santilli, J.Moscow Phys.Soc. 3,255 (1993). (30) R. M. Santilli, Chinese J.Syst.Ing. & Electr.6, 177 (1996). (31) R. M. Santilli, Found. Phys. 27,635 (1997). (32) R. M. Santilli, Found. Phys. Letters 10, 307 (1997). (33) R. M. Santilli, Acta Appl. Math. 50, 177 (1998). . (34) R. M. Santilli, contributed paper to the Proceedings of the International Workshop on Modern Modified Theories of Gravitation and Cosrnology, E. I. Guendelman, Editor, Hadronic Press, p. 113 (1998). (35) R. M. Santilli, contributed paper to the Proceedings of the VIII M . Grossrnann Meeting on General Relativity, Jerusalem, June 1998, World Scientific, p. 473 (1999). (36) R. M. Santilli, contributed paper in Photons: Old Problems in Light of New Ideas, V. V. Dvoeglazov, editor, Nova Science Publishers, pages 421-442 (2000). (37) R. M. Santilli, Found. Phys. Letters 2, 1111 (2002). [5] GENOTOPIC FORMULATIONS: (38) R. M. Santilli: Hadronic J. 1,574 and 1267 (1978). (39) R. M. Santilli, Hadronic J. 2 , 1460 (1979) and 3, 914 (1980). (40) R. M. Santilli, Hadronic J. 4, 1166 (1981). (41) R. M. Santilli, Hadronic J. 5, 264 (1982). (42) H. C. Myung and R. M. Santilli, Hadronic J. 5, 1367 (1982). (43) R. M. Santilli, Hadronic J. Suppl. 1, 662 (1985). (44) R. M. Santilli, Found. Phys. 27, 1159 (1997). (45) R. M. Santilli, Modern Phys. Letters 13, 327 (1998). (46) R. M. Santilli, Intern. J. Modern Phys. A 14, 3157 (1999). [6] HYPERSTRUCTURAL FORMULATIONS: (47) R. M. Santilli, Algebras, Groups and Geometries 1 5 , 473 (1998) [7] MONOGRAPHS: (48) R. M. Santilli, Foundations of Theoretical Mechanics, Vol. I, Springer-Verlag, Heidelberg-New York (1978). (49) R. M. Santilli, Lie-admissible Approach to the Hadronic Stmcture, Vol.1, Hadronic Press, Palm Harbor, Florida (1978). (50) R. M. Santilli, Lie-admissible Approach to the Hadronic Structure, Vol. 11, Hadronic Press, Palm Harbor, Florida (1981). (51) R. M. Santilli, Foundations of Theoretical Mechanics, Vol. 11, Springer-Verlag, Heidelberg-New York (1983). (52) R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities, Vol. I, Hadronic Press, Palm Harbor, Florida (1991). (53) R. M. Santilli, Isotopic Generalizations of Galilei and Einstein Relativities, Vol. 11, Hadronic Press, Palm Harbor, Florida (1991). (54) R. M. Santilli, Elements of Hadronic Mechanics, Vol I, Ukraine Academy of Sciences,

215

Kiev, Second Edition (1995). (55) R. M. Santilli, Elements of Hadronic Mechanics, Vol 11, Ukraine Academy of Sciences, Kiev, Second Edition (1995). (56) C. R. Illert and R. M. Santilli, Foundations of Theoretical Conchology, Hadronic Press, Palm Harbor, Florida (1995). (57) R. M. Santilli Isotopic, Genotopic and Hyperstructural Methods in Theoretical Biology, Ukraine Academy of Sciences, Kiev (1996). (58) R. M. Santilli, The Physics of New Clean Energies and Fuels According to Hadronic Mechanics, Special issue of the Journal of New Energy, 318 pages (1998). (59) R. M. Santilli, Foundations of Hadronic Chemistry with Applications to New Clean Energies and Fuels, Kluwer Academic Publishers, Boston-Dordrecht-London (2001). (60) R. M. Santilli, Isodual Theory of Antimatter, with Applications to Cosmology, Antigravity and Spacetime Machines, In preparation. (61) R. M. Santilli, Elements of Hadronic Mechanics, Vol. 111, in preparation. (62) H. C. Myung, Lie Algebras and Flexible Lie-Admissible Algebras, Hadronic Press (1982). (63) A.K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Santilli’s Lie-isotopic Generalization of Galilei’s and Einstein’s Relativities, Kostarakis Publishers, Athens (1991). (64) D. S. Sourlas and G. T. Tsagas, Mathematical Foundations of the LieSantilli Theory, Ukraine Academy of Sciences, Kiev (1993). (65) J. Lhhmus, E. Paal and L. Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, FL, USA (1994). (66) J. V. Kadeisvili, Santilli’s Isotopies of Contemporary Algebras, Geometries and Relativities, Second Edition, Ukraine Academy of Sciences, Kiev , Second Edition (1997). (67) R. M. Falcon Ganfornina and J. Nunez Valdes, Fondamentos de la Isoteoria de Lie-Santilki, (in Spanish) International Academic Press, America-Europe-Asia, (2001), also available in the pdf file http://www.i-br.org/docs/spanish.pdf (68) Chun-Xuan Jiang, Foundations of Santilli’s Isonumber Theory, with Applications to New Cryptograms, Fermat’s Theorem and Goldbach’s Conjecture, International Academic Press, America-Europe-Asia (2002) also available in the pdf file http://www.i-b-r.org/docs/jiang.pdf [8] CONFERENCE PROCEEDINGS AND REPRINT VOLUMES:

(69) H. C. Myung and S. Okubo, Editors, Applications of Lie-Admissible Algebras in Physics, Volume I, Hadronic Press (1978). (70) H. C. Myung and S. Okubo, Editors, Applications of Lie-Admissible Algebras in Physics, Vol. 11, Hadronic Press (1978). (71) H. C. Myung and R. M. Santilli, Editor, Proceedings of the Second Workshop on Lie-Admissible Formulations, Part I, Hadronic J . Vol. 2, no. 6, pp. 1252-2033 (1979). (72) H. C. Myung and R. M. Santilli, Editor, Proceedings of the Second Workshop on Lie-Admissible Formulations,Part 11, Hadronic J. Vol. 3, no. 1, pp. 1-725 (1980. (73) H. C. Myung and R. M. Santilli, Editor, Proceedings of the Third Workshop on Lie-Admissible Formulations,Part A, Hadronic J. Vol. 4, issue no. 2, pp. 183607 (19881) (73). (74) H. C. Myung and R. M. Santilli, Editor, Proceedings of the Third Workshop on Lie-Admissible Formulations, Part B, Hadronic J. Vo. 4, issue no. 3, pp. 608-1165 (1981). (75) H. C. Myung and R. M. Santilli, Editor, Proceedings of the Third Workshop on Lie-Admissible Formulations, Part C, Hadronic J . Vol. 4, issue no. 4, pp. 1166-1625 (1981). (76) J. Fronteau, A. Tellez-Arenas and

216

R. M. Santilli, Editor, Proceedings of the First International Conference on Nonpotential Interactions and their Lie-Admissible Treatment, Part A, Hadronic J., Vol. 5, issue no. 2, pp. 245-678 (1982). (77) J. F’ronteau, A. Tellez-Arenas and R. M. Santilli, Editor, Proceedings of the First International Conference on Nonpotential Interactions and their Lie-Admissible Treatment, Part B, Hadronic J. Vol. 5, issue no. 3, pp. 679-1193 (1982). (78) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, Proceedings of the First International Conference on Nonpotential Interactions and their Lie-Admissible Treatment, Part C , Hadronic J. Vol. 5, issue no. 4, pp. 1194-1626 (1982). (79) J. Fronteau, A. Tellez-Arenas and R. M. Santilli, Editor, Proceedings of the First International Conference on Nonpotential Interactions and their Lie-Admissible Treatment, Part D, Hadronic J. Vol. 5, issue no. 5, pp. 16271948 (1982). (80) J.Ronteau, R.Mignani, H.C.Myung and R. M. Santilli, Editors, Proceedings of the First Workshop on Hadronic Mechanics, Hadronic J. Vol. 6, issue no. 6, pp. 1400-1989 (1983). (81) A. Shoeber, Editor, Irreversibility and Nonpotentiality in Statistical Mechanics, Hadronic Press (1934). (82) H. C. Myung, Editor, Mathematical Studies in Lie-Admissible Algebras, Volume I, Hadronic Press (1984). (83) H. C. Myung, Editor, Mathematical Studies in Lie-Admissible Algebras, Volume 11, Hadronic Press (1984). (84) H. C. Myung and R. M. Santilli, Editor, Applications of Lie-Admissible Algebras in Physics, Vol. 111, Hadronic Press (1984). (85) J.Fronteau, R.Mignani and H.C.Myung, Editors, Proceedings of the Second Workshop o n Hadronic Mechanics, Volume I Hadronic J. Vol. 7, issue no. 5, pp. 911-1258 (1984). (86) JRonteau, R.Mignani and H.C.Myung, Editors, Proceedings of the Second Workshop on Hadronic Mechanics, Volume 11, Hadronic J. Vol. 7, issue no. 6, pp. 1259-1759 (1984). (87) D.M.Norris et al, Tomber’s Bibliography and Indez in Nonassociative Algebras, Hadronic Press, Palm Harbor, FL (1984). (88) H. C. Myung, Editor, Mathematical Studies in Lie-Admissible Algebras, Volume 111, Hadronic Press (1986). (89) A.D.Jannussis, R.Mignani, M. Mijatovic, H. C.Myung B. Popov and A. Tellez Arenas, Editors, Fourth Workshop on Hadronic Mechanics and Nonpotential Interactions, Nova Science, New York (1990). (90) H. M. Srivastava and Th. M. Rassias, Editors, Analysis Geometry and Groups: A Riemann Legacy Volume, Hadronic Press (1993). (91) F. Selleri, Editor, Fundamental Questions in Quantum Physics and Relativity, Hadronic Press (1993). (92) J. V. Kadeisvili, Editor, The Mathematical Legacy of Hanno Rund, Hadronic Press (1994). (93) M. Barone and F. Selleri Editors, Frontiers of Fundamental Physics, Plenum, New York, (1994). (94) M. Barone and F. Selleri, Editors, Advances in Fundamental Physics, Hadronic Press (1995). (95) Gr. Tsagas, Editor, New Frontiers in Algebras, Groups and Geometries , Hadronic Press (1996). (96) T. Vougiouklis, Editor, New Frontiers in Hyperstmctures, Hadronic Press, (1996). (97) T. L. Gill, Editor, New Frontiers in Hadronic Mechanics, Hadronic Press (1996). (98) T. L. Gill, Editor, New Frontiers in Relativities, Hadronic Press (1996). (99) T. L. Gill, Editor, New Frontiers i n Physics,, Volume I, Hadronic Press (1996). (100) T. L. Gill, Editor, New Frontiers in Physics,, Volume 11, Hadronic Press (1996). (101) C. A. Dreismann, Editor, New Frontiers in Theoretical Biology, Hadronic Press (1996). (102) G. A., Sardanashvily, Editor, New

217

Frontiers i n Gravitation, Hadronic Press (1996).(103)M. Holzscheiter, Editor, Proceedings of the International Workshop on Antzmatter Gravity, Sepino, Molise, Italy, May 1996, Hyperfine Interactions, Vol. 109 (1997). (104) T. Gill, K. Liu and E. Trell, Editors, Fundamental Open Problems in Science at the end of the Millennium, Volume I, Hadronic Press (1999).(105)T. Gill, K. Liu and E. Trell, Editors, Fundamental Open Problems in Science at the end of the Millennium, Volume 11, Hadronic Press (1999).(106)T. Gill, K. Liu and E. Trell, Editors, Fundamental Open Problems in Science at the end of the Millennium, Volume 111, Hadronic Press (1999).(107) V. V. Dvoeglazov, Editor Photon: Old Problems in Light of New Ideas, Nova Science (2000).(108)M. C. Duffy and M. Wegener, Editors, Recent Advances in Relativity Theory Vol. I, Hadronic Press (2000).(109)M. C.Duffy and M. Wegener, Editors, Recent Advances in Relativity Theory Vol. I, Hadronic Press (2000).

[9]EXPERIMENTAL VERIFICATIONS: (110)F. Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys. 18, L61 (1992).(111) F.Cardone, R. Mignani and R. M. Santilli, J. Phys. G: Nucl. Part. Phys. 18, L141 (1992).(112)R. M. Santilli, Hadronic J. 15, Part I: 1-50 and Part 11: 77134 (1992).(113)Cardone and R. Mignani, JETP 88,435 (1995).(114)R. M. Santilli, Intern. J. of Phys. 4,1 (1998).(115)R. M. Santilli Communications in Math. and Theor. Phys. 2, 1 (1999).(116)A. 0.E.Animalu and R. M. Santilli, Intern. J. Quantum Chem. 26,175 (1995).(117)R. M. Santilli, contributed paper to Frontiers of Fundamental Physics, M. Barone and F. Selleri, Editors Plenum, New York, pp 4158 (1994).(118) R. Mignani, Physics Essays 5, 531 (1992). (119)R. M. Santilli, Comm. Theor. Phys. 4, 123 (1995). (120) Yu. Arestov, V. Solovianov and R. M. Santilli, Found. Phys. Letters 11,483 (1998).(121)R.M.Santilli, contributed paper in the Proceedings of the International Symposium on Large Scale Collective Motion of Atomic Nuclez, G. Giardina, G. Fazio and M. Lattuada, Editors, World Scientific, Singapore, p. 549 (1997).(122)J.Ellis, N.E. Mavromatos and D.V.Napoulos in Proceedings of the Erice Summer School, 31st Course: From Superstrings to the Origin of SpaceTime, World Sientific (1996).(123)Borghi, C.Giori and A. Dall’OIlio Russian J. Nucl. Phys. 56, 147 (1993) (123).(124)N. F. Tsagas, A. Mystakidis, G. Bakos, and L. Seftelis, Hadronic J. 19,87 (1996). (125) R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy 24,943 (1999).(126)R. M. Santilli and D., D. Shillady, Intern. J. Hydrogen Energy 25, 173 (2000).(127)R. M. Santilli, Hadronic J. 21, pages 789-894(1998).(128)M.G. Kucherenko and A.K. Aringazin, Hadronic J. 21, 895 (1998). (129) M.G. Kucherenko and A.K. Aringazin, Hadronic Journal 23, 59 (2000). (130)R.M. Santilli and A.K. Aringazin, ”Structure and Combustion of Magnegases” , e-print http://arxiv.org/abs/physics/Ol12066, to be published.

[lo] MATHEMATICS PAPERS: (131)S. Okubo, Hadronic J. 5,1564 (1982).(132)J. V. Kadeisvili, Algebras, Groups and Geometries 9,283 and 319 (1992).(133) J. V. Kadeisvili, N. Kamiya, and R.

218

M. Santilli, Hadronic J. 16, 168 (1993). (134) J. V. Kadeisvili, Algebras, Groups and Geometries 9 , 283 (1992). (135) J. V. Kadeisvili, Algebras, Groups and Geometries 9, 319 (1992). (136) J. V. Kadeisvili, contributed paper in the Proceedings of the International Workshop on Symmetry Methods in Physics, G. Pogosyan et al., Editors, JINR, Dubna, Russia (1994). (137) J. V. Kadeisvili, Math. Methods in Appl. Sci. 19 1349 (1996). (138) J. V. Kadeisvili, Algebras, Groups and Geometries, 15, 497 (1998). (139) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and Geometries 12, 1 (1995). (140) G. T. Tsagas and D. S. Sourlas, Algebras, Groups and geometries 1 2 , 67 (1995). (141) G. T. Tsagas, Algebras, Groups and geometries 13, 129 (1996). (142) G. T. Tsagas, Algebras, Groups and geometries 13, 149 (1996). (143) E. Trell, Isotopic Proof and Reproof of Fermats Last Theorem Verifying Beals Conjecture. Algebras Groups and Geometries 15, 299-318 (1998). (144) A.K. Aringazin and D.A. Kirukhin,, Algebras, Groups and Geometries 12, 255 (1995). (145) A.K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Algebras, Groups and Geometries 7,211 (1990). (146) A.K. Aringazin, A. Jannussis, D.F. Lopez, M. Nishioka, and B. Veljanoski, Algebras, Groups and Geometries 8, 77 (1991). (147) D. L. Rapoport, Algebras, Groups and Geometries, 8, 1 (1991). (148) D. L. Rapoport, contributed paper in theProceedinys of the Fifth International Workshop on Hadronic Mechanics, H.C. Myung, Edfitor, Nova Science Publisher (1990). (149) D. L. Rapoport, Algebras, Groups and Geometries 8, 1 (1991). (150) C.-X. Jiang, Algebras, Groups and Geometries 15, 509 (1998). (151) D. B. Lin, Hadronic J. 11, 81 (1988). (152) R. Aslaner and S. Keles, Algebras, Groups and Geometries 14, 211 (1997). (153) R. Aslander and S. Keles,. Algebras, Groups and Geometries 1 5 , 545 (1998). (154) M. R. Molaei, Algebras, Groups and Geometries 115, 563 (1998) (154). (155) S. Vacaru, Algebras, Groups and Geometries 14, 225 (1997) (155). (156) N. Kamiya and R. M. Santilli, Algebras, Groups and Geometries 13, 283 (1996). (157) S. Vacaru, Algebras, Groups and Geometries 14, 211 (1997). (158) Y . Named, Hadronic J. 5, 1734 (1982). (159) R. Trostel, Hadronic J. 5, 1893 (1982). [ll]PHYSICS PAPERS:

(160) J. P. Mills, jr, Hadronic 3. 1 9 , 1 (1996). (161) J. Dunning-Davies, Foundations of Physics Letters, 1 2 , 593 (1999). (162) E.’Trell, Hadronic Journal Supplement 1 2 , 217 (1998). (163) E. Trell, Algebras Groups and Geometries 15,447-471 (1998). (164) E. Trell, ”Tessellation of Diophantine Equation Block Universe,” contributed paper to Physical Interpretations of Relativity T h e o v, 6-9 September 2002, Imperial College, London. British Society for the Philosophy of Science, in print, (2002). (165) J. Fronteau, R. M. Santilli and A. Tellez-Arenas, Hadronic J. 3, 130 (1979). (166) A. 0. E. Animalu, Hadronic J.7, 19664 (1982). (167) A. 0 . E. Animalu, Hadronic J. 9, 61 (1986). (168) A. 0. E. Animalu, Hadronic J. 10, 321 (1988). (169) A. 0. E. Animalu, Hadronic J. 16, 411 (1993). (170) A. 0. E. Animalu, Hadronic J. 17, 349 (1994). (171) S.Okubo, Hadronic J. 5, 1667 (1982). (172) D.F.Lopez, in Symmetry Methods i n Physics, A.N.Sissakian, G.S.Pogosyan and X.I.Vinitsky, Editors ( JINR, Dubna, Russia (1994) . (173) D. F. Lopez, Hadronic J. 16, 429 (1993). (174) A.Jannussis

219 and D.Skaltsas,Ann. Fond. L d e Broglie 18,137 (1993). (175) A.Jannussis, R.Mignani and R.M.Santilli, Ann.Fonnd. L d e Broglie 18, 371 (1993). (176) A. 0. Animalu and R. M. Santilli, contributed paper in Hadronic Mechanzcs and Nonpotential Interactions M. Mijatovic, Editor, Nova Science, New York, pp. 19-26 (1990). (177) M. Gasperini, Hadronic J. 6,935 (1983). (178) M. Gasperini, Hadronic J. 6, 1462 (1983). (179) R. Mignani, Hadronic J. 5 , 1120 (1982). (180) R. Mignani, Lett. Nuovo Cimento 39, 413 (1984). (181) A. Jannussis, Hadronic J. Suppl. 1, 576 (1985). (182) A. Jannussis and R. Mignani, Physica A 152, 469 (1988). (183) A. Jannussis and LTsohantis, Hadronic J. 11, 1 (1988). (184) A. Jannussis, M. Miatovic and B. Veljanosky, Physics Essays 4, (1991). (185) A. Jannussis, D. Brodimas and R. Mignani, J. Phys. A: Math. Gen. 24, L775 (1991). (186) A. Jannussis and R. Mignani, Physica A 187, 575 (1992). (187) A.Jannussis, R.Mignani and D.Skaltsas Physics A 187, 575 (1992). (188) A.Jannussis et al, Nuovo Cimento B103, 17 and 537 (1989). (189) A. Jannussis et al., Nuovo Cimento B104, 33 and 53 (1989). (190) A. Jannussis et al. Nuovo Cimento B108 57 (1993). (191) A. Jannussis et al., Phys. Lett. A 132, 324 (1988). (192) A.K. Aringazin, Hadronic J. 12, 71 (1989). (193) A.K. Aringazin, Hadronic J. 13, 183 (1990). (194) A.K. Aringazin, Hadronic J. 13, 263 (1990). (195) A.K. Aringazin, Hadronic J. 14, 531 (1991). (196) A.K. Aringazin, Hadronic J. 16, 195 (1993). (197) A.K. Aringazin and K.M. Aringazin,Invited paper, in the Proceedings of the Intern. Conference >Frontiers of Fundamental Physics ’, Plenum Press, (1993). (198) A.K. Aringazin, K.M. Aringazin, S. Baskoutas, G. Brodimas, A. Jannussis, and K. Vlachos, contribuuted paper in the Proceedings of the Intern. Conference ’Frontzers of Fundamental Physics’> Plenum Press, (1993). (199) A.K. Aringazin, D.A. Kirukhin, and R.M. Santilli, Hadronic J. 18, 245 (1995). (200) A.K. Aringazin, D.A.Kirukhin, and R.M. Santilli, Had-ronic J. 18, 257 (1995). (201) T. L. Gill, Hadronic J. 9, 77 (1986). (202) T . L. Gill and J. Lindesay, Int. J. Theor, Phys 32, 2087 (1993). (203) T. L. Gill, W.W. Zachary, M.F. Mahmood and J. Lindesay, Hadronic J. 16, 28 (1994). (204) T.L. Gill, W.W. Zachary and J. Lindesay, Foundations of Physics Letters 10, 547 (1997). (205) T.L. Gill, W.W. Zachary and J. Lindesay, Int. J. The0 Phys 37, 22637 (1998). (206) T.L. Gill, W.W. Zachary and J. Lindesay, Foundations of Physics, 31, 1299 (2001). (207) D. L. Schuch, K.M. Chung and H. Hartmann, J. Math. Phys. 24, 1652 (1983). (208) D. L. Schuch and K.-M. Chung, Intern. J. Quantum Chem. 19, 1561 (1986). (209) D. Schich 23, Phys. Rev. A 23, 59, (1989). (210) D. Schuch, contributed paper in New Frontiers zn Theoretical Physics, Vol. I, p 113, T. Gill Editor, Hadronic Press, Palm Harbor (1996); D. L. Schuch, Hadronic J. 19, 505 (1996). (211) S.L. Adler, Phys.Rev. 17, 3212 (1978). (212) CLGeorge, F. Henin, F.Mayene and LPrigogine, Hadronic J. 1, 520 (1978). (213) C.N.Ktorides, H.C.Myung and R.M.Santilli, Phys. Rev. D 22, 892 (1980). (214) R. M. Santilli, 13, 513 (1990). (215) R. M. Santilli, 17, 311 (1994). (216) A.J. Kalnay, Hadronic J. 6, 1 (1983). (217) A. J. Kalnay and R. M. Santilli, Hadronic J. 6, 1873 (1983). (218) M. 0. Nishioka, Nuovo Cimento A 82, 351 (1984). (219) G.Eder, Hadronic J. 4, (1981) and 5, 750 (1982). (220) H. E. Wilhelm, contributed paper in Space, Time and Motzon: Theory and Experiments,

220

H., E. Wilhelm and K. Liu, editors,Chinese J. Syst Eng. Electr., 6, issue 4 (1995). (221) C.A.Chatzidimitriou-Dreismann,T. Abdul-Redah, R.M.F. Streffer, J.Mayers Phys. Rev. Lett. 79,2839 (1997). (222) C.A.Chatzidimitriou-Dreismann,T. AbdulRedah, B. Kolaric J. Am. Chem. Soc. 123, 11945 (2001). (223) C.A.ChatzidimitriouDreismann, T. Abdul-Redah, R.M.F. Streffer, J.Mayers J. Chem. Phys. 116, 1511 (2002) (see also www.ISIS.rl.ac.uk/molecularspectroscopy/EVS). (224) R. M. Santilli, Hadronic J. Suppl. 1, 662 (1985)

Dedicated to the memory of B. H. Neumann

Semisimple Clifford semirings M. K. Sen and S. K. Maity' Department of Pure Mathematics, University of Calcutta,

35, Ballygunge circular Road, Kolkato-700019, India

e-mail: [email protected] and

K. P. Shum Department of Mathematics, The Chinese University of Hong Kong China, ( S A R ) Abstract It is well known that a semigroup S is a Clifford semigroup if and only if S is a strong semilattice of groups. In the paper [ll]we have extended this important result from semigroups to semirings by showing that a semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings. In this paper we introduce the notions of Nilpotent Clifford semiring and Semisimple Clifford semiring. Some structure theorems of these semirings are obtained.

AMS Mathematics Subject Classification (2000) : 16A78, 20M07. Key words : Skew-ring, Clifford semiring, Nilpotent Clifford semiring, Semisimple Clifford semiring. 1. Introduction

+, .) is a type (2,2) algebra whose semigroup reducts

Recall that a semiring (S,

( S ,+)

and ( S ,.) are connected by distributivity, that is, a(b+c) = ab+ac and (b+c)a = ba+ca for

+, .) additive regular if for every element a E S S such that a + x + a = a. Additive regular semirings were

all a, b, c E ( S ,+, .). We call a semiring ( S , there exists an element x E

first studied by J. Zeleznekow [13] in 1981. We call a semiring (S,t,.) an additive inverse

semiring if (S,+) is an inverse semigroup. Additive inverse semirings were first studied by Karvellas [7] in 1974. *The research is supported by CSIR, India.

22 1

222 In this paper we call an element a of a semiring ( S ,+, .) completely regular if there exists

(i) a + z + a = a ,

an element x E S such that

(22)

a+a;=x+a

and (iii) a(a

+ x) = a + 2.

+, .) completely regular if every element a of S is com-

Naturally, we call a smiring (S,

pletely regular. In fact, we can easily see that the condition (iii) can be replaced by the condition

+

(a 2 ) a = a

(222)’

+ 2.

According t o M. P. Grillet [5], a semiring ( S ,+, .) is called a skew-ring if its additive reduct (S,+) is a group. By using the concept of skew-ring we have proved the following theorem in [lo].

Theorem 1.1. The following conditions on a semiring ( S ,+, .) are equivalent

:

(I) a is a completely regular element of S . (11) There exists a unique element y E Vf(a) such that a

(111) H,’ aE

is a skew-ring, where H,’

is the

+ y = y + a , a(a + y) = a + y.

H+ -class on the semiring ( S ,+, .)

containing

s. The unique element in V+(a)satisfying the condition I1 of the Theorem 1.1 is denoted

by a’. Also we have proved the following theorem in

[lo].

Theorem 1.2. Let S be a completely regular semiring. Then for any a, b E S and e E E+(S) we have (i) (a’)’ = a , (ii) ab‘ = (ab)’ = a’b , (iii) ab = a‘b’ and (iv) e’ = e and e2 = e. Recall that an ideal I of a semiring S is a k-ideal of S if a E I and either a 2

+ x E I or

+ a E I for some x E S implies x E S. If S is a completely regular semiring as well as an additive inverse semiring then E + ( S ) ,

the set of all additive idempotents of S is an ideal of S but E+(S)may not be a k-ideal of

S , for instance, if we let S

= ( 0 , a,b } be a semiring with the following Cayley tables:

b

b

b

b

Then we can easily see that the additive reduct ( S ,+) is an inverse semigroup. It is also easy to see that ( S ,+, .) is a completely regular semiring because a(a

+ a ) = a0 = 0 = a + a

223

+ b) = bb = b = b + b hold. In this example, E+(S)= (0, b} is clearly an ideal of S but since a + b = b E E+(S),and a $ E+(S),E + ( S )is not a k-ideal of S . and b(b

In view of the above example, we call a completely regular semiring S a Clifford semiring if S is an additive inverse semiring such that E+(S)is a distributive sublattice of S as well

as a k-ideal of S. In our paper [ll],we have also proved the following theorem.

Theorem 1.3. A semiring S is a Clifford semiring if and only if S is a strong distributive lattice of skew-rings.

By using Theorem 1.3 we see immediately that, if S is additive commutative then S is a Clifford semiring if and only if S is strong distributive lattice of rings. In sections 2 and 3, we let S be a semiring such that ( S ,+) and (S,.) are both commutative. In section 2, we define nilpotent element in a Clifford semiring and show that the set of all nilpotent elements of a Clifford semiring forms a k-ideal of S. Finally we show that a semiring S is a nilpotent Clifford semiring if and only if S is a strong distributive lattice of nilpotent rings. In section 3, we define nilpotent ideal and Jacobson radical of a Clifford semiring with

1 such that 1 $ E+(S).We define semisimple Clifford semiring, Artinian semiring and the concept of unit element is introduced. We show that in a Artinian Clifford semiring S with 1, radS forms a nilpotent ideal of S. 2. Nilpotent Clifford semirings

We now define nilpotent element in a Clifford semiring.

Definition 2.1. Let S be a Clifford semiring. An element a E S is called nilpotent if there exists a positive integer n suh that an = a

+ a’.

Naturally, we call a Clifford semiring S a nilpotent Clifford semiring if every element of the Clifford semiring S is nilpotent.

Example 2.2. We consider the semiring S

=

Z4 x D , where D is a distributive lattice.

Then S is a nilpotent Clifford semiring. We can easily verify that the direct product of a nilpotent ring and a distributive lattice is a nilpotent Clifford semiring. The concept of Clifford semiring and nilpotent semiring is mutually independet. For instance, we consider the following examples:

224

Example 2.3. Let R be a non nilpotent ring. Then R is a Clifford semiring but not a nilpotent semiring.

Example 2.4. Let I be a non empty set with 111 2 2. We define a

+ b = ab = a for all

a, b E S. Then it is easy t o verify that I is an idempotent semiring. Let R be a nilpotent ring and S be the direct product of R and I ( i.e., S = R x I =

u( R x { a } ) ) where , addition

aEI

and multiplication are defined componentwise. Clearly S is a nilpotent semiring but not a Clifford semiring. In Clifford semiring S , we prove the following lemmas.

Lemma 2.5. If a, b E S are two nilpotent elements then so is a

+ b.

Proof. Since a, b E S are nilpotent elements, there exist positive integers m, n such that am = a+a' and b" = b + b'.

+ + bm+"

Since, ( a + b)m+n= am+n + ... + m+n~Tam+n--l. b ...

. . . . (A)

and,

..+(a+a')b'+. .. b+b') +...+(b+b') This shows that ( a + b) is nilpotent.

So from (A), we deduce that (a+b)m+n= a+a'+. =a

+ a' + b + b' = ( a + b) + ( a + b)'.

Lemma 2.6. Every element of E + ( S )is nilpotent. Proof. This follows from e2 = e for all e E Ef(S). We now prove the following theorem.

Theorem 2.7. Let S be a Clifford semiring. Then N(S),the set of all nilpotent elements of S, forms a k-ideal of S. Proof. Let a, b E N ( S ) and c E S. Then by Lemma 2.5, a + b E N ( S ) . Since a E N ( S ) , there exists a positive integer n such that a" = a c"(a

+ a'.

Now, (ca)"

=

cnan = c"(a

+ a') =

+ a')" = (ca + car)" = ca + ca' = ca + (ca)' and thereby, ca E N ( S ) . Hence N ( S ) is an

ideal of S. To show that N ( S ) a k-ideal, we let a

+ b, b E N ( S ) . Then b' E N ( S ) . Hence a+ b + b' E

N ( S ) . Consequently, there exists a positive integer m such that ( ~ + b + b ' )=~ (a+b+b') (a+b+b')' , i.e., am+"c,am-'(b+b')+

...+( b+b')

of S , we have am E E+(S). Hence am = am

= a+a'+b+b'.

+ (am)' = am + am-'a'

and so a E N ( S ) . Consequently N ( S ) is a k-ideal of S.

+

Since E+(S)is a k-ideal = a"-'(.

+ a') = a + a',

225 By Theorem 2.7, we can define the following relation on a Clifford semiring.

Definition 2.8. Let S be a Clifford semiring. Then E+(S)# q5 and because E + ( S )G N ( S ) , we see that N ( S ) # only if a

4.

We now define a relation p on a Clifford semiring S by a p b if and

+ b’ E N ( S ) .

We now prove the following theorem.

Theorem 2.9. Let S be a Clifford semiring. Then the relation p defined in Definition 2.8 is the smallest congruence on S such that S / p is a ring with no non zero nilpotent element.

Proof. Clearly p is a congruence on S . Let e , f E E+(S).Then, e + f‘ = e + f E E+(S)

N ( S ) . This shows that e p f and S / p clearly contains a single additive idempotent. Hence, S / p must be a ring. Next we let up be a nilpotent element of S/p. So there exists a positive integer n such that (ap)” = e p for all e E E + ( S ) ,i.e., an

+ e’ E N ( S ) , i.e., an + e E N ( S ) . Since N ( S ) is a

k-ideal of S , an E N ( S ) , i.e., a 6 N ( S ) . This means that a p is the zero element of S/p. Let o be any other ring congruence on S such that S / o contains no non zero nilpotent element element. Let a p b. Then a

+ b‘

E N ( S ) and so ( a

+ b’)o is a nilpotent element in

SIC. This leads t o (a+b’)o(a+a‘)o(b+b’). Now, a = (a+a’+a)a(b+b’+a)o(b+b‘+b)

= b.

Hence p is the smallest ring congruence on S with no non zero nilpotent element. We now prove the following characterization theorem for nilpotent Clifford semirings.

Theorem 2.10. A semiring S is a nilpoten Clifford semiring if and only if S is a strong distributive lattice of nilpotent rings.

Proof. Let S be a nilpotent Clifford semiring. Then S is a strong distributive lattice D of rings R,(a E D). We shall show that each Ra(a E D ) is nilpotent. For this purpose, we let a E R, for some

LY

n such that an = a

E

D. Then a E S. Since S is nilpotent there exists a positive integer

+ a’, which is the zero element of R,.

Hence each Re is nilpotent and

therefore S is a strong distributive lattice of nilpotent rings. Conversely, let S be a strong distributive lattice D of nilpotent rings R,(a E D ) . Then

S is a Clifford semiring. Let a

E

S. Then a

E

R, for some a E D. Since R, is nilpotent,

there exists a positive integer n such that a” is the zero element of R,, i.e., an = a+a’. This shows that a is nilpotent and hence S is a nilpotent Clifford semiring.

Definition 2.11. Let S be a Clifford semiring. We define a relation 8 on S by 8 = { ( a ,b) E

S x S : a + b‘ E E+(S)}.One can easily verify that p is in fact a congruence on S such that Sf0 is a ring.

226

Theorem 2.12. An additive inverse semiring S is a subdirect product of a nilpotent ring and a distributive lattice if and only if S is a nilpotent Clifford semiring.

Proof. First, we suppose that S is a nilpotent Clifford semiring. Then by Definition 2.11, the relation 6' defined on S is a congruence on S such that ,916' is a nilpotent ring. Again, since S is a nilpotent Clifford semiring, S is a strong distributive lattice D of nilpotent rings

R a , a D. We define a mapping ?i, : S + S/O x D by a$ = (ae,a ) ,

aE

R,.

Clearly $ is a monomorphism. Also the projection homomorphisms map S 9 onto D and ,916' respectively. This shows that S is a subdirect product of a nilpotent ring and a distributive lattice. Conversely, we assume that an additive inverse semiring S is a subdirect product of a nilpotent ring R and a distributive lattice D. Then, we may consider S a E D , let R,

=

R x D . For each

( R x { a } )n S. Then, we can easily show that R, is a nilpotent ring for

u R,.

each a E D and S =

Also, we can show that S is a strong distributive lattice D of

aED

nilpotent rings R, . Consequently, S is a nilpotent Clifford semiring.

3. Semisimple Clifford semirings Recall that a subsemiring I of a semiring S is called an ideal of S if S I

c I and I S

I.

An ideal I of a semiring S containing the set of all additive idempotents of S is called a full ideal of S. An ideal A4 of a semiring S is called a maximal ideal of S if there exists no ideal I of S such that A4;I;S. Throughout the section, S is always a semiring whose additive reduct ( S ,+) and multiplicative reduct ( S ,.) are both commutative. We first give some definitions. Definition 3.1. An ideal I of a Clifford semiring S is called nil if every element of I is nilpotent. Definition 3.2. An ideal I of a Clifford semiring is called nilpotent if there exists a positive integer n such that I"

c E+(S).

Example 3.3. We consider the ring S = 2 4 x D , where D is a distributive lattice. Then

I = { [ O ] , [2]} x D is a non-trivial nilpotent ideal of S. L e m m a 3.4. Every non-trivial nilpotent ideal of a Clifford semiring S is a nil ideal of S.

Proof. Let 1 be any nilpotent ideal of a Clifford semiring S . Then there exists a positive integer n such that I" = E + ( S ) .Let a E I . Then a" E I" & E+(S). So an = an

+ (an)' =

227 an +

of

a1 = a"-'(a

+ a') = a + a'. This shows that a is nilpotent and hence I is a nil ideal

s. We now prove the following theorem.

Theorem 3.5. The sum of finite number of nil ideals of a Clifford semiring is again a nil ideal.

Proof. It suffices to prove that the sum of two nil ideals of a Clifford semiring is also nil and then by induction. The proof is easy and we omit the proof. We now give the folkowing definition.

Definition 3.6. Let S be a Clifford semiring with 1 such that 1 6 E+(S).An non additive idempotent element a E S is said to be left invertible if there exists an element r E S such that ra

+ 1 + 1' = 1. In this case, we call r is called the left inverse of a. Similarly we can

define right invertible element in a Clifford semiring. An element is said t o be invertible if it is left invertible as well as right invertible. If a is invertible, we say that a is a unit of S.

Lemma 3.7. An element a in a Clifford semiring S with 1 is not a unit if and only if a is contained in a maximal full ideal of S.

Proof. We first suppose that a is not a unit. Then, we can see that E+(S)+Sa is a full ideal

+ + sa for some e E E+(S)and s E S i.e., sa+ e + 1= 1+ 1 i.e., sa + 1 = 1+ 1 i.e., sa + 1 + 1' = 1 + 1 + 1' = 1. This shows that a is left invertible.

of S. If E'( S) Sa = S then 1 = e

Similarly, we can show that a is right invertible. Consequently, a becomes a unit, which is a

+

contradiction. Thus E + ( S ) Sass.

+

Let C = { I : I is a proper full ideal of S containing a and E + ( S ) Sa & I } . Then, by using Zornis lemma, we see that C has a maximal element J . Thus J is a maximal full ideal of S containing a. Conversely, let a be an element in a maximal full ideal M of S and suppose that a is a unit. Then there exists

T

E

S such that ra + 1 + 1' = 1. Now a E A4 implies ra E M . Also,

1+ 1' E M. This leads to, 1 = ra + 1+ 1' E M ,a contradiction. Hence a can not be a unit. Next we give the following definition.

Definition 3.8. Let S be a Clifford semiring. Then the Jacobson radical of S , denoted by radS, can be defined by radS = n{M : M is a maximal full ideal of S } . By using Lemma 3.7, We are now able to prove the following important theorem.

Theorem 3.9. In any Clifford semiring, an element a E radS if and only if 1 invertible for each s E S .

+ s'a

is

228 Proof. We first let a E radS. Suppose if possible let 1+ ~ ’ ais not a unit of S for some T E S . Then 1 + T’U is contained in a full maximal ideal M of S. This leads to a E radS

M , and

so r a E M . Hence, we have 1 = 1 + ~ ’ +ara E M , a contradiction. This shows that 1

+ s’a

must be a unit for each s E S . Conversely, let 1 + s’a be a unit for all s E S . If a $ radS, then there exists a full maximal ideal M of S such that a

4

M . Let M’ =< M, a >. Since m = m + ( m + m’)a

for all m E M , we have MSM‘. Since M is maximal so S = M‘. This leads to 1 = m

+ sa,

+ sa + s’a E M . Since 1 + s’a is invertible, there exists an element c E S such that c ( l + s’a) + 1 + 1’ = 1. Now 1 + 1’ E M and c(l + s’a) E M.So 1 E M, a contradiction. Hence a E radS.

for some m E

M ,s

E

S. Hence, we have 1 + s‘a

=

m

We now give the following definition.

Definition 3.10. A Clifford semiring S is called Artinian Clifford semiring if any descending chain of full ideals of S terminates i.e., for any descending chain of full ideals 11

2 1 2 2 ....

there exists a positive integer n such that I, = I,+’ = I,+, = .... We can easily prove that a semiring S is Artinian if and only if any non empty collection of full ideals contains a minimal element. We need some lemmas.

Lemma 3.11. If a is a nilpotent element in a Clifford semiring S with 1 then (1 (1

+ a) are both invertible elements of S.

Proof. Since a is nilpotent, there exists a positive integer n such that a” = a T

+ a’) and

= 1+ a + a 2

+ ... +an-’.

Then ~ ( +a’) 1

+ 1 + 1’ = 1+ a + a Z + .. +an-’

+ a’.

Let

+a’+aa’+ ...

+

+

+ 1 + 1’ = 1. Similarly, (1 + U’)T + 1+ 1’ = 1. Hence (1 + a’) is an invertible element of S.

a”-’a’ 1 + 1’ = 1 + a + a‘

Again, let t = 1+a’+(a’)2+...+(a’)n-’.

a

Then t ( l + a ) + l + l ’ = 1+a’+ ...+(a’)“-l+

+ aa‘ + ...+ a(a’)”-’ + 1+ 1’ = 1+ a + a’ + 1+ 1’ = 1. Similarly (1 + a)t + 1 + 1’ = 1. So (1 + a) is also invertible element of S.

Lemma 3.12. Every nil ideal is contained in T U ~ where S S is a Clifford semiring with 1 such that 1 $ E+(S).

Proof. Let N be a nil ideal of S and a E N. Then for every s E S , sa E N. So sa is nilpotent. Now by Lemma 3.11, we see that 1

+ (sa)’ is invertible, i.e., 1 + s‘a is invertible

for all s E S . This shows that a E r a d s and so N contained in T ~ S .

C ~ a d S .Consequently every nil ideal is

229 We now prove the following theorem.

Theorem 3.13. If S is a Artinian Clifford semiring with 1 then radS forms a nilpotent ideal of S .

Proof. We consider the descending chain of full ideals (radS) 2 ( ~ a d S 2) ~.... Since

S is Artinian, there exists a positive integer n such that (radS)” = (radS)”+’ = .... Let radS and I’ = I . Clearly, E + ( S )C (radS)” = I . To complete the

I = (radS)”. Then I

proof, it suffices t o show that I

E+(S).Assume that I

C = { J : J is a full ideal of S such that J

g E + ( S )and consider the collection

s I and J I g E+(S)}.Then C # 4, since I E C.

Hence C has a minimal element K and thereby, K I

g E+(S). Hence a1 g E + ( S )for some

non additive idempotent element a E K . Thus, we have (a1

a1 + I

E + ( S )with (a1 + E’(S))

K

+ E + ( S ) ) IC a1 + E + ( S ) I

I . IIence by the minimality of K , we have

aI+E+(S)= K . Clearly, we can see that there exists an element b E I and e E E+(S)such that a = e+ab. But b E I

radS implies (1+ b’) must be an invertible element of S . So there exists an

+

element c E S such that (1+b’)c+ 1 1’ = 1. Hence, we have a

(e+ab+ab’)c+a+a’

E

E + ( S ) ,and so aZ

= a.1 = (a+ab’)c+a+a‘

=

C E + ( S ) ,contradicting the fact that aI g E+(S).

This proves that I = (radS)” 5 E + ( S ) ,as required.

Corollary 3.14. Let S be an Artinian Clifford semiring with 1 such that ( S ,.) is commutative. Then every nil ideal of S is nilpotent.

Proof. Let I be a nil ideal of S . Then by Lemma 3.12, we have I nilpotent, there is a positive integer n such that (radS)“

I”

C: (radS)” C E + ( S )and therefore I

radS. Since radS is

E+(S). Consequently, we have

is a nilpotent ideal of

S.

Lemma 3.15. Let S be a Clifford semiring with 1 such that ( S ,+) and ( S ,.) is not commutative. Then every non additive idempotent element of S is left invertible if and only if every non additive idempotent element of S is right invertible.

Proof. Let a E S be such that a $ E+(S). Then, there exists an element r E S such

+ 1 + 1’ = 2. Clearly, T $ E+(S). Hence, there exists an element t E S such that tr + 1 + 1’ = 1 i.e., tra+ a + a’ = a i.e., tra+ a +a’ + 1 = a + 1 i.e., tra+ 1 = a + 1. Also from ra + 1+ 1’ = 1, we deduce that tra + t + t’ = t , i.e., tra + 1 = t + 1. This leads to a + 1 = t + 1 so that ar + = tr + r i.e., ar + r + r’ + 1 = tr + r + + 1 i.e., ar + 1 = tr + 1 i.e., ar + 1 + 1’ = tr + 1 + 1’ = 1. Hence, a is right invertible. The converse part follows that t a

T

dually.

T’

230

Lemma 3.16. Any multiplicative idempotent element in radS is additive idempotent. Proof. Let a E radS be such that a’ = a. Then 1 + l’a = 1 + a’ is invertible. Hence,

+ + 1+ 1’ = 1. Consequently, we deduce that a = a.1 = ( a + aa’)b + a + a’ = ( a + a’)b + a + a’ = a + a’. This shows that a is additive

there exists an element b E S such that (1 a‘)b

idempotent.

Lemma 3.17. Every full maximal ideal of a Clifford semiring S with 1 is a k-ideal S . Proof. Let S be a Clifford semiring with 1 and M be a maximal full ideal of S . Let

a,a

+b

E M . We prove that b E M . Suppose if possible that b

+ ( m + m’)b, so MSM’. Since M is a maximal ideal, Hence,, 1 = m + sb, for some m E M , s E S . This leads to

M’ =< M , b >. Since m we must have M’ = S .

M . Then, we let

=

m

l f s ‘ b = m+sb+s’b E M . Again a E M implies s’a E M and consequently, l+s’b+s’a E M

i.e., l + s ’ ( a + b ) ~ M .A l s o , w e h a v e s ( a + b ) E M . Hence, l = l + s ’ ( a + b ) + s ( a + b ) E

M,

a contradiction. This shows that b E M so that M is a k-ideal of S . From the above lemma we immediately deduce the following corollary.

Corollary 3.18. Let S be a Clifford semiring with 1. Then radS is a k-ideal of S.

Lemma 3.19. Let S be a Clifford semiring with 1. Then S/radS is a ring with identity. Proof. Straightforward. We now give the following definition.

Definition 3.20. A Clifford semiring S is called a semisimple Clifford semiring if radS =

Ef(S). Clearly, E+(S) C radS. For semisimple Clifford semiring, we only need to show that radS

E+(S).

Example 3.21. We consider the Clifford semiring S = Z x D , where D is a distributive lattice. Then S is clearly a semisimple Clifford semiring.

Theorem 3.22. For any Clifford semiring S with 1, the ring S/radS is a semisimple ring. Proof. Let radS = I . Let a+I E rad(S/I). Then for every s E S , we can see that (l+s‘a)+l

+

+ +

+

is invertible in S / I . Hence, there exists b+ I E S / I such that (( 1 s’a)b+ 1 1’) I = 1 I

+ s’a)b + 1+ 1’+ 1’ E I or equivalently, (1 + s’a)b + 1’ E I = radS. This shows that 1 + (1 + s’a)b + 1’ is invertible in S , and consequently, there exists an element c E S such

i.e., (1

that c+(1+s’a)bc+c‘+1+1’=1i.e., ( l + s ’ a ) b c + l + l ’ = 1. Hence, (l+s‘a) isinvertible in

s for every s E s and so, a E r a d s = I . Thus, a + I = ( a + a’) + I is the zero element of

S/radS so that S/radS is semisimple. The proof is complete.

23 1 References 1. Bandelt, H. J. and Petrich, M.,Subdirect product of rings and distributive lattices.

Proc. Edinburgh. M a t h .

, 25 (1982), 155-171.

2. Burton, D. M., A First Course in Rings and Ideals. Addison-Wesley Publishing Company, 1970. 3. Ghosh, Shamik, A characterization of semirings which are subdirect products of a

distributive lattice and a ring. Semigroup Forum. Vol. 59, (1999), 106-120. 4. Golan, J. S., The Theory of Semirings with Applications in Mathematics and Surveys

in Pure and Applied Mathematics. 54, Longman Scientific and Technical, 1992. 5. Grillet, M . P., Semirings with a completely simple additive semigroup. J. Austral. Math. SOC.20 (Series A) (1975), 257-267. 6. Howie, J. M., Introduction t o the theory of semigroups. Academic Press , 1976. 7. Karvellas, P. H., Inverse semirings. J. Austral. Math. SOC.18(1974), 277-288. 8. Pastijn, F.,and Guo, Y . Q., The lattice of idernpotent distributive semiring varieties.

Science in China ( Series A ) 42(8) 1999, 785-804. 9. Sen, M. K., Guo, Y . Q. and Shum, K. P., A class of idempotent semirings. Semigroup Forum 60, (ZOOO), 351-367. 10. Sen, M. K., Maity, S. K. and Shum, K. P., O n Completely Regular Semirings. (Submitted).

11. Sen, M. K., Maity, S. K. and Shum, K. P., Cliflord semirings and generalized Clifford semirings . (Submitted). 12. Sen, M. K., Ghosh, Shamik and Mukhopadhyay, P., Congruences o n inverse semir-

ings. Algebras and Combinatorics, Proceedings ICAC 97 (HK), Springer-verlag (1999), 391-400. 13. Zeleznekow, J., Regular semirings. Semogroup Forum, 23 (1981), 119-136.

TIGHT INVERSE SEMIGROUPS

BORIS M. SCHEIN Department of Mathematical Sciences University of Arkansas Fayetteville, A R 72701, USA E-mail: [email protected]

To the memory of B. H. Neumann A semigroup is called tight if each of its congruences is uniquely determined by each of its congruence classes. This is a survey of old and new results about such semigroups with the main emphasis on inverse semigroups. The last section gives an easy description of all inverse subsemigroups of the bicyclic semigroup.

1. Preliminaries

This article can be read by those of us who wish their familiarity with semigroups were somewhat greater than it really is. For this reason I explain many elementary notions, thus risking to bother those of us who may wish their familiarity with semigroups were somewhat less than it really is. Some standard results that are not proved here can be found in every standard book on semigroup theory (for example, in [l]). Let h : S + T be a surjective (that is, “onto”) homomorphism of a semigroup S onto a semigroup T . The inverse images of the elements of T partition S into equivalence classes. The corresponding equivalence relation ker(h) is called the kernel of h. If S is a group, the equivalence classes of ker(h) are cosets of a normal subgroup. Each of these cosets uniquely determines ker(h). One of the cosets is distinguished because it contains the identity element of S , it is the only coset that is a subgroup. This is why, in group theory, this coset is called the kernel of h. However, if S is not a group, the situation is very different. An equivalence class C of ker(h) need not determine ker(h). Distinct kernels may share a same equivalence class. For example, if S is a left zero semigroup

232

233

(that is, zy = z for all z, y E S) then every equivalence on S is a kernel of a homomorphism, and hence every nonempty subset of S whose complement contains at least two distinct elements is a class of at least two different kernels of homomorphisms. The kernels are precisely those equivalence relations on S that are stable with respect to multiplication (that is, they are subsemigroups of S x S ) : a G b and c E d imply ac E bd for all a,b,c,d E S. Here z E y means that z and y belong to the same equivalence class of the kernel. A stable equivalence is called a congruence. For example, the identity relation AS and the universal relation S x S are congruences on every semigroup S. If S has no other congruences, it is called congruence-fp-ee. A congruence E on S is called tight if it is uniquely determined by each &-class (that is, if C is an &-class,then every congruence on S for which C is a class coincides with E ) . A semigroup S is called tight if all congruence relations on S are tight. For example, groups are tight. Congruence-free semigroups are also tight. In particular, a semigroup with at most two distinct elements is necessarily tight. For this reason we assume in the sequel that all semigroups have more than two elements. Thus every semigroup can be embedded in a tight semigroup (because every semigroup is embeddable in a congruence-free one). 2. Initial results

A semigroup S is called simple if it has no proper (two-sided) ideals, that is, S is its only ideal. If S has a zero element 0 then ( 0 ) is an ideal and S cannot be simple. If S has no ideals besides ( 0 ) and S, it is called 0-simple. Theorem 2.1. (See [4]) A tight semigroup without zero is simple. A semigroup with zero is tight i f and only if it is congruence-free. Proof. Let S be a tight semigroup that is not simple. Choose an ideal I of S such that I # S. Then RI = ( I x I ) U A s is the so-called Rees congruence on S; its congruence classes are I and singletons. If s 4 I , then {s) is both an RI-class and also a As-class. Therefore, RI = A s , that is, I is a singleton. The only ideal consisting of a single element is ( 0 ) . If S has no zero, the assumption that I # S leads us to a contradiction, and hence S is simple. If the semigroup has zero, we see that it is 0-simple. Let E be a congruence on a tight semigroup S with zero. The &-class~ ( 0 ) that contains 0 is necessarily an ideal. Therefore, this class is either ( 0 ) or

234

S. In the former case E and A s share the class {0}, and hence E = A s . In the latter case E = S x S . Therefore, S is congruence-free. Obviously, the 0 converse is true: every congruence-free semigroup is tight. An element s E S is called disjunctive if A s the only congruence with the congruence class {s}.

Theorem 2.2. (See [4]) Homomorphic images of tight semigroups are tight. A semigroup is tight i f and only i f all elements in all homomorphic images of this semigroup are disjunctive. Proof. To prove the first claim let E be a congruence on a tight semigroup S and let a’ be a congruence on the factor semigroup S / E . Then a‘ = a / & , where a is an appropriate congruence on S (it is the inverse image of a‘ under the canonical homomorphism h : S + S / E ) .If C‘ is an a’-class, then A = h-I(A’) is an a-class, and hence it uniquely determines a because S is tight. Also, a determines a’ because a’ = a/&.Thus A‘ determines a‘ uniquely, which means that a’ is tight. It follows that S / E is tight. Every homomorphic image of S is isomorphic to S / E for an appropriate congruence E , and hence it is tight. The second claim of the theorem consists of two parts. Necessity. All elements of a tight semigroup S are necessarily disjunctive because each of them is a class of the identity congruence As. Since homomorphic images of tight semigroups are tight, all elements in all homomorphic images of a tight semigroup are disjunctive. Suficiency. Suppose that all elements of all homomorphic images of S are disjunctive. Let C S be a class of two congruences a and ,8 on S . Then C is a class of the congruence 6 = anp. Therefore, C as an element of S / a , S/P, and ,916, and hence {C} is a class of the congruences As/J, a / & , and ,B/S on the semigroup S/6. All elements of S/6 are disjunctive, and so AS/J = a/6 = P / S . It follows that a = p. Therefore, S is tight. 3. Disjunctive elements in inverse monoids

If S is a semigroup and a, b E S , we say that a is a left divisor of b whenever a = b or az = b for some z E S. If a and b are left divisors each of the other, we say that a and b are right associates and write (a,b) E R. Clearly, R is an equivalence relation on S . Analogously, define right divisors and left associates. Let L be the equivalence relation for left associates. Also, let 3t = RnL. It is well known that the equivalence relations R and L commute

235

with respect t o the relative product of binary relations: R o C = C O R= V , and hence 2) is the join R V C of R and C in the lattice of all equivalence relations on S . The equivalence relations R, C, N ,and V are called Green's relations on S . A semigroup S is called (von Neumann) regular if for every s E S there exists t E S such that sts = s. In this case there exists u E S such that both sus = s and usu = u are true (just take u = tst). We say that u is an inverse of s (in which case s is obviously an inverse of u).In other words, a semigroup is regular if each of its elements possesses an inverse. A semigroup S is called inverse if each of its elements possesses a unique inverse, which we denote by s-'). It is well known that inverse semigroups are precisely the regular semigroups with commuting idempotents (that is, if e2 = e and f 2 = f are idempotents, then ef = f e . ) Inverse semigroups are the most important class of semigroups besides groups. Clearly, an inverse semigroup S is a group precisely when S has a single idempotent. However, inverse semigroups are not necessarily tight. For example, it can be proved that an idempotent inverse semigroup (that is, a semilattice) with more than two elements is never tight. Green's relations on an inverse semigroup S can be expressed in a somewhatmore compact form. For example,

R = { ( s , t ) lss-1 = t t - I } , c = { ( S i t ) [ s-ls

=t-It},

and

V

= { ( s , t ) J(3u)[ss-l = uu-l and u-lu = t-lt}.

Every inverse semigroup S is canonically or naturally ordered by the (partial) order relation s


s = ss-lt H s = s t - l s H s = ts-ls.

This order relation is stable with respect to both multiplication and inversion, that is, s1 4 tl and s2 4 t 2 imply s1s2 < t l t 2 and s < t implies s-1 4 t - 1 . A congruence E on an inverse semigroup S is called a group congruence if the factor inverse semigroup S / E is a group. Every inverse semigroup S has a (necessarily unique) least group congruence y. Here sG

A congruence

t

(mod y)

(j

(I.)[. 4 s and u < t].

E is a group congruence precisely when all idempotents of S are contained in the same &-class.

236

We say that a subset B C S of an inverse semigroup S is a (majorant) base if every element s of S is the 1.u.b. (that is, the least upper bound with respect to <) of a subset C B . We write s = C in this case. If S is an inverse monoid (that is, an inverse semigroup with an identity element e), let D1 denote the 2)-class of S that contains e. The following technical proposition is a result of collaboration between Ms. H. Y. Wu and the author.

v

Proposition 3.1. Let S be an inverse monoid. Its identity element e is disjunctive if and only if D1 is a base of S . Proof. Necessity. Suppose that e is disjunctive and define the following binary relation E on 5':

s

= t H (Vz,y

E S)[zsy = e @ z t y = el.

Obviously, 3 is a congruence. If s e then eee = e implies s = ese = e, and hence {e} is a congruence class of E. For each s E S define a set A, = { a E D1 I a 4 s}. Observe that s is an upper bound of A, (that is, a 4 s for each a E A s ) . Assume that t E S is another upper bound of A,. Let z s y = e for some x , y E S. Then

2z-l = z z - l e = zz-lzsy = z s y = e. Analogously, y-ly = e. Thus z , y - l E D1. Now, z-ly-lyz = 2-12,

(z-ly-l)(z-ly-l)-l

and hence x-ly-'

E D1. Also, y-lz-l

- y-lez-l -

- z -1 xsyy-1

so that z-ly-l E A , . It follows that y-lz-l e = zz-ly-ly

<s

4 t, whence

4 zty.

Thus e = e(zty) = zty. We have proved that (Vz, y E S)[zsy = e

+ zty = e l .

(1)

Substituting zss-' for z in (1) and using (zss-')sy = z s y we obtain ( ~ z , yE ~ ) [ z s y= e

+ zss-'ty

= el.

Suppose that zss-'ty = e. Then e = e-l = (zss-lty)-l = (y-'t-')s(s-'z-').

(2)

237

BY (21, = (y-lt-l)t(s-lz-l) = y-l(t-lt)s-lz-l

Therefore, e = e-l 4 ( y - l s - l z - l ) - l

4

? p s - l 2 -1 .

= zsy. This and (2) imply

(vz,~ E ~ ) [ z s= y e H xss-lty = el ,

(3)

that is, ss-lt s. However, e is disjunctive, and hence ss-lt = s, that is, s < t. It follows that E is the identity congruence A s (because { e } is its congruence class). Thus s = V A,, the 1.u.b. of A,. Suficiency. Suppose that each element s of S is the 1.u.b. of a subset B, of D1. Then B, A,, and hence s = V A,. Analogously, t = At. Let s t for some s, t E S. Suppose that a E A , . Then a E D1 , and hence uu-l = e and u-lu = a-la for some E S . Also a < s, whence a = aa-'s. Therefore,

v

e = uu-l

z u(u-lu)u-'

= (ua-l)au-l = Ua-l(aa-1s)u-l = ua-lsu-l.

Thus uu-l = e = ua-ltu-ll and hence a = aa-la = au-'u

= av-leu = au-lua-ltu-lu

4

t.

It follows that a E At. We have proved that A , C A t . Analogously, At 5 A , , whence A , = A t . Therefore, s=

v A, v =

At = t

=

and is the identity congruence A,. Let E be a congruence on S and {e} its congruence class. Suppose that s E t (mod E ) . If zsg = e for some x,g E S then e = m y xty (mod E ) , and hence zty = e. Similarly, xty = e implies xsy = e. Therefore, s t , so that s = t. It follows that e is a disjunctive element of S. 0

=

4. Bisimple inverse semigroups

A semigroup S is called bisimple if V = S x S , that is, if all elements of S are 2)-equivalent (for every a , b E S there exists c E S such that a and c are left (right) associates, while b and c are right (left) associates). It is easily seen that an inverse semigroup S is bisimple exactly when every two idempotents e and f of S are V-related, that is, for every idempotents e and f .there exists u 6 S such that uu-l = e and u-lu = f . Theorem 4.1. Bisimple inverse semigroups are tight.

238

Proof. It is known that a homomorphic image of an inverse semigroup is an inverse semigroup. It is easy t o see that a homomorphic image of a bisimple inverse semigroup is bisimple. Thus by Theorem 2.2 it suffices to prove that every element c of a bisimple inverse semigroup S is disjunctive. Suppose that E is a congruence on S , { c } is its class, s t (mod E ) for some s , t in S , ss-l = uu-l, and c-lc = u-lu. Then

c = cc-lc = cu-lu = cu-1(uu-1)u = (cu-')s(s-lu)

(cu-l)t(s-1u).

Since { c } is an &-class, c = cu-lts-lu. Now, s = ss-ls = uu-ls = u(u-1u)u-ls = uc-lcu-ls

= u(u-lst-1uc-1)cu-15

= SS-lSt-luu-luu-lS

= st-luu-ls = st-lss-ls = st-ls,

and hence s 4 t. Analogously, t 4 s, so that s = t. Thus c is disjunctive.^ Theorem 4.1 was first proved in [6] in a rather complicated way. A much simpler proof was found in [4]. 5. Tight inverse semigroups that are neither bisimple nor congruence-free

If a semigroup is bisimple but not inverse, it need not be tight. For example, a left zero semigroup is not tight although it is both bisimple and regular (and even idempotent). Thus the class BSZ of bisimple inverse semigroups is contained in the class 7 2 of tight inverse semigroups, while in turn 'TIis contained in the class SZ of simple inverse semigroups: BSZ g 7 2 2 SZ. Also, there exist congruence-free (and hence tight) inverse semigroups without 0 that are not bisimple (see [3]). A natural question is: are there tight inverse semigroups that are neither bisimple nor congruence-free? This question has remained open since 1971. It has been answered in the affirmative in [5], where the first example of such semigroups and monoids was constructed. A simpler example is given below. Recall that a semigroup B generated by two elements, p and q , and subject to a single defining relation p q = 1 is called bicyclic. Its elements are all products of the form qmpn where m,n E Nf (here Nf is the set of all nonnegative integers). We can denote qmpn by (rn,n),in which case

239

B = N+ x

N+ and the multiplication

in B is given by the following rule:

If we define m 2 n = max{m - n, 0}, then (ml,nl)(m2,n2) = ( m 1 +(m2 -n1),722+(n1 - m 2 ) ) .

Clearly, (m,n)-' = (n,m). The idempotents of B are the elements of the form (m,m ) , They form a chain (0,O) + (1,l) + (2,2) + . . . . Any congruence on B is either AB or a group congruence. Also, elements ( m l ,nl) and ( ~ ~ 2 , 7 2 2 )of B are R-related precisely when ml = mz, and they are L-related precisely when n1 = 722. It follows that 3t = AB. Example 5.1. The set S = {(m,n)l(m,n) E B; m n (mod 2)) is an inverse subsemigroup of B. The inverse semigroup S is tight but neither congruence free nor bisimple (it has two 2)-classes). Proof. It follows from the definition of multiplication and inverses in B that S is closed under both multiplication and inversion, that is, it is an inverse subsemigroup of B. It has two 2)-classes. One of them, D1,consists of all (m,n)such that m and n are even nonnegative integers, while the other 2)-class, D2, consists of all (m,n)with both m and n odd. Each of these 2)-classes is an inverse subsemigroup of S isomorphic t o the bicyclic semigroup B. Also, S is a fuZZ inverse subsemigroup of B (that is, S contains all idempotent elements of B). Define an equivalence relation 77 on S as follows: the equivalence classes of 77 consist of the two elements (2m,2n) and (2m + 1,2n l), for every m,n 2 0. It is easy to see that 77 is a congruence relation on S . We call it the halving congruence. To describe all congruences on S (there are infinitely many of them) suppose that E is such a congruence.

+

Case 1 . Suppose that E is idempotent-separating (that is, no two distinct idempotents belong to the same &-class). If s, t E S , then s 3 t (mod E ) implies that ss-' = tt-' (mod E ) and also s-'s t-lt (mod E ) . Since s s - l , tt-', s-'s, and t-lt are idempotents and E is idempotent-separating, ss-l = tt-' and s-ls = t-lt. It follows that s and t are 3t-related in B, and hence s = t , that is, E = A s . Case 2. Now suppose that E is not idempotent-separating. Then it is possible t o find two distinct idempotents (m,m) and (n,n ) in S such that

240

( m , m )E (n,n) (mod E ) . Since m # n, we can assume without loss of generality that m < n. Consider two subcases.

+ 1 for some m and n. Then ( k , m + l ) ( m , m ) ( m + l , k=) ( k , m + l ) ( n , n ) ( m + l , k )( m o d € ) .

Subcase 2.1. Let n > m

Clearly, ( k , m + l ) ( m , m ) ( m + l , k= ) (k,k).

+ 1) = d > 0. Then ( k ,m + 1)(n,n)( m+ 1,k ) = ( k + d , k + d ) , and hence ( k ,Ic) & ( k + d , Ic + d ) for all k > 0. Replacing k here by k + d we obtain ( k + d, k + d) & ( k + 2d, k + 2d). Repeatingthis process we obtain ( k , Ic) ( I c + Id, k + Ed) for all k , I 2 0. In particular, for k = 0, we obtain Let n - ( m

(0,O) 6 (Id, Id). It follows from 1 5 Id that (0,O)

(Id, Ed) = (Id, Id)(l,I)

& (O,O)(Z,

1) = (1,E)

for all idempotents (1,l) of S . Therefore, all idempotents of S belong t o the same &-class,and thus E is a group congruence.

Subcase 2.2. Now suppose that if (m,m) (n,n) (mod E ) for m < n , then n = m 1. Let (a,u ) G ( a 1,a 1) (mod E ) for some a 2 0. Then

+

+

L (k,a)(a + 1,a + l ) ( a ,I)

( k ,1 ) = ( k ,a ) ( u ,a)(.,I) for all k , I 2 0. Therefore,

+

E

= (k

+ 1,I + 1)

is the halving congruence 7 .

It follows that every congruence on S is either A s (with singleton classes), or a group congruence (with infinite classes), or the halving congruence (with classes consisting of two elements each). Thus A s and 17 are tight congruences, and it remains to prove that all group congruences on S are tight. To do this I reproduce an argument used in the proof of Lemma 7 in [5], which, in turn, was adopted from [4]. Let EI and ~2 be group congruences on S and let C be a congruence class of both congruences. The natural homomorphism S -+ S / E is ~ the product of two natural homomorphisms h : S -+ S/y,where y is the least group congruence on S , and hl : S/y + S/c1. The image of C under h is C/y, and it is the set of y-classes whose union is C. Since C is an El-class, the image of C/r in S / E is ~ a single element, and hence C/y is a class of the kernel congruence ~1/yof hl. Repeating the same argument with ~2

241

instead of E ~ we , see that C/r is a class of the kernel congruence of the homomorphism Sly -+ S / E Z .Therefore, C/y is a class of t w o congruences on the group Sly. A congruence on a group is uniquely determined by each of its congruence classes, and hence ~1I-y = E ~ / Y which , implies ~1 = € 2 . It follows that S is a tight inverse semigroup, since all congruences on S are tight. 0 Our description of all congruences on S shows that the set of all congruences distinct from A s has a least element, the halving congruence 7. Therefore, S is subdirectly irreducible (that is, if S can be isomorphically embedded in a direct product of a family of semigroups, then it can be isomorphically embedded in one of these semigroups) .

In Example 5.1 we considered an inverse subsemigroup of B . The bicyclic semigroup B is one of the “standard” semigroups. Yet no one has described all subsemigroups of B (in fact, even inverse subsemigroups of B remain unknown). Such a description is not difficult and, with many a thousand of semigroup articles already existing, a separate paper on that specific topic might be considered too shallow to see the light of day. Yet a list of all inverse subsemigroups of B may deserve a page and a half. These subsemigroups are only slightly more complicated than our semigroup S. 6. Inverse subsemigroups of the bicyclic semigroup Every nonempty subset of EB,the chain of all idempotents of the bicyclic semigroup B , is an idempotent subsemigroup of B. Both EB and B are countably infinite. Thus B has 2N0 inverse subsemigroups. Thus the set of all inverse sebsemigroups of B has the cardinality of continuum. By Theorem 6.1, B has No inverse subsemigroups that are not subsemigroups Of

EB.

Choose a positive integer s and a sequence bo < bl < ... < b, of s + 1 nonegative integers. Let d = b, - bo. Obviously, d > 0. Define P(bo,..., b,) = { ( b i + m d , b i + n d ) ) Z = O , l , ..., s - l a n d m , n = 0 , 1 , 2 ,... }. Observe that i # s. An easy argument shows that P(b0,. . . ,b,) is an inverse subsemigroup of B, and its D-classes are P ( b o ) , . . . , P ( b s - l ) . Let a1 < . . . < a, be another sequence of nonnegative integers, where T 2 0 and a, < bo (if T = 0, our sequence is empty). Define the set P(ai,...a,;bo,...,bs) = {(ai,ai),...,(a,,a,)}

UP(bol...,bs).

Clearly, P(a1,. . .a,; bo, . . . ,b,) is an inverse subsemigroup of B.

242

Theorem 6.1. Every inverse subsemigroup of B is either a nonempty subset of EB or one of P(a1,. . . a,; bo, b l , . . . ,b 8 ) .

Proof. Let S be a n inverse subsemigroup of B and suppose that S EB. Then S contains elements of the form (m,n)with m # n. Therefore, S contains elements of the form ( m , n )with m > n (because, if m < n, then ( n , m )= (m,n)-l E S). Let bo be the smallest nonegative integer such that (bo,b) E S for some b > bo. Of all such b choose the smallest and call it a. Then d = a - bo > 0. It is easy to see that (b0,a)" = (bo,bo n d ) and (bo,a)-" = (bo + nd,bo) for every n > 0. Therefore, k = qd and (bo + m d , bo n d ) = (bo, a)-"(bo, a)" E S , and hence P(b0,a ) C S . Suppose that (bo,bo k ) E S for some k 2 0. By the minimality of a , d 5 k . Divide k by d with a remainder: k = qd s, where 0 5 s < d. Thus (bo, bo + s) = (bo, bo + k)(bo + qd, bo) E S . It follows from the minimality of a that s = 0. Therefore, (bo,bo k ) = (b0,qd) E P(b0,a). Let S\P(bo,a) contain a non-idempotent element (z,y). Then it also contains (y, z) = (z,y)-' and so, without loss of generality, we may assume that z < y. Clearly, bo < z. If bo kd 5 z < bo + ( k + l ) d for some k 2 0 then (z - k d , z) = (bo, bo k d ) ( z ,z) E S . Let z - k d = z'. Then bo 5 z' < bo d. Therefore, (z', y) = (z',z)(z, y) E S . Now, (bo, bo d - z' y) = (bo, bo d ) ( z ' ,y) E S and hence d is a divisor of y - z' + d , that is, dly - z'. Let y - z' = Id. Then (z,y) = (z' k d , z' Id) E P(z',z' d). Collect all numbers z' such that bo 5 z' < bo d. These numbers form a sequence bo < . . . < b,-l. Let b, = a. Then P(b0, . . . , b,) 2 S and all nonidempotent elements of S belong to P(bo,. . . ,b8). Let (u, u)E S\P(bo, . . . ,b s ) . If (u, w) E S for some v # u,then (u, u)= (u,v)(u,v)-' E P(b0,.. . , b,) because ( u , v ) E P(bo,.. . , b,). Thus ( u ,v ) $! S for all v # u. If bo 5 u then (u,u d ) = (u,u)(bO,bO d ) E S , which is impossible. Thus u < bo. There are only finitely many such u. Let them be a l , a 2 , . . . ,a,. Then S = S ( a 1 , .. .a,; bo, . . . , b s ) . 0

+

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+

Recall that an inverse subsemigroup T of an inverse semigroup U is called full if T contains all idempotents of U . The following Corollary is an immediate consequence of Theorem 6.1.

Corollary 6.1. a The full inverse subsemigroups of the bicyclic semigroup B are EB and P(O,l,..., a - 1;a , a 1,a 2 , . . . , a d ) for all a 2 0 and

+

+

+

&I am grateful to Prof. P. R. Jones who informed me that another description of full inverse subsemigroups of B can be obtained from the arguments he gave in [2].

243

dL1.

In other words, every full inverse subsemigroup of B is EB or has the following f o r m . Let a 2 0 and d 2 1 be fixed integers. O u r subsemigroup consists of a finite (possibly empty) chain { ( O , O ) , (1,l),( a - 1,a - 1)) of idempotents and of all elements of the f o r m ( m , n ) ,where m and n belong t o a n arithmetic progression with difference d and the first t e r m 2 a . References 1. A. H. Clifford and G. B. Preston, T h e Algebraic Theory of Sermigroups, Vols. I and 11. Mathematical Surveys #7, American Mathematical Society, Providence, R. I., 1961, 1967. 2. P. R. Jones, Distributive inverse semigroups. Journal of the London Mathematical Society (2) 17(1978), 457-466. 3. W. D. Munn, Fundamental inverse semigroups. Quarterly Journal of Mathematics (Oxford) (2) 71(1970), 157-170 4. Boris M. Schein, A remark concerning congruences o n [0-Ibisimple inverse semigroups. Semigroup Forum 3(1971), 80-83. 5. B. M. Schein and H. Y. Wu, Tight inverse semigroups that are n o t bisimple. Submitted. 6 . G. I. Zhitomirsky, On imprimitivity relations of generalized groups of transformations. Matematicheskii Sbornik 73(1967), 500-512 [in Russian].

MOOR-PENROSE GENERALIZED INVERSES OF MATRICES OVER DIVISION RINGS

ZHE-XIAN WAN* Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 100080 Beijing, China and Institute of Mathematics, Suzhou University, 21 5006 Suzhou, Jiangsu, China

Dedicated to the memory of Bernhard H. Neumann 1. Introduction

Let D be a division ring with an involution -. By an involution - of D we mean a bijective maps -: D 4 D ,a + 6 which satisfies a+b = zi 6, ab = 66, and Z = a for all a, b E D. Let A be an m x n matrix over D. An n x m matrix X over D is called a Moor-Penrose generalized inverse of A , (or an M-P inverse of A for short), if the following four conditions are satisfied:

+

AXA = A , XAX = X , =AX, =XA.

A is said to be M-P invertible if A has an M-P inverses. It follows immediately from the definition that if X is an M-P inverse of A , A is also an M-P inverse of X and is an M-P inverse of %.

tx

This purpose of this note is to study the condition for the existence of an M-P inverse of a given rn x n matrix A over D , the uniqueness of the M-P inverse of A and the explicit expression of the M-P inverse of A if it exists. *Supported by National Science Foundation of China.

244

245

2. SomeLemmas Let D be a division ring and denote the set of n x n invertible matrices over D by GL,(D), which forms a group with respect t o the matrix multiplication, called the general linear group of degree n over D. Let

D " = { ( z ~,..., Z,);Z~ € D , i = l , 2,..., n } . D" is an n-dimensional left vector space over D with respect t o the addition (21,.

. . , x n )+ (Yl,.. . ,Y") = (21 + Yl,. . . , X " + for all xi, yi E D , i = 1 , 2 , 3 , . . . , n ,

Yn)

and scalar multiplication. x(x1,.

. . , 2") = (2x1,.. . ,xx,) for all x, xi E D,i = 1 , 2 , . . . ,n .

Let P be an r-dimensional subspace of D" and let of P. We use the r x n matrix

211,

. . . ur be a basis

t o represent the subspace P , call it a matrix representation of P , and write

P=

(:)

,

"Jr

i.e. we use the same letter P t o denote a matrix representation of the subspace P. It should be noted that a matrix representing an r-dimensional subspace of D" is an r x n matrix of rank r. It is clear that two r x n matrices P and Q both of rank r represent the same r-dimensional subspace if and only if there is an element T E GLr(D)such that P = T Q . Let A be an m x n matrix over D. Denote the subspace spanned by the rows of A over D by L ( A ) . Then L ( A ) C D". Let r > 0 and denote by M ( r ,n) the set of r-dimensional subspaces of D". For any R E M ( r ,n) we fix a matrix representation of the subspace R, which will also be denoted by R. Denote by M ( m x n , ~the ) set of m x n matrices of rank r over D. For fixed R E M ( r l n )and C E M ( r , m ) let

M ( m x n, r, R, C ) = { A E M ( m x n, r ) : L ( A ) = R and L(tA) = C } .

246

Then M ( m x n, r ) is partitioned into the following disjoint union:

M ( m x n,r)=

U

M ( mX n,r,R,C)

7

( R , C ) E M( r ~x M ) (r,m)

Lemma 2.1. Let r > 0 , A E M ( m x n,r) and R E M ( r ,n ) ,C E M ( r ,m ) . Then A E M ( m x n , r ,R, C ) i f and only i f A = tCTR for some T E GL,(D). Moreover, the map T H t c T R is a bijection from GL,(D) to M ( m x n, r, R , C ) . Proof. If A = tCTR for some T E GL,(D), then L ( A ) = L(R) = R and L(%) = L ( C ) = C. Therefore A E M ( m x n, r, R, C ) . Conversely, assume that A E M ( m x n , r ,R , C ) , then rank A = r , L ( A ) = R and J ~ ( ~=AC). There exist P E GL,(D) and Q E GL,(D) such that

A=P(':'

:)Q,

where I(') denotes the r x r identity matrix. Thus

R = L ( A ) = L((I(') 0)Q) , hence (I(') 0)Q = T1R for some TI E GL,(D). Similarly,

c = L ( 2 )= L ( ( I ( ' ) , O ) t P ) , hence (I(') 0) t P

= T2C

for some T2 E GL,(D). Therefore,

where T = tT2T1 E GL,(D). Assume that A = t c T R = t c S R where T , S E GL,(D). Then t C ( T - S ) R = 0. Since R is an r x n matrix of rank r , t(7(T - S ) = 0. Since C is an r x m matrix of rank r , T - S = 0. This proves the injectiveness. 0

Lemma 2.2. Let r > 0; R E M ( r , n ) , C E M ( r , m ) , and A E M ( m x n,r, R, C ) . Assume that A is M-P invertible and let X be a n M-P inverse of A . Then X E M ( n x m, T , C,R ) . Proof. By the definition of an M-P inverse we have

L ( A ) = L (A X A ) C L (X A ) = L(t(XA)) 5 L ( t X ) and

L ( X ) = L (X A X ) C L ( A X ) = L ( t ( A X ) ) 5 L(tA).

247

Thus dim(L(A)) C dim(L(tX)) = ranktX = rankX = dim(L(X)) C dim(L(9)) = rank(%) = rank(A) = dim(L(A)) .

Therefore, dim(L(A)) = dim(L(tX)) and

dim(L(X)) = dim(L(%)),

which by (2.1) and (2.2), imply

R = L(A) = L(tX) and C = L(X) = L ( % ) , respectively. Hence X E M ( m x n , r, C, R).

0

Let P be a subspace of D". Define

PL = {z E D" : P t 3 = O } , P' is also a subspace of D" and dim PL = n - dim P. Lemma 2.3. Let T > 0 , R E M ( r , n ) ,C E M ( r , m ) , and A n , r, R, C ) . Then the following statements hold: (i) (ii) (iii) (iv)

E

M(mx

rank(AtA) = rank(RtR), rank(tAA) = rank(CtC), dim(L(A) n L(A)'-) rank(RtR) = r , dim(L(tA) n L(%)'-) rank(CtC) = r.

+ +

Proof. From Lemma 2.1 we can assume that A = tC?TR for some T E GLr ( F )* (i) We have AtA = t C T R t R t T C . Since T E G L r ( F ) ,rank(RtR) = rank(TRtRtT). But C is of T x m matrix of rank r , therefore rank(tCTR t R t?%') = rank(TR tR "i?). Hence rank(AtA) = rank(RtR). (ii) Let B = %, then B = t R t T C . By Lemma 2.1 B E M ( n x m,r,C,R). Thus rank(tAA) = rank(BtB) = rank(CtC), where the last equality follows from (i). (iii) We have L(A) n L(A)' = {z E L(A) =

=R :Rt3=0)

{x = yR : y E D", R t R t j j = 0 ) .

248

SO, dim(L(A) n L(A)l) = dim ({x = yR : y E D", R t R t g = 0)) = dim({y E D"

: R t R t y = 0))

= r-rank(RtR).

(iv) Let B = tA, then B E M ( n x m , r , C , R ) . Thus dim(L(tA) n L('A)') = dim(L(B) n L(B)') = r-rank(CtC), where the last equality follows from (iii). 3. Existence, Uniqueness and Expression of M-P Inverses

Theorem 3.1. Let r > 0, R E M(r,n), C E M(r,m), and A E M ( m x n, r, R, C). Then the following statements are equivalent: (i) (ii) (iii) (iv)

The matrix A is M-P invertible, rank(RtR) = rank(Ctc) = r, rank(At& = rank(tAA) = r, L(A) n L ( A ) l = (0) and L(%) n L(%)I = (0).

Proof. By Lemma 2.1 we can assume that A = tCTR for some T E GL,(D). (i) + (ii). Let X be an M-P inverse of A. By Lemma 2.2 we can assume that X = tRSC for S E GL,(D). Then tCTR = A

= AXA = tC'TRtRSCtC'TR.

If any one of R t R and C t c is of rank < r, TRtRSCtC'T is of rank < r . Since R i s r x n a n d o f r a n k r a n d C i s r x m a n d o f r a n k r , t C T R t R S C t c T R is of rank < r, which contradicts that A is of rank r. Therefore both R t R and C '6 are of rankr. (ii) ($ (iii). Follows from (i) and (ii) of Lemma 2.3. (ii) H (iv). Follows (iii) and (iv) of Lemma 2.3. (ii) + (i). We have C A t R = C(tCTR)t R = (CtC)T(RtR)E GL,(D). Let X = tR(CA tR)-lC. Then it is easy to see that X is an M-P inverse of A by checking the conditions (1.1)-(1.4). 0

Theorem 3.2. Let r > 0, R E M ( r , n ) , C E M ( r , m ) and A E M ( m x n , r , R , C ) . If A is M-P invertible, then C A t R E GL,(D) and X = tR(CAtR)-lC is the unique M-P inverse of A. Proof. Assume A is M-P invertible. By the proof (ii) + (i) in the proof of Theorem 3.1, C A t R E GL,(D) and X = tR(CAtR)-'C is an M-P inverse of A.

249

Assume XI is any M-P inverse of A. By Lemma 2.2, X1 = t R S ~ C for some S1 E GL,(D). As in the proof (i) + (ii) of the proof of Theorem 3.1, tC?TR = t c T R t R S I C t C T R and T R t R S I C t C T E GL,(D). By Lemma 2.1, T = TRtRSICtCT, which implies T-' = R t R S I C t C . Therefore,

x1= %sic = t R ( ~ t R ) - l ~ - l ( ~ t C ) - l ~ = tR[(CtC)T(RtR)]-lC =tR(CAtR)-lC = X .

0

4. Examples

Example 4.1. Let D = C the complex field and - be the complex conjugation. Let A be any m x n matrix of rank r > 0 over C, and let L(A) = R, L(tA) = C. Then C E M ( r ,m ) , R E M ( T n), , and A = tCTR for some T E GL,(@). Clearly, both R t R and C tC are T x r positive definite Hermitian matrices, thus rank(R = rank(C ' C ) = r. By Theorem 3.1, A is MP invertible. By Theorem 3.2, C A t R E GL,(D) and X = tR(CAtR)-lC is the unique M-P inverse of A. This is the classical case ([l],[4]).

"a>

Example 4.2. Let D = R the real field and - be the identity map of R. Let A be an m x n matrix of rankr > 0 over R. By Theorem 3.1, A is M-P invertible if and only if rank(AtA) = rank(tAA) = r . If these two conditions are fulfilled, A has a unique M-P inverse given by Theorem 3.2. This case is also known ([I]). Example 4.3. Let D = W be the division ring of real quaternions and be defined by a bi c j dk = a - bi - c j - dk for all a , b, c, d E R. Let A be an m x n matrix of rankr > 0 over W,and let L(A) = R, L(%) = C. Then C E M ( r , m ) ,R E M ( r , n ) ,and A = tCTR for some T E GL,(W). Clearly, both R t R and C tC are T x r positive definite Hermitian matrices, thus rank(RtR) = rank(Ct6) = r. By Theorem 3.1, A is M-P invertible. By Theorem 3.2, C A t R E GL,(D) and X = tE(CA tR)-lC is the unique M-P inverse of A.

+ + +

Example 4.4. Let D = Fq2 be the finite field with q2 elements, where q is a prime power, and let - be defined by 7i = a4 for all a E Fq2. Let A be any m x n matrix of rankr > 0 over F,z. By Theorem 3.1, A is M-P invertible if and only if rank(A tA) = rank(tAA) = r. If these two conditions are fulfilled, A has a unique M-P inverse given by Theorem 3.2. This is the case studied in [2].

250

Example 4.5. Let D

= IF, be the finite field with

q elements, where q is a prime power and let - be the identity map of IF,. Let A be any m x n matrix of r a n k r > 0 over IF,. By Theorem 3.1, A is M-P invertible if and only if rank(AtA) = rank(tAA) = r. If these two conditions are fulfilled, A has a unique M-P inverse given by Theorem 3.2. This is the case studied in [3].

References [l] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Robert E. Krieger Publishing Company, Huntington, New York, 1980. [2] Z. Dai and Z.-X. Wan, MP-invertible matrices and unitary groups over F,2, Science in China 45 (2002), 443-449. [3] Z. Dai and Y.-F. Zheng, Partition, construction and enumeration of M-P invertible matrices over finite fileds, Finite Fields and their Applications 7 (200l), 428-440. [4] R. Penrose, A generalized inverse for matrices, Proc. Camb. Philos. SOC. 51 (1955), 406-413.

Properties and Characterization Theorems of

S AP-rings Wu Zhixiang Department of Mathematics Zhejiang University, Zhejiang 310027 P. R. China

K. P. Shum* Science Faculty The Chinese University of Hang Kong Shatin, N . T.,Hongkong China (SA R) Dedicated to the memory of Professor B H Neumann Abstract

We call the rings whose simple modules are absolutely pure the SAP-rings. The relationships between SAP-rings,V-rings, and von Neumann regular rings are now explored. Some recent results obtained by Huynh-Jain-Lopez Permauth are generalized and the results recently obtained by Wu-Xia are strengthened. Keywords. Absolutely Pure Modules; SAP-rings; V-rings; von Neumann regular rings. AMS Mathematics Subject Classijkation(2002). l6A30,16A40,l6A50,16A64

1

Introduction Throughout this paper, the ring

R is a n associative ring with identity, and all

R-modules are unital. We call a n R-module absolutely pure if M is pure in every Rmodule containing M. Consequently, an R-module M is absolutely pure if and only if

M is pure in its injective hull. According to Stenstrom [8], a n absolutely pure module 'The second author is partly supported by a UGC(HK) great 2060187 (2002/3)NSFC

25 1

252 is also called FP-injective module. We now call a ring R SAP-ring if every right simple R-module is absolutely pure. We call a ring R a right V-ring if every right simple module is injective. Clearly, every right V-ring is a SAP-ring. Also, it is known that a ring is von Neumann regular if and only if every right module is absolutely pure [6], in particular, every von Neumann regular ring is a S A P ring. However, the converse is not necessarily true, because it is known that there exist V-rings which are not von Neumann regular. On the other hand, there also exist some S A P rings which are not von Neumann regular. Thus, the class of SAP-rings is in fact a bigger class which contains both the class of right V-rings and the class of von Neumann rings as its proper subclasses. The class of SAP-rings has been recently studied by Wu and Xia in [9]. They have given some basic properties of SAP-rings such as every S A P ring is semiprimitive; every homomorphic image of a S A P ring is still a SAP-ring; the direct sum of all absolutely pure minimal submodules of a module is a fully invariant submodule and etc. However, the relationship between SAP-ring; I/-rings and von Neumann regular rings are not clear. In this paper,we will establish the relationships between these kinds of rings. We first investigate the von Neumann regular rings by using the properties of SAP-rings. Some conditions for group rings and PI-rings to be S A P ring will be given. We also determine whether a ring is a SAP-ring by considering its subrings. In particular, the results of Armendariz-Fisher [l]and Huynh-Jain-Lopez Permouth [4] are generalized. For V-rings, we prove that a SAP-ring R is a V-ring if and only if the injective hull E ( S ) of every simple R-module S is finitely generated. As a consequence, we are able to decompose some S A P rings into the direct sum of simple SAP-rings. This result extends the decomposition theorem of Faith to SAP-rings [3]. SI-rings were firstly studied by Goodearl [5]. By a SI-ring, we mean a ring R whose singular R-modules are all injective. In this aspect, Huynh-Jain-Lopez Permouth [4] have proved that if every right cyclic singular module is injective then the ring

R is a right SI-ring. By using these result, we further characterize the SAP-rings having Krull dimensions. We also consider the O-rings which are the rings whose cyclic modules are either non-singular or absolutely pure, and thereby we prove that the O-ring R is equivalent to the special non-noetherian V-rings studied by HuynhJain-Lopez Permouth if R/Soc(R) is a division ring 141. In fact, this result identifies the SAP-rings and V-rings. Unless stated otherwise, a SAP-ring always means a right a SAP-ring and all mappings between the R-modules will be R-homomorphisms. We denote the submodule N of a module M by N 5 M . Also we use M , E ( M ) , Soc(M) and Kdzm(M) to denote the injective hull, socle, and Krull dimension of the module M respectively. For

253 other notations and terminologies not given in this paper, the reader is refereed to the texts of McConnell and Robson [7] and Stenstrom [8].

2

Properties of SAP-rings

In this section, we investigate the relationships of SAP-riiigs and von Neumann regular rings. Various kind of SAP-rings are considered. We first prove the following characterization theorem of von Neumann regular rings.

Theorem 2.1. Let P be either left or right primitive ideal of a ring R . Suppose that R I P i s a right Artinian ring. Then R is a won Neumann regular ring i f and only if R satisfying the following conditions: (a) R i s a left nonsingular SAP-rings. (ii) every principal left ideal is either a projective module or a maximal left ann.ihilator. Proof. +) The necessity part is clear. This is because that the conditions (i) and (ii) holds by the regularity of the ring R.

+)To prove the sufficiency part, we need to prove that every left simple module T is flat. For this purpose, we denote the annihilator of T by P = Ann(T). Then P is a two-side ideal and the quotient ring R I P is right Artinian and semiprimitive. This implies that R I P , regarded as both left and right RIP-module, is semisimple. Consequently, R I P itself is particularly a semisimple left R-module. This lead to T is isomorphic to a minimal left ideal of the quotient ring R I P . Now, for any Rmodule M, we denote H o m ( M , E ) by M+, where E is an injective cogenerator of the category of Abelian groups. Since (R/P)+P = 0, we have ( R I P ) + = C@T,, where each T i is a simple right R-modules. Because R is a SAP-ring, by a result of Wu-Xia in ( [9] , theorem 6), we immediately see that R is a right V ring. Hence, we obtain E e t k ( M ,( R I P ) + ) = rIExt;(M,T,) = 0. Because Extk(A4,(RIP)+) cv (Torf(R/P,M ) ) + , we see that Torf(R/P, M ) = 0 for all right R-modules M . This shows that R J P is a flat module, and thereby T is flat. We now show that R is a von Neumann regular ring. Suppose on the contrary that the SAP-ring R is not a von Neumann regular ring. Then, there exists a principal left ideal R a which is not projective. Since by condition (i), R is left non-singular, Ra can not be an essential left ideal. Hence there exists c E R such that R a n Rc = 0. Because R. This means that there exists a Ra is not projective as well, we have Ra @ Rc

254

+

maximal left ideal I 2 Ra Rc. However, because R / I is flat, there exists an element d E R such that a = ad and c = cd. Thus a ( 1 - d ) = 0 and c(1- d ) = 0. This implies a and c belong t o the left annihilator l(1- d ) of R. On the other hand, because Ra is a principal left ideal of R , we can denote Ra by the left annihilator l(b), for some element b E R. Hence, we have l(b) = R a C l(1 - d ) . Since R a is not projective, by condition (ii), l(b) is maximal and so l(b) = l(1 - d ) . This lead to c E l(b) = Ra, contradicts to Ra n Rc = 0. Thus, R must be a von Neumann regular ring. Our proof is completed. Corollary 2.2. Suppose that the ring R has a n essential socle and that every

singular left module i s flat. T h e n R is von Neumann regular ring if and only if R i s a SAP-ring satisfying condition (ia) above. Proof. We only need t o prove the sufficiency part. Let Z be the singular ideal of R and S be the socle of the module RR. To show that R is a left non-singular ring, we need to prove that Z(RR) = 0. Because R is a SAP-ring, R/12 is semiprimitive for any ideal I of R. This leads to I = I' for any ideal. Consequently, if I,K are any two ideals of R, then I K I n K = ( I n K)' C I K . Hence, we have I K = K I = I n K. By using this result to the module RR, we have SZ = ZS = S n Z = 0, because S is a direct sum of idempotent minimal right ideal of R. Consequently, Z(RR) = 0 as S is essential. This shows that R is indeed left singular. Then by using the argument in the proof of Theorem 2.1 we see that R is a von Neumann regular ring. We call a ring R a left IF ring if every left injective R-module is flat. Thus, if 0 --t A --t B C 0 is an exact sequence of modules whose B , C are flat, then A is also a flat module. Consequently, for any left IF-ring R, every absolutely pure left R-module is flat. With this observation, we obtain the following corollary. -+

-+

Corollary 2.3. Suppose that every left simple module i s absolutely pure. T h e n R is a von Neumann regular ring i f and only i f R is a left IF ring satisfying the conditions (i), (ii) in Theorem 8.1. It was proved by Kaplansky [6] that a commutative ring is von Neumann regular if and only if it is a V-ring. This celebrated result was extended from commutative rings to left commutative rings, that is, a ring R with Ra 5 aR for any a E R, by Wu and Xia in [9]. We now extend the Kaplansky's Theorem to PI-rings.

Theorem 2.4. Let R be a PI ring. T h e n the following conditions are equivalent: (i) R i s a von Neumann regular ring; (iz) R as a left (right) V-ring; (iii) R i s a SAP ring.

255 Proof. It is suffices to prove that (iii)e (ii)because ( i ) @ (ii)and (ii)+ (iii) are quite obvious. Suppose that (iii) holds. Then by using the argument in Corollary 2.2, we see that every ideal of R is globally idempotent. Now let P be right (left) primitive ideal of R and C be the center of the quotient RIP. Then C is von Neumann regular domain. Invoking, the theorem of Kaplansky in [6], we know that RIP is finitely dimensional space over C . Hence RIP is a semisimple Artinian ring. , see that R is a von Neumann regular Hence, by applying Theorem 1 in [ l ] we PI-ring. The proof is completed. The following theorem consider the group rings over a SAP-ring.

Theorem 2.5. ( 1 ) Let R be a left commutative ring and G be a group. T h e n the following statements hold. (i)If R[G]i s a SAP-ring, then G is a torsion group and R is a SAP-ring and is uniquely divisible by the order of every element in G,i.e. R = nR f o r every n, which is an order of an element of G. (ii) Suppose G i s a locally finite group, then the group algebra R[G]is a S A P ring if and only if R is a S A P ring uniquely divisible by the order of each element of G. (2) If G is a finite group and R is also a PI ring then the group algebra R[G]is a S A P ring i f and only i f R i s a S A P ring uniquely divisible by the order of G. Proof, For the proof of (l), we recall first a result, which has been recently proved by Wu and Xia (see Corollary 4 in [9]), that a left commutative ring R is a S A P ring if and only if R is a von Neumann regular ring. It is easy to prove that R[G]is a left commutative ring if R is a left commutative ring. Then, by using these result and a classical result of Auslander [2], we see that (1) holds. (2) By a result stated by McConell and Robson (see Corollary 13.4.9 in [7]),we see that if R is a PI ring and G is a finite group, then R[G]is a PI ring. Now, if R[G] is a SAP-ring, then by Theorem 2.4, R is a SAP-ring. Because G is a finite group, it is clear that the order of R is unique divisible by the order of G. The converse part is trivial. Remark. (i) It is clear that a group algebra R[G]over a field is not necessarily a S A P ring because R[G]is in general not semiprimitive. However, we do not know whether Theorem 2.5 still holds if R is not left commutative or R is not a PI-ring. (ii) If R is either a left commutative ring or a PI ring, then we know that R is a SAP-ring if and only if R is von Neumann regular ring. Thus, by using the trace functor, we can show that if the order of a finite group G is invertible in a SAP-ring

256

R and G is an automorphism group of R, then the fixed subring RG is a SAP-ring whenever R is either a left commutative ring or a PI ring. In closing this section, we give an example to show that RG is not necessarily a SAP-ring.

Example 2.6. Let F be a Jeld with charF = 2, the 3 x 3 matrix ring over F b y R = A43(F). Then we can check that R = M3(F) is a PI and S A P ring. Now let G be the subgroup of G L 3 ( F )generated by the matrices

i:::)h::I 1 0 0

1 0 0

Then, b y using G that acts on R by conjugation, we obtain the following

[; ;

x o o

RG=(

j~x...iiF)

It is obvious that RG is not a S A P ring because it is not semiprimitive.

3

SAP rings, SI rings and V-rings

In this section, we study the relationships between S A P rings, SI rings and V rings. In particular, we characterize the SAP-rings which are SI-rings. We start with the following characterization theorem of V-rings.

Theorem 3.1. A S A P ring R i s a V ring aj and only af the injective hull of every simple module is finitely generated. Proof. We only need to prove the sufficiency part for the necessity part is trivial. Since R is a SAP-ring, every simple R-module is absolutely pure. Now let T be an absolutely pure simple R-module, and E ( T ) the injective hull of T . If E ( T ) is finitely generated, then E ( T ) / T is finitely presented. Hence the short sequence f f o m ~ ( E ( T ) , T+) Homfi(T,T) 0 is exact, thereby T is a directed summand of E ( T ) .This leads to E ( T ) = T and R is a V-ring. The proof is completed. The following lemmas give some interesting properties of SAP-rings. ---f

Lemma 3.2. Any absolutely pure simple R-module S is divisible.

257

Proof. To show that the absolutely pure R-module S is divisible, we need to show that S = Sb for any right regular element b E R. For this purpose, we let w E S. Since S is absolutely pure, there exists a homomorphism h from bR into S such that h(br) = vr, for all r E R. Since v E S , there exist an element m E E ( S ) such that 'u = mb. This lead to w E E(S)bn S = Sb, and thereby S = Sb. This proves that S is divisible. Lemma 3.3. Let I is a proper ideal of a SAP-ring R . Then x R is not isomorphic to R for anyx E I .

Proof. Let M be a maximal right ideal containing the ideal I . Then S N R I M is a simple right R-module, which is pure in the injective hull E ( S ) of S. Suppose on the contrary, that x R N R for some x E I . Let f be an isomorphism from R t o xR such that f(1) = y = xu E xR. We next prove that y is right regular. For this purpose, let r E R satisfying yr = 0. Then f ( r )= f(1)r = 0. From this we obtain that r = 0. This proves that y is right regular. From Lemma 3.2 we see that S = Sy. Hence y @ I . However, this is absurd because x E I . Hence, we must have x R 'f4 R. The proof is completed. In his paper, Xue [lo] called a ring R a right quo ring if every maximal right ideal of R is an ideal of R. It is clear that a left commutative ring is always a right quo ring. Also we call a ring R a right quotient ring if every right regular element of R is invertible. If T is a subring of a right quotient of R and every element x in R can be expressed by x = ar-' for some a,r E T , where r is a right regular element of T , then we call T a right order of R. With the above notations, we immediately have the following corollary of Lemma 3.3.

Corollary 3.4. If a right quo ring R is a SAP ring, then R is a right quotient ring. In particular, R becomes a skew field if it is a domain. For the decomposition of SAP-rings, we have the following similar result of Faith [3] on prime right V-ring.

Theorem 3.5. Let R be a SAP ring which is a right order in a semisimple ring

Q. Then R can be decomposed into a finite directed sum of simple SAP-rings. Proof. Let the ring R be a right order in a semisimple ring Q. If Q is a direct sum of n simple right modules, then every set of globally idempotent right ideals of R has cardinality 5 n , where n is the right ideal dimension of the ring Q. Furthermore, we know that every essential right ideal of R contains a regular element x of R. Thus,

258

zR

R. This implies that R does not have proper ideals which are essential right ideals of R. Now, let B be a maximal ideal of R. Since B is not an essential ideal, B n I = 0 for some nonzero right ideal I . This leads to I B = 0 and consequently (BI)*= 0. The semiprimitive of R implies that BI = 0 so that the right annihilator of B,say r ( B ) = R1 is a nonzero ideal. Hence, by the maximality of B,we have R = R1 @ B. Keep on the same process until the maximal ideal of B,+, is 0. Thus R = R1 @ . . . @ R,-1@ B,. Let R, = B,. Since every Ri inherits the S A P property of R, the proof is completed. A module M has finitely uniform dimension if there exists a finite independent set of uniform submodules {Uz},",l such that C g , U, is an essential submodule of M . N

With this notation, we obtain the following corollary.

Corollary 3.6. Let R be a S A P ring with a right essential socle S and S is finitely generated, then R is a semisimple ring.

Proof. Let Z be the right singular ideal of RR. Then S Z = Z S = S n 2 = 0, because S is essential. Consequently, Z = 0. From the fact SR is finitely generated, we obtain that the uniform dimension of RR is finite. So R is a right Goldie ring. By Theorem 3.5, R = R1 + . . .Rt, where R, is a simple S A P ring, for i = 1,.. . ,t . Thus R is a semisimple ring, because every simple ring R, has an essential socle. The proof is completed. The following theorem links up the SAP-rings and SI-rings. Theorem 3.7. Let R be a S A P ring with Krull dimension. Then the following statements are equivalent: (i) R is a right SI-ring; (ii) R satisfies the RRM condition, that is, RII is artinian ring for any essential ideal of R; (iii) KdimR 5 1; (iv) all singular right R-module M are semisimple; (v) E ( R ) / R is a semisimple right R-module.

Proof. (i)+ (22). Suppose that R is a right SI-right. Then by Lemma 1 in [4], we see that R = K @ R1@. . . @ %, which is a direct sum of rings, where K/Soc(KK) is a semisimple ring and each R, is Morita-equivalent to a right SI-domain D iwhich is not a division ring. Also, Di is right Noetherian and right hereditary and so for each nonzero right ideal C, Di/C is semisimple. In other words, Ri is right Noetherian and right hereditary. This shows that R satisfies the RRM condition.

259

(ii) @ (iii). By a result of Wu-Xia ([S,Theorem 3]), we see that R is semiprime. Consequently, KdimRR = sup {K(R/E) +1IE is an essential right ideal}, by Proposition 6.3.11 in McConell and Robson [7]. Thus, if (ii) holds, then KdimR 5 1. On the other hand, if (iii) holds, then K d i m R j I 5 0 for all right essential ideals. This leads to the result that R j I is an artinian ring for all essential right ideal I of R. (ii) + (iw). By Proposition 6.3.5 in [7], we know that R is a right Goldie which is nonsingular. Hence, by our Theorem 3.5, R = R1 @ . ’ . @ R, for some n, where each R, is a simple SAP rings. Without loss of generality, we may assume that R itself is a simple SAP ring. If R contains a minimal right ideal, then R is a simple artinian ring. In this case, (iv) holds trivially. On the other hand, if the socle of R is equal to zero, then we can let M be a singular right R-module with 0 # x E M . In this case, we have xR N R/I, for some nonzero right ideal I of R. Thus, by (ii), xR is artinian. Consequently, Soc(xR) is an essential finitely generated module. Suppose that S is a simple submodule of Soc(zR) and L is an essential right ideal of R. Let h be a homomorphism from L onto S. Then there exists an element z in the injective hull E ( S ) of S such that h(a) = xu for any a E L. By using Corollary 3.4.7 in [7], we see that L is generated by a regular element c E L , that is, L = cR. This leads to h(c) = xc E S n E(S)c = Sc. Thereby, there exists an element y E S such that xc = yc. Now, define a homomorphism g from R into S by g(a) = ya. Then, we have g(cb) = ycb = h(cb) for every element b E R. This shows that S is indeed an injective R-module. However, this is impossible unless Soc(xR) = xR. Thus (iv)is proved. (iv)+ (ii) and (iv)+ (v) are obvious. We omit the details. ( u ) =+ (iv). Let R / I be a cyclic module with I # 0. Suppose c is a regular element such that I = cR. Then the isomorphism f : R -+ cR given by f ( a ) = ca induces an isomorphism between the injective hulls E(R) and E(cR). Thus, by our assumption, we have E(R)/cR rv E(R)/R so that E(R)/R is a semisimple right R-module. This leads to R / I 5 E ( R ) / I = E(R)/cR/R/cR, which is semisimple. Hence (iv)holds. (iv) H (i). This part follows similarly by using the arguments as (ii) + (iw). Thus, the cyclic proof is completed.

Corollary 3.8. If any one of the equivalent conditions in Theorem 3.7 holds, then R = R1 @ R2 @ . . . @ &, where R1 is a semisimple subring and the rest rings R, (for all i 2 2) are sample right noetherian hereditary rings.

Proof. If any one of the above condition holds, then by Theorem 3.5, R = B1 @

. . . @ B, for some n, where each B,is a simple SAP rings. It is obvious that each B, is a SI-ring. By Lemma 1 in [4], a right S I ring R can be expressed by a direct sum of rings such that R = K @ R1 @ . . . @ Rt, where K / s o c ( K ~ )is semisimple and each

260 Ri (i = 2, . . . , n) is Morita-equivalent to a right SI-domain Di which is not a division ring. Apply this result to each Bi, we complete the proof of this Corollary.

Corollary 3.9. Suppose that R is a PI-ring and also a right SI-ring. Then R/Soc(RR) is semisimple if R as a S A P rin,g.

Proof. Because R is a right SI ring and SAP-ring. Then by Lemma 1 in [4], R = K @ R1 @ . . . @ R,, where K/Soc(KK) is semisimple and every Ri is Moritaequivalent to a right SI-domain Di which is not a division ring. Since R is a PI-ring, every R,, is a PI ring. Thereby, Di is a right SI-ring and PI-ring, which is not a division ring. This implies that the center C, of Di is a field and Di is finitely dimensional over Ci. Thus, Di is a division ring, which is a contradiction. Hence, the corollary is proved. Corollary 3.10. If the S A P ring is also a SI-ring an,d PI-ring with Krull dimension, then R is semisimple. Proof. Since RR has Krull dimension, SOC(RR) is finitely generated. Thus this corollary follows from Theorem 3.7 and Corollary 3.9.

Theorem 3.11. A ring is a SAP-ring if and only i f every Lowey module is an a6solutely pure module. Proof. Let M be a Lowey module with a Lowey chain: Mo G M I G . . . s M, C . . .. Suppose that E ( M ) is the injective hull of M . Then Mo is clearly pure in E ( M ) . Assume that Mp is pure in E ( M ) for any ordinal /? 5 a. We need to prove that M, is also pure in E ( M ) . It has been well known that M, is pure in E ( M ) if (Y is a limited ordinal. If a = /?+1,then M,/Mp is pure in E ( M ) / M pwhenever R is a SAP-ring. Let I be any left ideal of R and x E M, n E ( M ) I . Then, Z E M,/Mp n E l M p I . Express x = C Eiai for some ei E Ma and ai E I . Then we have that z - C eiai = C mj6, for some mj E Mp and 6j E I. This implies that x 6 M,I. Consequently M, is pure in E ( M ) . The converse part is trivial. This completes the proof.

Corollary 3.12. If RR is considered as a right semiartinian R-module, then the following statements are equivalent: (i) R is a SAP-ring;

(ii) R is a won Neumann regular ring.

Proof. The equivalence of (i) and (ii) follows directly from Theorem 3.11

26 1

4

R-rings

We call a ring R a right (respectively, left) R-ring if every cyclic right (respectively, left) R-module is either nonsingular or absolutely pure. On the other hand, HuynhJain-Lopez Permouth consider the right E-ring in [4]. By a right E-ring R, it means that a ring R whose cyclic right modules are either non-singular or injective. Similarly, we call a ring R a left E-ring if every cyclic left R-module is either nonsingular or injective. Clearly, all right (respective, left) E-rings are right (respectively, left) Rrings. We now show that , under certain conditions, E-rings and O-rings are equivalent. This generalizes a result of Huynh-Jain-Lopez Permouth ( see Theorem 6 in[4]).

Theorem 4.1. The following rings are equivalent: (i) R is a non-Noetherian right O-ring such that R/Soc(RR) is a division ring; (ii) R is a SAP ring with infinitely generated right socle SOC(RR)such that R/Soc(RR) is a division ring; (iii) R is a right V-ring with infinitely right generated socle Soc(R) such that R / s o c ( R ~ )is a division ring; (iv) R is a non-Noetherian right (E)-ring; (v) R is a von Neumann regular ring with infinitely generated right socle SOC(RR) such that R ISOC(RR)is a division ring; (a’) R is a non-Noetherian left O-ring such that R ISOC(RR)is a division ring; (ii’) R is a ring with infinitely generated socle SOC(RR)such that R/Soc(,R) is a division ring and any left simple module is absolutely pure; (iii’) R is a left V-ring with infinitely generated left socle SOC(RR)such that R/Soc(RR) as a division ring. (ivy R is a non.-Noetherian left (E)-ring; (v’) R is a von Neumann regular ring with infinitely generated left socle SOC(RR) such that R/SOC(RR)is a division ring.

Proof. ( i )+ (ii). Let R is a semiprimary R-ring. We first show that R is semisimple, that is, we need to show that the Jacobson radical J ( R ) is zero. If J ( R ) # 0, then there exists a primitive idempotent element e E R such that e J ( R ) # 0. Hence e R / e J ( R ) becomes a singular simple R-module. This implies that R / J ( R ) is not non-singular as an R-module. Because R is a O-ring, R / J ( R ) is an absolutely pure R-module. In other words, J ( R ) contains an absolutely pure minimal right ideal I . However as a minimal right ideal and is absolutely pure, we must have I = 1’. This is impossible. Hence J ( R ) = 0.

262 Now, let J be the sum of all globally idempotent minimal right ideals of R. We now claim that J is an ideal of R. For this purpose, we let S = eR be a minimal idempotent right ideal of R , with e = e2 E R. Set L = (1 - e)R. Since L/SOC(LR) is singular and is nonzero, R / s o c ( L ~ )must be absolutely pure. Because R is an Rring, S R is absolutely pure. This shows that every idempotent minimal right ideal of R is an absolutely pure simple R-module. Therefore J , as the sum of all idempotent minimal right ideals of R, is a two-sided ideal of R by Theorem 8 in [9]. Moreover, for any nilpotent right ideal N of R, Soc(R)N = 0. This leads to (Soc(RR))’ C J . By this result and the fact R/Soc(RR) is a semisimple ring, we can see that R / J is a semiprimary ring. Thus it is a semisimple ring. It is now easy to verify that R is a S A P ring. (iz) + (iii). Assume that R is a right S A P ring and R/Soc(R) is a division ring. If I is a right essential ideal in R, then I = Soc(R). Let h be any homomorphism from I into a simple R-module S . Then, we can easily observe that h induces an isomorphism between S and T , which is a minimal right ideal of R contained in I . Because R is a SAP-ring, there exists an idempotent element e E R such that T = eR. Now, we can define a homomorphism f from R to S by f(x) = g(ez) for all z E R. This shows that S is an injective module. (iii) G (iv)follows from Theorem 6 in [4]. (ii) @ (v) follows from Corollary 3.12. We omit the details. (iv) (2). This part is obvious. (v) + (v’). If (v), then R is semiprime. Thus the left socle of R is equal to the right socle S of R. If S is a finitely generated left ideal, then R is a left Noetherian ring. Consequently, it is semisimple. This is contradiction the assumption (v). So (v’) holds. Similarly we can prove (v’) + (v). By this left-right symmetry of (v) and (v’),we complete the cyclic proof. Remarks. (i) We remark here that all the rings satisfying the above equivalent conditions in Theorem 4.1 are all rings without Krull dimension because Soc(RR) is not finitely generated. We have already described the SAP-rings with Krull dimension 1 (see Theorem 3.7) and some S A P rings without Krull dimension. In closing this paper, we propose the following questions: Can we describe the SAP-rings with Krull dimension greater than I? (ii) Corollary 3.6 characterizes S A P rings with finitely generated essential socle but can we characterizes some S A P rings with infinitely generated essential socle? Can we describe the SAP-rings without socle?

263 (iii) Wu and Xia proposed the following question in [9]. Is there exist nonabsolutely-pure left simple module over a SAP ring?. This problem is still unsolved. However, our Theorem 4.1 shows that left simple modules over any non-Noetherian right E-rings are absolutely pure.

References [l]E. P. Armendariz, Joe W. Fisher, Regular P.I. rings, Proc. AMS, (39)1973, 247-

251. [2] M. Auslander, On regular group rings, Proc. AMS, 8(1957), 658-664 [3] C. Faith , A general Wedderburn theorem, Bull. AMS, 77 ( 1967), 338-3428, [4] D. V. Huynh, S. K. Jain and S. R. Lopez-Permouth, On a class of non-noetherian V-rings, Comm. Algebra, 24(9), 1996, 2839-2850. [5] K. R. Goodearl, Singular torsion and splitting properties, Memoirs of the Amer. Math. SOC.,124(1972). [6] I. Kaplansky, Rings with a polynomial identity, Bull. AMS, 58(1948), 575-580. [7] J. C. McConnell, J. C. Robson, Noncommutative noetherian rings, John Wiley and Sons, 1987.

[8] B. Stenstrom, Rings of quotients, Die Grundle Math.Wiss.Bd 217, SpringerVerlag, 1975. [9] 2. Wu, Q. Xia, Rings whose simple modules are absolutely pure, Comm. Algebra, 29(4), 2001, 1477-1458. [lo] W. Xue, Rings with Morita duality, LNM 1523, Springer-Verlag, 1992.

PARAMETRIZATION OF G * ( ~ , z )ON PL(F,) MUHAMMAD ASHIQ AND QAISER MUSHTAQ ABSTRACT.Let PL(F,) denote the projective line over a Galois field Fq, where q is a prime power. Let G ( 2 , Z ) be a free product of two cyclic groups < u > and < 21 > of order 3. We have parametrized the conjugacy classes of nondegenerate homomorphisms a with the non-trivial elements of Fq. Also, we have developed a useful machanism by which one can construct a unique coset diagram (attributed to G. Higman) for each conjugacy class, depicting the action of G ’ ( 2 , Z ) on PL(F,).

1. INTRODUCTION

It is well known [l,51 that the modular group PSL(2,Z) is generated by the and y : z --+ which satisfy the linear-fractional transformations z : z --+ relations

5

z2 = y 3 = 1

(1.1)

&

Let v = zyz, and u = y. Then (2). = and u3 = v 3 = 1. So the group G =< u,v > is a proper subgroup of the modular group PSL(2,Z). Let G(2,Z) = < u.w : u3 = v 3 = 1 > be the group of linear-fractional transformations of the form z-+-

(1.2)

az

cz

+b +d

where a, b, c, d E Z and ad - bc = 1. Specifically, the linear-fractional transformations of G(2,Z) are u : z -+ and v : z -+ f$ which satisfy the relations

5

PI As u and v have the same order, there is an automorphism interchanging u and v and this yields the split extension G*(2,Z).The linear-fractional transformation t : z -+ inverts u and v, that is, t2 = (ut)’ = (vt)’ = 1 and so extends the group G(2,Z) to G*(2,Z). The extended group G*(2,Z) is then the group of transformations of the form

2

z+-

az

cz

+b +d

where a, b, c, d E Z and ad - bc = f l and its defining relations are of the form 1991 Mathematics Subject Classification. Primary 20F05; Secondary 20G40. Key words and phrases. Linear-fractional transformations, Non-degenerate homomorphisms, Conjugacy classes, Parametrization, and Projective line.

264

265

(1.5)

G*(Z,Z)=< U,v,t : u3 = v 3 = t2 = (ut)2= (vt)' = 1 > .

Let PL(Fq)denote the projective line over the Galois field Fq where q is a prime power. The points of PL(Fq) are the elements of Fq together with the additional point 00. The group G*(2,q ) is then the group of linear-fractional transformations where a,b,c,d E Fp and ad - bc # 0, while G(2,q) is of the form z + its subgroup consisting of all those linear-fractional transformations of the form z -+ where a, b, c, d E Fq and ad - bc is a non-zero square in Fq. We use coset diagrams for the group and study its action on PL(F,). Our coset diagrams consist of triangles; they are called coset diagrams because the vertices of the triangles are identified with cosets of the group. These diagrams are defined for a particular group which has a presentation with three generators. The coset diagrams defined for the actions of G' ( 2 , Z ) on PL(Fq) are special in a number of ways (see [3]). First, they are defined for a particular group, namely, G* (2, Z ) , which has a presentation in terms of three generators t, u and v. Since there are only three generators, it is possible to avoid using colours as well as the orientation of edges associated with the involution t. For u, and v both have order 3, there is a need to distinguish u from u2 and v from v2. The three cycles of the transformation u are denoted by three (blue) unbroken edges of a triangle permuted anti-clockwise by u and the three cycles of the transformation v are denoted by three (red) broken edges of a triangle permuted anti-clockwise by v. The action o f t is depicted by the symmetry about vertical axis. Fixed points of ti and v, if they exist, are denoted by heavy dots. A general coset diagram of the action of G* ( 2 , Z )on PL(Fq)will look as follows

s,

3,

7

11

In this paper, we have parametrized actions of G*(2,Z)on PL(Fq),except for a few uninteresting ones, by the elements of Fq. We show that any homomorphism from G(2,Z) into G(2, q ) can be extended to a homomorphism from G*(2,Z)into G*(2,q).We have developed a useful mechanism by which one can construct a unique coset diagram, attributed to Graham Higmam [4],for each conjugacy class of these non-degenerate homomorphism which depict the actions of G ' ( 2 , Z ) on PL(F,).

266

2 . CONJUGACY CLASSES O F THE NON-DEGENERATE HOMOMORPHISMS

9

2

generate The transformations u : z -+ , v : z -+ and t : z + G * ( 2 , Z ) ,subject to defining relations (1.5). Thus to choose a homomorphism a : G * ( 2 , Z )-+ G * ( 2 , q )amounts to choosing 3 = ua,?j = va and Z = ta,in G*(2,q) such that

(2.1)

3 - 3 - 4 -

u

-7,)

-

--2 t - (Ut)

--2 - (vt) - 1.

We call a to be non-degenerate homomorphism under the condition that orders of u and v are the same as orders of ua and va,if neither of the generators u,v of G'(2,Z) lies in the kernel of a , so that the images 3 and ;ii are of orders 3. Two homomorphisms a and /3 from G'(2,Z) to G*(2,q ) are called conjugate if there exists an inner automorphism p of G"(2,q ) such that p = pa. Both G ( 2 , Z ) and G ' ( 2 , Z ) have index 2 in their automorphism groups. Let 6 be the automorphism on G * ( 2 , Z )defined by ub = ut,vb = v, and t b = t. The homomorphism a' = ba is called the dual homomorphism of a. This, of course, means that if a maps u,v,tto Ti,iJ,j, then a'maps u,v,tto i i Z , is, 2 respectively. Since the elements Z, v, Z as well as Ti& v, 3 satisfy the relations (Z.l), therefore the solutions of these relations occur in dual pairs. Of course, if a is conjugate to /3 then a'is conjugate to f l The parameter of a , or of the conjugacy class containing a, is the parameter of 35. Thus for each 8, which is a square in Fqrthere exists a unique coset diagram. It is unique for 6' in Fq in the sense that the diagram is the same except for the labels for any element in the conjugacy class that, it represents; only the vertices vary. Hence if we know 8, we can find some homomorphism a and hence a coset diagram.

3. PARAMETERS FOR THE CONJUGACY CLASSES O F G * ( 2 , q )

If the natural mapping GL(2,q ) -+ G*(2,q ) maps a matrix A4 to the element of g of G*(2,q ) then 6' = (tr(A4))2 / det(A4) is an invariant of the conjugacy class of g [ 5 ] . We refer to it as the parameter of g or of the conjugacy class. Of course, every element in Fq is the parameter of some conjugacy class in G * ( 2 , q ) .For instance, the class represented by a matrix with characteristic polynomial z2 - Oz + 6' if 6' # 0 or z 2 - 1 if 6' = 0. If q is odd and g is not an involution, then g belongs to G ( 2 , q ) if and only if 6' is a square in Fq. On the other hand g : z -+ where a , b, c, d E F,, has a fixed point Ic in the natural representation of G'(2,q) on PL(Fq)if and only if the discriminant, a2+d2-22ad+4bc, of the quadraticequation k 2 c + k ( d - a ) - b = 0 is a square in Fq. Since we have the determinant ad-bc is 1 and the trace a+d is r, then the discriminant is (6' - 4).Thus, g has fixed point in the natural representation of G*(2,q ) on PL(Fq)if and only if (6' - 4) is a square in F,. It is an easy fact that if U and V are two non-singular 2 x 2 matrices corresponding to the generators 3 and 5 of G'(2,q) with det(UV) = 1 and trace r, then for a positive integer Ic

H,

267

( u v ) ~=

{(

k-1

{(k

-

2 ) F

(

k-2

- (k

)rk-3

;

+ ...Iuv

3)Tc-4

+ ...}I

Furthermore, if we let

then the polynomial f ( r ) is homogeneous in r and r2 = 0. We can substitute 0 for r2 in f ( r ) and obtain a polynomial f(0) in the form of 0. Thus, one can find a polynomial for a positive integer k such that: 1. if k is prime number and q G f l ( m o d k ) , then one can find f(0) by the equation (3.2), 2. if k is a positive odd integer and q = f l ( m o d k ) , then first find g(0) by putting k in equation (3.2) and then g,/,(0), d is least prime divisor of k , by putting k / d in equation (3.2) instead of k , then f(0) = and 3. if k > 2 is an even integer and q = f l ( m o d k ) , then find g,l2(0). If k / 2 is a prime, we are in situation 1. If k / 2 is odd, we are in situation 2. If k / 2 is even, the polynomial g k l Z ( 0 ) splits into factors gk/4(0) and g,(O) = k 2(e) We continue in this way until we get no more factors. In the following we list conditions for k = 2 to 20.

s,

%.

I k I Minimal Eauation satis

Lemma 1. If M is an invertible 2 x 2 matrix whose entries from the set of integers then M 3 = XI, X E Z if and only if{trace(M)}’ = det(M).

268

Proof. Let M =

[+: +]

then

a3 2abc bcd a2b+ b2c + bd2 + abd M3 = a2c + bc2 + cd2 + acd abc + 2bcd + d3 If M 3 = X I then, a3 2abc bcd = 1 and d3 abc 2bcd = 1 implies that (a3-d3+abc-bcd=Oor (a-d)(a2+d2+ad+bc) =0.Anda2b+b2c+bd2+abd=0 implies that b(a2 + bc + d2 + ad) = 0 and a2c + cd2 + bc2 + acd = 0 implies that c(a2 + d2 + bc + ad) = 0. Let a2 + d2 + bc + ad = 0. This implies that ( a + d)' - (ad - bc) = 0 or {(trM)}2 = (det M). I

[

+

4.

Let U =

+

+

+

PARAMETRIZATION OF THE ACTIONS

[: ]

be an element of GL(2, q ) which yields the element 21 of

a3 = 1, U 3 is a scalar matrix, and hence the det(U) is a square in Fq. Thus, replacing U by a suitable scalar multiple, we assume that det(U) = 1. By lemma 1, we may assume that tr(U) = a d = -1 and det(M) = 1. Thus G*(2,q).Then, since

u=

[

a c

-a-1

].

+

Similarly, v =

[

Now let a matrix corresponding to -2

t

= 1, the trace of

f -e-l].

t, be represented

by T =

[ A 7 1.

Since

T is zero. So, upto scalar multiplication, we can assume

that the matrix representing

t has the form

trace@?) = 0 and so b = kc.

[ ik] .

Because

(Tit)' = 1, the

Thus we can take the matrices corresponding to generators 21, V and Z of G*(2, q ) as:

I,.=

c - akc -1 where a , c, e, f , k E Fq. Then, (4.1) and

[

f

1

+ a + a2 + kc2 = 0

- ekf -1

],

+

1+ e + e 2 k f 2 (44 because the determinants of U and V are 1. The matrix UV has the trace

(4.3)

r = a(2e+ 1)

and T =

=0

+ 2kfc+ ( e + 1)

If trace(UVT) = k s , then (4.4)

s = 2af - c(2e

+ 1)+ f

So the relationship between (4.3) and (4.4) is (4.5)

r2+ks2-r-2=0

[

0

-k

]

respectively,

269

We set 8 = r2

(4.6)

Now for each conjugacy class of pairs (n,ii)we can draw a coset diagram. By D(8,q) we shall mean a coset diagram associated with the conjugacy class of non-degenerate homomorphisms a of G*(2,Z) into G*(2, q ) corresponding to 8 E Fq.

Example 1. Let us consider the coset diagram D(8,q) f o r 8 = 3 and q = 13. By (4.6), 8 = r2 and so r2 = 3 = 16 implies that r = f 4 . Let us take r = 4. Now substitute the value of r in (4.5) and suppose that k = 1. W e get s2 = 3 = 16 implying that s = f 4 . Let us choose s = 4. If we suppose e = 0 in equation (4.2) we have f 2 = -1 = 25, that is, f = f 5 . Suppose f = 5 and substitute the value of r, s, d , k and f in equations (4.3) and (4.4), to obtain 3 = a + 1Oc and -1 = 10a - c. Solving these equations for a and c, we get a = -2 and c = 7. Thus U =

1,

[ &,

[

0

-1

]

L

.I

. So, our 21, ii and f will be 21 : z -+ -,

and T = and t : z -+ $ respectively, where z E PL(Fl3). The linear-fractional transformation n, ii and act respectively as:

V= v : z -+

l:

(0 7 10)(1 12 5)(2 8 6)(9 11 m)(3)(4), (0 8 m)(l 11 9)(3 5 4)(7 12 10)(2)(6), and (0 w)(l 12)(2 6)(3 4)(7 11)(9 10)(5)(8) yielding the coset diagram D(3,13)

Fig3 W e ote that each vertex of the diagram is fixed by (ii;Ij)6. As o non-trivial linear -fractional transformation can f i x more than two vertices therefore ('iiii)6 = 1. Thus, the above coset diagram D(3,13) is a homomorphic image of A(3,3,6).

REFERENCES [l] M. D. E. Conder, Generators for alternating and symmetric groups, J. London Math. SOC. 2(1980), 75

-

86.

270

"4

H.S.M. Coxeter and W. 0. J. Moser, Generators and relations for discrete groups, 4th. ed., Springer-Verlag, Berlin, 1980 [31 Q. Mushtaq, Coset diagrams for Hurwitz groups, Comm. Algebra, 18,11(1990), 3857 - 3888. [41 Q. Mushtaq and Gian-Carlo Rota, Alternating groups as quotients of two generator groups, Adv. Math., 1,96(1993), 113 - 121. [5] W. W. Stothers, Subgroups of the (2,3,7)-triangle group, Manuscripta Math., 20(1977), 323 334. DEPARTMENT OF MATHEMATICS, COLLEGE TECHNOLOGY, RAWALPINDI, PAKISTAN E-mail address: ashiqjavedmyahoo .co .uk

OF

E & ME, NATIONALUNIVERSITY OF SCIENCES

AND

DEPARTMENT OF MATHEMATICS, QUAID-I-AZAM UNIVERSITY, ISLAMABAD, PAKISTAN E-mail address: [email protected]

Non Isomorphic Trees with First Three Characteristic Numbers Equal T. Bier,T.-H. Tan email: [email protected] August 22, 2002

1

Characteristic Numbers of Graphs and Trees: The Problem

Assume that G is any graph of order v and with the degree sequence 15 d l 5 d2 5 d3 5 .... 5 d,

(1)

We are counting the number of induced subgraphs on vertex subsets S of size s which contain p2 edges inside and p1 edges with one end point in S. This number is denoted by K ( P 1 , PZ; s).

We ask the question of finding non isomorphic trees T’.T” which have partially equal numbers, that is we want the existence of an integer t such that K’(p1, p2; 3 ) = K ” ( P 1 , pa; 5).

(2)

for s = 1 , 2,..., t. This question has gained in relevance since the authors discovery [l]of several pairs of regular graphs G’,G” which are not isomorphic, yet have the same numbers (2) for all s = 1 , 2 , . . . , v. It is also shown in [l] that in certain cases the characteristic numbers are useful to distinguish non isomorphic copies of graphs. Finally recently [2] the existence of two trees of order eleven T310. T316 of [5] with equal characteristic numbers K ( p 1 ,pa: s ) for all s = 1 , 2 , . . . u - 1 was established. Assume that

vd

is the number of vertices of the tree T which have degree d.

First we remark that in the case s = 1 we get for p1 = d the equality K ( d , O ; 1) = ~ l d .

(3)

Hence the number of p0int.s V d of degree equal to d is determined by the integers K ( d ;0: 1). The next case s = 2 has a similar description. For any edge e E E ( T ) with let deg(6) = deg(s) deg(y).

+

27 1

E

=

{x.y} we

272

This information may be summarized in the array of numbers

a, =I

{E

: deg(E) = r}

I

(r = 1,2, ..., v

- 1).

(4)

Hence assuming the numbers K ( r ,0 , l ) already known, and thus the degree sequences already determined, we then get the numbers K ( r ,1; 2) = a, as the number of edges E with deg(E) = r. The sequence of numbers K ( r ,0; 2) will then be given as the set of all pairwise sums deg(u) + deg(v) with u, v E V(T), u # v which have not yet been used for the edges. These pairwise sums are completely determined by the complements of (4) in all pairs of (3). Thus our question generalizes the well known exercise [6] of showing that there are several trees which have the same degree and edge degree sequences. Next we consider the case of s = 3. We show in this paper that (for v 2 14) there exist N, non isomorphic trees with all their characteristic numbers K(p1,p2; s) equal for s = 1,2;3; where the number N , is given as:

For odd v = 4k

+ 1 we have Nu =

while for odd v = 4k

(2ky')

+ (k

-

3 ) ( k - 2);

+ 3 we have N u = ( 2k

-

3) + ( k - 2 ) 2

We remark that it is simple to show that there also exist pairs of trees of orders 10,11,12; 13 which are not isomorphic, yet have the same characteristic numbers up to s = 3. One such pair of order 10 is the pair T185,T186 of [5]. One such pair of order 11 is the pair T421,T427 of [5]. Indeed any of these mentioned pairs also form certain infinite families of trees which have the same characteristic numbers up t o s = 3; but yet are not mutually isomorphic. We do not give these families here for two reasons. They are asymptotically not as big (for a given large v ; that is we get less non isomorphic copies of trees, as in the family described here,) and the proofs and computations with the other families are more complicated. Indeed we are happy that for the family given here all explanations can be presented in a comparatively short, simple and conceptual way. We also remark that very recently we showed (for any h 2 6) the existence of a pair of non isomorphic trees on v = 2h vertices which have their Characteristic numbers equal for all values of s 5 h - 3.

2

Some general facts on the characteristic numbers of trees

We first note that for any values of p1, p2 that satisfy the inequality p1 have vanishing K ( p 1 ,p2; s ) . that is

K ( p 1 - p ~s): = 0 for p1

+

p2

< s.

+ p2 < s we must (5)

273 For the proof of ( 5 ) we first observe that clearly

K ( T 0; , s) = 0 for

T

< s.

The equivalent contrapositive statement is

K ( T 0; , s) > 0 implies

T

2 s.

(7)

This follows from the fact that a set S = { q , z 2 ,...,zs}of s independent points in a connected graph has deg(zi) 2 1, so that the set S must have at least %(xi)

+ deg(z2) + ... + deg(x:,) 2 s

edges incident with it, and each such edge contributes t o the count of the integer pl The general case now follows by contracting the connected components of the induced subgraphs into single points, thereby reducing the number of g - 1 edges to 0 edges, and the number of g vertex points t o one single vertex point. The number p1 is not changed during this contraction. Considering all the c different connected components in turn, and denoting their sizes by g1,92, ...,gc we see that p2 = g1 +g2

+ ... +g,

-

c = s - c,

(8)

that is after contraction into the tree T’ we have pk = 0 , p ; = p1, while before we have p2 = s - c. After contraction we apply (7) to get p l = pi 2 c. This implies p1 2 s - p2 by (8). Hence we can reduce the general case t o (6). This completes the proof of (5). Then we study the case p1

K ( s ,0 ;s) =

+ p2 = s. Here we get the number

();

(9)

For the proof of (9) we see that any set of s vertices of the tree constitutes an induced subgraph with pi = s; and with p2 = 0 if and only if the set contains only vertices of degree one, that is a set which contains only the leaves of the tree.

From the connected property of the graph we obtain for s < u that K ( 0 .s;s) = 0; since each proper subset must have at least one edge connecting it to the complement. On the other hand consider the number K(1,s - 1:s)

K ( 1 ,s - 1;s) counts the induced subtrees of order s

(10)

which have only one edge connecting it t o the complement,. And in general for 7~ = q = p2 satsifying p q = s, we see that K ( p ;q ; s) counts the number of p component. induced subforests where each component contains a leaf vertex, and the removal of each component in turn results in another smaller tree. Let us denote by e , the number of induced subpath of length T + 1 which contain one leaf of the tree, and which have the other end vertex of degree two. For example we get for the number of leaves of the tree eo = u1.

+

274

We then get a further formula for the characteristic numbers as follows:

3

Some Linear Sum Relations for s

=

3.

We now show that the degree sequence of any connected triangle free graph G on the ,. also determines the sum of the numbers K ( p 1 , 2 ;3) by the vertex set V = ( ~ 1 ~ x 2. .xu} formula

This follows from the fact that each path P3 has a unique midpoint, and there are (degJz'"7') possibilities for the choice of a path with this midpoint. We now show that the edge degree sequence of any connected triangle free G on the vertex set V = { X I , 2 2 , . . .xu} also determines the sum of the numbers K ( p 1 , l ;3) by the formula

u-1 -

1) -

V(W-

1) -

= W(W

=

deg(ti) deg(zjI2 j=1

Let F3 be the graph consisting of one edge an one isolated point. Then it is clear by considering the number of choices for the isolated point in F3 if the edge is chosen as one of the edges E, that there are II - deg(e,) choices for the isolated point. Then we sum over all edges. Note that the two equations given also imply the following fact: If two triangle free graphs G, G' have their numbers K(p1.p2; s) equal for s = 1, and for s = 2,then the number of subsets of size 3 containing exactly e induced edges 0 5 e 5 2 is the same for G and for G'. Of course that is a weaker property than the equality of the numbers K(p1,p2: 3). For the case of trees in particular we may revise and extend the above arguments by stating them in terms of degree sequences.

3.1

Some Properties of Degree Sequences

Assume that T is a tree wit,h v vertices and e = v - 1 edges. We let dl, d2, . . . , d, be the vertex degrees of T . For the v - 1 edges z j y j of T assume Ohat, deg(tj) = deg(zj) deg(yj)(l 5 j 5 w - 1) is the edge degree sequence of the tree. We

+

275 note that

c4 c v-1

2)

=

deg(9).

(14)

j=1

i=l

In general for any s, 1 5 s 5 ‘u - 1 we consider all induced subtrees of T. Let their number be M = M S . For the subset of vertices S of the subtree we get a relationship with the numbers p2(5’) defined above as follows: pz = s - 1, p1 = dl

+ dz + . . . + d,

-

2(s

-

1).

(15)

For s = 3 there is only one type of tree, namely the path P3, hence we find the number M3 to be

For s = 4 there are two types of trees on 4 vertices, namely the path and for the star we find the number

P4

and the star

5 (:). z=1

1

= -[xdeg(ey)’J=1

Ed?] x d e g ( E , ) + (v -

i= 1

-

1)

j=1

Hence the total number N4 turns out t o be

4

Trees with Four Long Ends

We describe a family of trees which have equal characteristic numbers up to s = 3 . We assume that we have a tree of degree sequence 142’-‘3’ such that the two points 41, 42 both of degree 3 form an edge. We also assume that the four paths with endpoints ~ 1 . 4 2 all have edge length larger than two. Note that this implies v 2 14. In this case by computation we see this tree has the following characteristic numbers K(p1, pz; 3) for the values of pl and p2 as given:

276

We now consider all possible trees arising in this way. Assume that the ends are of length al,a2,a3, ad.Since there is a two fold symmetry, obtained by swapping the edge 4142; and then by swapping the two paths at each end, we may classify the trees in question by which satisfy the inequalities 3 5 a1 5 a 2 , 3 5 a3 5 all integer sequences (al,az,a3,~4) ad,a1 5 a3 and the integer equation a1 a2 a3 a4 = v - 2. From standard counting techniques we see that for any value of v 2 14 we get N u non isomorphic trees with the same numbers K(p1,p ~s ); for s _< 3, where for even ZI = 2h we have

+ + +

Nu= ( h i 4 ) , and for odd v = 4k

+ 1 we have Nu =

while for odd u = 4k

(,',*) +

+ 3 we have N u = ( 2k

5

(k - 3)(k- 2);

-

3)+(k-2)'.

The Proof: Some Counting Arguments for subgraphs up to size 3.

We now proceed to give the proof for the calculations t,hat lead to the table of the previous section. We use the considerations on degree sequences given above.

5.1

Induced Subpaths of Length 3.

Lemma 1 Assume t.hat T is any tree which has the degree sequence 322L'-614.Then T has precisely v induced subpaths P3. This is clear from (16). since 2(;)

+ (v

-

6)(i) = v.

277

Lemma 2 Assume that T is any tree which has the degree sequence 322v-614,and assume further that the two points 41, q2 of degree 3 form an edge. If each of the four neighbour vertices of 41, qz has degree larger than one, then T contains precisely 4 induced subpaths P3 with p1 = 4. These are precisely the paths

P3

which contain the edge 4142.

Lemma 3 Assume that T is any tree which has the degree sequence 322"-614, and such that the two points ~ , q of2 degree 3 form an edge. If each of the four paths with end vertices 41, q2 has edge length ai > 2 larger than two, then T contains precisely 6 induced subpath P3 with p1 = 3. From the connectedness of the tree it follows that any induced subpath p1 > 0. Hence from lemmas 1 and 3 we obtain:

Pk

with k < v has

Lemma 4 Assume that T is any tree which has the degree sequence 322"-614. Assume that the two points 41, 42 of degree 3 form an edge. If each of the four paths with end vertices q1,q2 has edge length ai > 2 larger than two, then (as v 2 14) K ( 4 , 2 ;3) = 4, K(3,2;3) = 6, K ( 2 , 2 ;3) = v

-

14, K(1,2;3) = 4.

Remark: If z of the integers ai are equal t o 2, and 4 - z of them are then we get accordingly K ( 4 , 2 ;3) = 4, K(3,2;3) = 6 - Z, K ( 2 , 2 ;3) = v

5.2

-

14

aj

> 2, ( 0 5 z 5 4),

+ 22, K ( 1 , 2 ;3) = 4

- Z.

(21)

Induced Subgraphs with Two Components

Now we consider the case of an induced subgraph F3 on 3 points that is a union of an edge and an isolated point. It has already been shown that

The next easiest case is that of p1 = 6.

Lemma 5 Assume that T is any tree which has the degree sequence 322"-614. Assume that the two points 41,q 2 of degree 3 form an edge. If each of the four paths with end vertices q1,q2 has edge length larger than one, then T has precisely v - 10 induced subgraphs F3 with p1 = 6. Indeed such a subgraph must contain the edge 4192.The isolated point must be of degree 2. Hence the result follows by considering the number of choices for this isolated point. Next we are counting the subgraphs F3 with p1 = 3.

Lemma 6 Assume that T is any tree which has the degree sequence 322"-614. Assume that the two points 4 1 , of ~ degree 3 form an edge. If each of the four paths with end vertices q1 42 has edge length larger than two, then T has precisely 8v - 76 induced subgraphs F3 with p1 = 3.

278 Either the isolated point, or one of the two non isolated points of F3 must be a leaf.

+

In the first case,if we choose the leaf on the path of length a ] , we obtain (a1 - 3) (a2 - 2) (a3 - 2) (a4 - 2) choices for the edge of F3. Similarly for the paths of length azra3,a4 respectively. Hence the total number of subgraphs of type F3 in this case is + a3 Q) - 36 = 4~ - 44. 4(al In the second case, if we choose the edge containing the leaf on the path of length a l , there are (a1 - 3) (a2 - 1) (a3 - 1) (a4 - 1) choices for the isolated point of F3. Similarly for the paths of length az, a3, a4 respectively. Hence the total number of subgraphs of type F3 in this case is 4 ( a l + a2 a3 a4) - 24 = 4v - 32.

+

+

+

+

+

+

+

+ +

In total we obtain a number K ( 3 , l ;3) = 8v

-

76.

Lemma 7 Assume that T is any tree which has the degree sequence 322"-614. Assume that the two points q1,qz of degree 3 form an edge. If each of the four paths with end vertices q l , q z has edge length larger than two, then T has precisely 6v - 56 induced subgraphs F3 with p1 = 5. There are precisely three possibilites: Firstly, either the isolated point is a leaf, and the edge is q1q2. There are 4 possibilities for this situation. In the second case, the edge of F3 in T consists of two points of degree 2. In this case the isolated point must be equal t o q1 or 42. The number of possibilites for the choice of the edge is 2[(a1 - 2) (a2 - 2) (a3 - 2) (a4 - 2)] - 4 = 2~ - 24.

+

+

+

accordingly. In the third case; we choose the edge containing either one of q1 or 42. If we assume t,hat this edge is chosen on the path of length a ] , then there are (a1 - 3 ) (a2 - 2 ) + (a3 - 1)+ (a4- 1) choices for the isolated point of F3. Similarly for the paths of length 4,ag,a4 respectively. Hence the total number of subgraphs of type F3 in this cme is 4 ( a l + a2 a3 a d ) - 28 = 4~ - 36.

+

+ +

In total we obtain a number K ( 3 , l ;3 ) = 6v

~

56.

Note that the total number of subgraphs of order 3 containing precisely one edge can be calculated by (13). It is V(V

- 1) - [18

+4

( -~ 6)

+ 41 = v2

-

5v

+ 2.

(22)

Note that if K ( p i ; 1;3) > 0, then p1 E {2,3,4,5,6}. This follows from the fact, that the trees are connected, and all have degree sequence 3 ' Y 6 1 4 . This enables us to give the remaining number as K ( 4 ; 2; 3 ) = u2 - 5v

+ 2 - K(2,2; 3) - K ( 3 ;2: 3)

-

K ( 5 . 2 ;3 ) - K ( 6 , 2 ;3 ) = u2 - 19v + 96.

This completes the calculation of the second column of the table.

279

5.3 The 3-Cocliques We now count the independent sets of size 3 in the trees under consideration. The proof for K(3,O;3) = 4 has already been given above in (9). Next we show that K(4,O; 3) = GV

- 48.

(23)

Indeed a subset of size 3 with p1 = 4, p2 = 0 must consist of two points of degree one, that is a leaf, and one point of degree 2 all of which are independent. This is easy to count: For any of the 6 = possible pairs of leaves, there are precisely two forbidden points of degree 2, and all other points of degree 2, that is u - 8 such points may be chosen. This shows (23). We now show that

(3

K(5,O;3 ) = 2v2 - 34v + 168.

(24)

Indeed there are now two possibilities for the set, indicated as 311 and as 221. Either one point is of degree three, and two others are leaves, or two points are of degree two, and one is a leaf. The first kind is easily counted t o be 6 . 2 = 12. The second kind is a little tricky, and we may proceed as follows: Assume as before that there are 4 paths ending in 41, q2 of length a l , a2,a3,a4 such that a1 a2 a3 a4 = v - 2. If the leaf happens to be situated in the first path, then we see that the number of independent pairs of points of degree two can be counted as follows, distinguishing the case where the points are in the same path, given first, from the other case, given last:

+ + +

+(a1

-

2)(a2 - 1) l)(Q- 1)

+ (a1 - 2)(Q + (a2 - l ) ( Q

-

1)

1)

+ (a1 + (a3

-

2)(Q

-

l)(Q

-

1)

1) 13 1 = -(v - 2)’ - -(v - 2) + 24 = -(v2 - 1711 78). (25) 2 2 2 Now this expression is independent of the (first initially chosen) path, and hence the same calculation applies to the other t.hree paths. Thus we obtain a total of 2v2 - 34v 156 such sets, which together with the other 12 gives the proof of (24). We now deal with the case p1 = 7 and we show that +(a2 -

1

-

-

+

+

K(7,O;3 ) = v2 - 19v

+ 96.

(26)

Here the only case to consider is 322, one point of degree 3 and two independent points of degree 2. Assuming that the first and the second path are attached to 41; the chosen point of degree 3, we get similarly as above

(y3)

+(

y 3 )

+ (a,;z) + r 2 - 2 )

280 - 2)(U2 - 2) f

+(a1

(Ul

- 2)(Q

-

1)

+ (a1 - 2)(U4

-

1)

+

+(a’ - 2)(Q - 1) (a2 - 2)(Q - 1) 4- (U3 - l ) ( Q - 1) 1 15 1 = -(v - 2)’ - -(u - 2) 31 = -(u’ - 19v + 96). (27) 2 2 2 Considering the other choice 42 for the point of degree 3, the result is doubled. This proves (26). Now everything is shown except the cubic term for p1 = 6, p2 = 0. This can be dealt with arithmetically since we see from (12) (13) that

+

K(i,0;3) = i>l

(3 2 (3 + -

(deg:z’))

-

v(v - 1) +

j=1

=

- 212

2

deg(zj)2

j=1

4v

-

2.

Note that if K(p1,O;3) > 0 , then p1 t {3,4,5,6,7}. This follows from the fact that the trees are connected, and all have degree sequence 322”-614, and the two points q1,q2 of degree 3 are adjacent. Substituting all the other terms into

K(6,O; 3) =

(3

-

v2 + 4u - 2 - K(3,O;3) - K(4,O; 3) - K(5,O;3)

we find the last expression

1 K(6,O; 3) = -(v3 6

-

27v2

+ 308v)

-

222.

~

K(7,O; 3)

281

References [1] T. Bier, T.-H. Tan, On some characteristic numbers for graphs, submitted paper.

[2] T.Bier, T.-H. Tan, Two non isomorphic trees of order eleven with equal characteristic numbers, submitted paper. [3] T. Bier, Pendekatan berkombinatorik kepada model Ising, talk and accepted paper, Majlis PERSAMA Ogos 2001, UKM, Bangor, Malaysia [4] T. Bier, Non Isomorphic Trees with Equal Subgraph Numbers, submitted paper [5] R.C. Read and R.J. Wilson, An atlas of graphs, Oxford, Clarendon Press; New York, Oxford University Press, 1998, 454 p. [6] R. J. Trudeau, Introduction to Graph Theory, Dover Books on Mathematics, Dover Publications Inc.; New York, 1993; 209p. T. Bier , Kuhliyyah Sains dan Komputer Universiti Antarabangs Islam, Jalan Gombak, Kuala Lumpur Malaysia T.-H. Tan, Kolej Tunkku Abdul Rahman, Kuala Lumpur, Malaysia.

Sectionally pseudocomplemented lattices and semilatt ices I. Chajda, R. Halag * Abstract. We characterize lattices and semilattices with the greatest element 1 where for every element p the interval [p, 11 is a pseudocomplemented lattice or a semilattice, respectively. If for elements x,y the relative pseudocomplement x * y exists then it coincides with the pseudocomplement of x in the interval [xA y, 11. However, the pseudocomplement of x in [x A y, 11 can exist also when the relative pseudocomplement x * y does not. Thus our construction is a generalization of relative pseudocomplementation also to non-distributive lattices. Key words. Relative pseudocomplement, pseudocomplement, semidistributive lattice, semilattice, section pseudocomplement. MS classification: 06D15, 06D20. Let S = ( S ;A, 0) be a semilattice with the least element 0 and x E S. An element z* E S is called the pseudocomplement of IC if it is the greatest element of S such that x A x* = 0; in other words, for y E S we have x A y = 0 if and only if y 5 x* whenever x* exists. A semilattice S is pseudocomplemented (see e.g. [l], [3]) whenever x* exists for each x E S. Of course, every pseudocomplemented semilattice has also the greatest element 1 because it coincides with O*. A lattice L = ( L ;A, V, 0) with the least element 0 is pseudocomplemented if the semilattice ( L ;A, 0) has this property. Let S = ( S ;A) be a semilattice and 2,y E S. An element Q E S is called a relative pseudocomplement of x with respect to y, in symbols q = x*y, *The paper was prepared under the support of Czech Government Council No. 314/98: 153100011.

282

283

if it is the greatest element of S such that x A q 5 y; in other words z A z 5 y if and only if z 5 x* y whenever z * y exists. A semilattice S is called relatively pseudocomplemented or a Brouwerian semilattice if z * y exists for each z, y E S. A lattice 1: = (L;A,V) is called relatively pseudocomplemented or a Heyting algebra if the semilattice ( L ;A) is Brouwerian. It is well-known ([l] ,[3]) that every relatively pseudocomplemented lattice is distributive. Let us give a n example:

Example 1. Consider the five-element non-modular lattice

N5:

Then N5 is a pseudocomplemented lattice where O* = 1,1*= 0, z* = y* = z and z* = y. However, N5 is not relatively pseudocomplemented since e.g. y * z does not exists. On the other hand, for each p E N5 the interval [p, 11 is a pseudocomplemented lattice and hence N5 is sectionally pseudocomplemented. The previous example motivates us to introduce the following concept:

Definition 1. Let S = (5’; A, 1) be a V-semilattice with the greatest element 1.S is called a sectionally pseudocomplemented semilattice if for each p E S the interval [p, 11 is a pseudocomplemented semilattice. A lattice 1: = ( L ;A, V, 1) with the greatest element 1 is sectionally pseudocomplemented if the semilattice (L;A, 1) has this property. Contrary to the case of relative pseudocomplementation, the section pseudocomplementation as defined above is not a binary operation since it is relative to a given interval. To avoid this difficulty, we introduce the following operation:

Definition 2. Let S = (S;A, 1) be a V-semilattice with 1. For z, y E S we define z o y to be the pseudocomplement of z in the interval [z A y, 11. If

284

L = ( L ;A, V , 1) is a lattice with 1, x o y is defined identically. Remark 1. It is immediately clear that a semilattice S = (S;A, 1) with 1 is sectionally pseudocomplemented if and only if x o y is a binary operation on S. Example 2. Considering the semilattice N5 from Example 1, then y o x exists and is equal to x although y * z does not exists. Remark 2. For a lattice C = ( L ;A, V , 1) with 1,also another binary operation x y can be introduced as a pseudocomplement of x V y in the interval [y, 11. This approach was treated by the first author in [2]. Of course, C is sectionally pseudocomplemented if and only if x 0 y is a binary operation on L. Contrary to Definition 2, this approach cannot be applied for semilattices.

A lattice 1: = (L ;V , A) is called A-semidistributive if a A b l = a A b z =+- a A b l = a A ( b l V b z )

for every a, bl, bz E L. To characterize sectionally pseudocomplemented complete lattices, we generalize this concept as follows:

Definition 3. A complete lattice C is called completely semidistributive if for every a E L, {bi;i E I } C L we have a A bi = y for all i E I =+ a A V{bi;i E I } = y. Let us note that for finite lattices the concepts of i-sernidistributivity and that of complete semidistributivity coincide.

Theorem 1. Let C be a complete lattice. The following are equivalent: ( a ) C is completely semidistributive; ( b ) L is sectionally pseudocomplemented. Proof. ( a ) + (b) : Let x, y E L. Let A = {ai E L; z A a* = x A y}. Clearly A. # 0 since y E A . Put a = V A . Then (a) implies xAa = xAy and, moreover, a is surely the greatest element of this property and hence a = x o y directly by Definition 2. (b) + (a) : Let a, bi E L for i E I and suppose a A bi = y for each i E I. Take b = V{bi;i E I } . Then y = V { a A bi;i E I } 5 a A V { b i ; i E I ) = a A b.

285

Evidently, a 2 y thus a A y = y whence a o y is the greatest element such that a A ( a 0 y) = y. Hence b, 5 a o y for each i E I thus also b 5 a o y. We have shown a A b 5 a A ( a o y ) = y. Altogether, a A b = y thus C is completely semidistributive.

W

We are going to axiomatize the operation x o y in the case of lattices and semilattices.

Lemma 1. Let S = ( S ;A, 1) be a semilattice with 1 and let o be a binary operation on S satisfying the identities (a) x o x = 1 (b) x A (X 0 9) = x A y (c) z A (Z0 y) = ((x A z ) o (Z A 3))A z Then for each x, y E S the element x o y is a pseudocomplement of x in the interval [z A y, 11.

Proof. Assume xAy 5 z 5 xoy. Then, by (b), xAy 5 xAz 5 xA(xoy) = x A y giving x A z = x A y. Conversely, if x A z = x A y then x A y 5 z and, by ( a ) , (x A z ) o (x A y) = 1. By (c) we have z A (x o y) = 1 A z = z whence z 5 zo y. We have shown that x o y is the pseudocomplement of x in [xA y, 11. I

Remark 3. Although the axiomatic system ( a ) ,(b), (c) of Lemma 1 is very simple, the converse assertion is not valid since not every sectionally pseudocomplemented semilattice satisfies (c), see the following Example 3. Consider the (semi)lattice S given by the diagram depicted in Fig.2. Then z A (x o y) # z = ((x A 2) o (x A 9))A z. On the other hand, the reader can easily check that S is sectionally pseudocomplemented.

286

XAZ

Fig.2

We can involve a simple axiomatization of the operation o in sectionally pseudocomplemented lattices:

Theorem 2. Let L = ( L ; V , A , l ) be a lattice with the greatest element 1.The following conditions are equivalent: (1) C is sectionally pseudocomplemented; ( 2 ) there exists a binary operation o on L satisfying the following identities: ( a ) x o x = l and 1 o x = x (b) (((x v Y) O Y) O Y) A .( v Y) = x v Y (c) (((x V z ) A ( y V 2)) 0 z ) A (((x V z ) A ((y V z ) o 2)) 0 z ) = (xV z ) o z (4 ((x v Y) 0 9)v = (xv Y) O Y (e) x o y = x o ( x A y ) . Proof. (1) + (2) : Let x o y be the pseudocomplement of x in the interval [x A y, 11. The identities ( a ) and ( e ) are immediately clear. For (d), one can easily see that (x V y) o y is the pseudocomplement of x V y in the interval [y, 11and hence (xV y) o y 2 y. Similarly, x V y 5 ((x V y ) o y) o y whence (b) is evident. It remains to prove ( c ) . Trivially, (xVz)A(yVz)<xVz (xV z ) A ((y V z ) o z ) 5

xVz

Thus also

((xV z ) A (y V z ) ) o z 2 (x V z ) o z ((xVz) A((yVz)oz))oz 2 (ZV z ) o z whence

(((x V z ) A (y V 2)) 0 z ) A (((x V z ) A ((y V z ) o 2)) o z ) 2 (x V z ) 0 z. Prove the converse inequality. By using of the properties of pseudocomplementation in the interval [z,11, we have

287

(z V z ) A (y V z ) A [((z V z ) A (y V 2)) o z] = z , i.e. (x V z ) A [((x V z ) A (y V z ) ) o z] 5 (y V z ) o z and hence (z V z )A [( ( x V z )A (y V z )) oz] 5 ( x V z )A ( (y V z )02) 5 [ ( ((z V z ) A (yV z ) )02)oz] oz thus (z V z ) A

(((x V z ) A (y V z ) ) o z ) A ((x V 2) A ((3 V z ) o 2))

= z.

Hence (((z V 2) A (y V z ) ) 0 z ) A

(((x V z ) A ((Y V 2) 0 z ) ) 0 2) 5 (z V 2)

0

z,

the converse inequality. Thus (c) is also valid. (2) + (1) : We prove that x o y is the pseudocomplement of z in the interval [z A y, 11. Taking of x = 1 and z = p in (c) we obtain ((3V P ) O P ) A ( ( ! l o p ) O P > O

P

= l o p = P.

By (b) we have ((Y V P ) O P ) A ( ! l o p ) 5 ((Y V P ) O P ) A ((Y V P ) O P ) O P = P. Conversely, y V p 2 p and, by (d), (y V p ) op 2 p thus the converse inequality is also valid. Together we have (I/ V P ) O P ) A (Y V P ) = P (*I Interchanging y and and taking p = x A y, we have (xo (x A y)) A x = z A ?J and, by (e),

(**I

(Z0 3 ) A

=5~ y .

Suppose now that z E [z A y, 11 is an element satisfying z A x = z A y. We are going to show that z 5 z o y. Together with (**) it means that x o y is the pseudocomplement of z in [z A y, 11. Substituting z = x A y in the identity (c) and applying (a) we obtain z A (y o

(xA y)) 0 (xA y) = z 0 (X A 9).

By setting y = z we get

(4

(xA ( z o (x A z ) ) ) o (x A z ) = 2 o (x A 2).

By (c) we have

((x v P) A ( z v PI) 0 P L .( v P ) 0 P and hence z 5 x implies z V p 5 x V p which gives (2 V P ) O P 2 .( V P ) O P . (* * *>

288

Clearly z A ( z o (z A z ) ) 5 z o (z A z ) and, by (* * *) and (**) for p = z A z we obtain ( z o (z A z ) ) o (z A z )

5 (z A ( z o (z A 2)))

o

(z A z ) = z o (z A z )

directly by ( A ) .We have shown ( z o (z A z ) ) o (z A z ) 5 z o (z A z). Applying (b) we obtain z

5 ( z o (z A z ) ) o (z A z ) 5 z o (z A z )

and due to z o (z A z ) =

o (z A y) = z o y (derived by

( e ) ) , we conclude

z5zoy.

In what follows, we set up a similar axiomatization for sectionally pseudocomplemented V-semilattices.

Theorem 3. Let S = ( S ;A, 1) be a A-semilattice with the greatest element 1.The following conditions are equivalent: (1) S is sectionally pseudocomplemented; ( 2 ) there exists a binary operation o on S satisfying the identities: (a)zoz=l (b) ( z o y ) A z = z A y (4((z A 3) 0 2) A ((z A (Y 0 .( A 4 ) )0 (Y A 2)) = z 0 (Y A z> (4 ((z 0 9)0 .( A Y)) A 2 = z ( e ) z o (z A y) = z o y.

Proof. (1) + (2) : The identities ( a ) and (b) are almost trivial since (z o y) A z = xxAYA z = y A z, where ub for a E [b,11 stands for the pseudo-

complement of a in the interval [b,11. For ( c ) we have (z A y) o z = (z A y)xAyAzand z A (y o (z A 2))) o (y A z ) = (z A yzAyAz) o (y A z ) = (Z A yxAyAz)zAyAzA~Z"v"z =(zAy

xAyAz xAyAz

)

thus the left side of (c) is equal to (z A y)xAyAzA (z A yxAyAz)xAyAz. The right hand side equals to z o (y A z ) = z x A y A z . For the sake of brevity, denote by * the pseudocomplement in the interval [z A y A z, 11. Then ( c ) can be read (ZA y)* A (ZA y*)* = z*,

which is valid in every pseudocomplemented semilattice, see e.g. [3] or [l]. To prove (d), we compute ( ( ~ O ~ ) O ( X A ~ )= )A ( z~ Z

Ay~(z~y= ))~ ( zzx A ~ ) z Z " v A (= xA ( ~x) ~~z ~ v =) 2.~ ~

289

Analogously, z o (x A y) = z z A Y = x o y proving (e). (2) + (1) : We need to show that z o y is the pseudocomplement of z in the interval [z A y, 11. By (b) we have (z o y) A x = z A y. Suppose now that z E [z A y, I] and z A x = z A y. We only need to show that z 5 I o y which proves that z o y is the pseudocomplement of z in [z A y,1]. By (e) we have (~Az)oy=(zAz)o(zAyAz)=(zAz)o(zA~)=l.

Substituting the above equality in (c) and interchanging y and z we obtain ( ( z A z ) o y ) A ( ( z A ( z o ( z A y ) ) ) o ( z A y A z ) ) = ( z A ( z o ( z A y ) ) ) ~ ( z A y )= z o (y A z ) = z o (x A y), where we used the identity (e), hence (z A ( z o (z A y))) o (z A y) = z 0 (z A 9). (*I In account of ( b ) we have ( z o (z A y)) A z = z A y A z = z A y, thus z A ( z o (x A y ) ) A y = z A ( Z O (ZA y ) ) A z = z A y ,

which yields (z A ( z 0 (z A y)))

0

(x A ( z 0 (z A Y) A Y)) = (x A (2 0 (z A 9)))0 (z A 9).

The identity (c) implies immediately

((x A y) o z ) A (y o (x A z ) ) = y 0 (xA z ) (**I since (c) gives (z A y) o z 2 z o (y A z ) and ( f ) follows by interchanging the variables z and y. We apply (**): (z A ( z 0 (z A y))) 0 (z A y) L ( z 0 (x A y)) 0 (x A y) and hence ( * * * ) ( z o ( z A ~ ) ) o ( ~ : A 5~ )( ~ A ( Z O ( Z A Y ) ) ) O ( ~ A=~~) o ( ~ A Y = Z) O Y by (*) and (w). Now we apply ( d ) to obtain ( z o (z A 3)) o (z A y A z ) = (2 0 (z A y)) 0 (X A y) 2 z. Together with (* * *) it yields z 5 z o y.

290

Reference [l] Birkhoff G.: Lattice Theory (3rd edition), Publ. Amer. Math. SOC.,Vol. 25, Providence, R. I., 1967.

[2] Chajda I.: An extension of relative pseudocomplementation to non-

distributive lattices, Acta Sci. Math. (Szeged), to appear. [3] Frink 0.: Pseudo-complements in semi-lattices, Duke Math. J., 29 (1962), 505-514.

Author’s address: Department of Algebra and Geometry Palackf University Olomouc Tomkova 40 779 00 Olomouc Czech Republic e-mail: [email protected] [email protected]

T-STABLE IDEALS OF REGULAR RINGS Huanyin Chen Department of Mathematics, Zhejiang Normal University Jinhua, Zhejiang 321004, People’s Republic of China E-mail: [email protected]. edu. cn Miaosen Chen Department of Mathematics, Zhejiang Normal University Jinhua, Zhejiang 321004, People’s Republic of China E-mail: miaosen@mail. j h p t t .z j . cn Jinqi Li Department of Mathematics, Zhejiang Normal University Jinhua, Zhejiang 321004, People’s Republic of China E-mail: lij inqi@mail. j h p t t . z j . cn

Abstract. In this paper we introduce the concept of T-stable ideals of regular rings. Let R be a regular ring, I an ideal of R. It is shown that if I is a T-table ideal of R then M,(I) is a T-stable ideal of M,(R). Some interesting applications are also obtained. 2000 Mat hematics Subject Classification: 16U99,16E50

Keywords: T-stable ideal; regular ring; diagonal reduction

A ring R has stable range one in case a R + bR = R implies that a + by E U(R) for a y E R. This stable range condition is very important in algebraic K-theory(see [3,4,6,12]). A regular ring R is said to be unit-regular if it has stable range one. We know that every unit-regular ring is directly finite. A natural problem is how to extend this stable range condition to indirectly finite rings. In [3], P. Ara, G.K. Pedersen and F. Perera generalize this condition to QB-rings so as to investigate stable range conditions for indirectly finite rings. In this paper we introduce the concept of T-stable ideals of regular rings so that we can extend stable range one condition to indirectly finite rings in a ‘This work was supported by the National Natural Science Foundation of China (Grant No. 19801012) and the Ministry of Education of China ([ZOOO] 65).

29 1

292

new route. We prove that if I is a T-table ideal of R then M,(I) is a T-stable ideal of M,(R). Some interesting applications are also obtained. Throughout, all rings are associative with identity and all modules are right modules. U ( R )denotes the set of units of R. The notation T(R) stands for the set { x E R I 3 left invertible u E R and right invertible v E R such that x = u v } . Let M,(R) be the ring of n x n matrices over R with identity I,. We say that R is regular provided that for any x E R there exists y E R such that x = xyx. If M and N are right modules over a regular ring. The notation M 5 N means that M is isomorphic to a submodule of N .

Definition 1. Let R be a regular ring, I an ideal of R. We say that I is a T-stable ideal of R an case a R + bR = R with a E 1 + I , b E R implies that a + by E T ( R ) for a y E R. Let R be a regular ring, I an ideal of R. We call R a T-stable ring if it is a T-stable ideal of R. Clearly, every ideal of T-stable rings is a T-stable ideal. If I is a maximal ideal of R, then I is a T-stable ideal of R if and only if R is a T-stable ring. Also we see that every ideal having stable range one is a T-stable ideal for any regular rings. Clearly, every T-stable regular ring is a generalized stable ring; hence, the natural homomorphism E(l(R)+ U(R) is surjective provided that R is a T-stable ring.

Lemma 2. Let R be a regular ring, I an ideal of R. Then the following conditions are equivalent: ( 1 ) I is a T-stable ideal of R. ( 2 ) For any x E 1+ I , there exist an idempotent e E R and a w E T ( R )such that x = ew. Proof. (1)+(2) Given any z E 1 + I , there exists y E R such that 5 = xyx. Fromzy+(l-zy) = 1 with z E l f l , we deduce that z + ( l - x y ) z = w E K(R) for a z E R. Set e = xy. Then e 6 R is an idempotent of R. In addition, we have x = x y ( x ( 1 - x y ) r ) = ew, as desired. ( 2 ) + ( l ) Given aR+bR = R with a E 1+I, b E R, then az+by = 1 for some 2, y E R. As a E 1 I , there exist an idempotent e E R and a w E K(R) such that a = ew. So we have ewz by = 1, whence ewx(1- e) by(1- e ) = 1 - e. Therefore a by(1 - e)w = ew by(1 - e)w = ( 1 - ewx(1 - e ) ) w = (1

+

+

+

+

+

ewx(1 - e))-'w E T(R), as required.

+

+ 0

Example 3. Let V be an infinitely dimensional vector space over a division ring D and S be a directly finite regular ring which is not unit-regular(see [9, Example 5.101). Set R = EndD(V) @ S and I = EndD(V) @ 0 . Then I is a T-stable ideal of R, while R is not a T-stable ring.

293

Proof. Clearly, R is regular. Given any z E 1 + I , we have y E Endo(V) such that 5 = ( y , l ) . Because EndD(V) is one-sided unit-regular, there are idempotent e E R and right or left invertible u E R such that y = eu. Hence z = ( y , l ) = ( e , l ) ( u , l ) . Set f = ( e , l ) and w = ( u , l ) . Then f E R is an idempotent and w E T(R). It follows by Lemma 2 that I is a T-stable ideal of R. Assume now that R is a T-stable ring. We easily check that 0 G3 S is a T-stable ideal of R. This implies that S is also a T-stable ring. Because S is directly finite, it is unit-regular, a contradiction. Therefore we conclude that 0 R is not a T-stable ring. Let R be a regular ring, I an ideal of R. Denote by FP(I) the set of finitely generated projective right R-module P such that P = P I . We say that 1 satisfies the comparability axiom provided that for any A, B E FP(I), either A 5 B or B 5 A.

Theorem 4. Let R be a regular ring, I an ideal of R. If I satisfies the comparability axiom, then I is a T-stable ideal of R. Proof. Given any 5 E 1 + I , there exists a y E R such that x = xyx. Since + I, we deduce that x = x y x E y x + I , so z - y z E I . Likewise, we have x - z y E I . Hence 1-yz = (1-x)+(x-yx) E I and 1-xy = ( 1 - x ) + ( z - z y ) E I . We see that ( 1 - y x ) R = (1- y x ) ( l - y x ) R = ( 1 - y x ) I . Since R is regular, every ideal of R is idempotent. Thus ( 1 - y z ) R = ( 1 - yz)12 = (1 - y z ) R I . That is, (1 - y x ) R E FP(1). Analogously, we claim that ( 1 - z y ) R E F P ( I ) . By the hypothesis, either (1 - zy) R 5 ( 1 - y x ) R or ( 1 - yz) R 5 ( 1 - x y )R. If ( 1 - x y ) R 5 ( 1 - gz) R, then ( 1 - z y ) R is isomorphic to a direct summand of ( 1 - y z ) R ; hence, we have an injection 11, : ( 1 - zy)R + (1 - yz)R. Clearly, R = y x R @ (1 - y z ) R = x y R @ ( 1 - x y ) R with i$ : x y R = z R 2 y x R given by i$(xr) = y ( x r ) for any z r E zR. Define u E EndR(R) so that u restricts to i$ and u restricts to $.I Then x = z u x with left invertible u E R. If (1- y z ) R 5 ( 1 - x y ) R , we also have right invertible u E R such that x = zuz. Consequently, x = xux for a right or left invertible u E R. zE 1

From zy + (1 - z y ) = 1, we have (uz)y + u(1 - z y ) = u. Set e = uz. Then e E R is an idempotent and e ( y + (u(1- z y ) ) (1 - e ) u ( l - x y ) = u. By the regularity of R, there exists c E R such that (1 - e ) u ( l - zy) = ( 1 - e ) u ( l - x y ) c ( l - e ) u ( l - zy). Set g = ( 1 - e ) u ( l - z y ) c ( l - e ) . Then e = e2,g = g 2 and eg = ge = 0. Furthermore, we have e ( y (u(1- zy)) =

+

+

294

eu, ( 1 - e ) u ( l - zy) = gu, and that ( e + g)u = u. Hence u ( z + ( 1 - z y ) c ( l - e ) ( l + e u ( 1 - z z ) c ( ~- el)>( I - e u ( 1 - z z ) c ( l - e ) j u = e + u ( 1 - z y ) c ( l - e ) ( l + eu(1- z z ) c ( l - e ) ) (1 - eu(1- z z ) c ( l - e ) j u = e ( 1 - eu(1 - z z ) c ( l - e ) ) u ( 1 - z y ) c ( l - e ) = e ( 1 - e ) u ( l - z y ) c ( l - e )j u = ( e g)u = u.

I

+

+

+

+ ( 1 - z y ) c ( l - e ) (1 + eu(1 - z z ) c ( l - e)> E R is right or left invertible. Set w = z + ( 1 - z y ) c ( l Since u E R is right or left invertible, we show that x

We have z = z y z = z y w with a y = ( ~ y )w~ E, T(R). 0 I is a T-stable ideal of R. As an immediate consequence of Theorem 4, we know that every regular ring satisfying the comparability axiom is a T-stable ring. Lemma 5. Let R be a regular ring, I an ideal of R. Then the following are equivalent: ( 1 ) R is a T-stable ideal of R.

+

+

( 2 ) Whenever a R bR = R with a E 1 I , b E I , there exists y E R such that a by E K(R). Proof. ( 1 ) + ( 2 ) is trivial. ( 2 ) j ( 1 ) Suppose that ac b = 1 with a E 1 I , c, b E R. Because of the regularity of R, there exists z E R such that a = aza. Clearly, a z + ( l - a z ) ( l a ) = az ( 1 - a z ) - ( 1 - az)a = 1 with a E 1 + I and ( 1 - a z ) ( l - a ) E I . So we can find y E R such that a ( 1 - a z ) ( l - a)y = w E T(R); hence, a = az (a ( 1 - a z ) ( l - a)y> = e w , where e = az is an idempotent of R. 0 Using Lemma 2, we show that I is a T-stable ideal of R.

+

+

+

+

+

+

+

The pair (a,b) is called right unimodular in case a R bR = R. We say that right unimodular ( a ,b) is right T-reducible if there exists y E R such that a by E T(R).

+

Lemma 6. Let ( a ,b ) be a right unimodular row in a ring R. Let u,v E U(R) and c E R. Then (vau vbc, vb) is also right unimodular. Furthermore, ( a ,b) is right T-reducible if and only if so is ( a u vbc, vb).

+

+

Proof. Because of [3, Lemma 6.31, (vau+vbc, vb) is right unimodular. Assume that (a,b) is right T-reducible. Then there is a y E R such that a+ by E T(R). Set z = y u - c. Then (vau vbc) (vb)z = v ( a by)u E T(R); whence,

+

+

+

295

+ vbc,vb) is right T-reducible. Conversely, assume that there exists a z E R such that vau + vbc + vbz E T(R). Then v(a + b(c + z)u-')u E T(R), and that a + b(c + 2)u-l E T(R). Therefore ( a ,b) is right T-reducible. 0 (au

Theorem 7. Let I be an ideal of a regular ring R. If I is a T-stable ideal of R, then M n ( I ) is a T-stable ideal of M n ( R ) for any n 2 1. Proof. By [ll,Theorem 1.71, Mn(R)is regular. Clearly, the result holds for n = 1. Assume inductively that the result holds for n. It suffices to show that the result also holds for n + 1. Suppose that

in

Mn+l

( R ) ,where

+

+ +

+

+

1 1 with all E 1 1. Obviously, allbll alzb21 . . . al(n+l)b(n+l)l ~ 1 = Since I is a T-stable ideal of R, there is a z1 E R such that all (a12b21 . . . al(n+l)b(n+l)l f c11)zl E T(R). Furthermore, we have all (anbzl+. . . al(n+l)b(n+l)l+ cn)z1 E 1 I since all E 1 I , a12,. . . ,q n + l ) , c11 E I . Using Lemma 6 , (*) is right T-reducible if and only if this is so for the row with

+

+

+

+

+

+ +

296

elements

1 0 ... 0 0 1 ... 0

b2121

b31q

.. .. . . . . . . .

i(n+l)lzl

+

c11

c12

C13

c2 1

c22

C23

... . ' '

C2(n+1)

c33

...

C3(n+l)

c31

c32

Cl(n+l)

0 1 0 .. . 0 0

z1 0 0 . ..

0 0 1 . .. 0

0 0 ... 1

..' ... 0

... 0 .. . 0.. 1 ... 1

and

Thus we assume that all E T(R) n (1 + I ) in (*). Since a l l E T(R), there exist u, w,s, t E R such that all = uw,su = 1 and vt = 1. So s q l t = 1. From all E 1 I , we see that st E 1 I . Clearly, we may assume that

+

+

297

It is easy to check that

0 ... 0 ... 0 1 ... 0

0

S

a11t 0

1 -u11ts

0

0

... .. . ..

..

. 0 0 t 1 -tsa11

... .-. ... ... .. .. . 0 ...

0 0 0 1

SUll

0 0

1 0 0 0

.. 1

and that S

0

l-allts

Ullt

0

0

0 ... 0 0 ... 0 1 ... 0

... .. . .. 0 ... 1 . .. 0 ... 0 ... 0 .. . ... 1

... 0

0

t 1 - tsa11 0 0 0

0 1

SUll

0

.. 0

0

0

0

a11t

0

0 0

1 - UlltS 0 ... 0 S 0 ... 0 1 ... 0 0 .. . . . .. 0 ... 1 0

sa11 1 -tsa11

0

.

0

0 0 t 0 0 1 ... ...

*.. 0

... ... ... 0 0 ...

-1

I

-1

0 0

.. 1

Thus (*) is right T-reducible if and only if this is so for the row with elements 1

bl2

b13

' ' '

b2l

b22

b23

' ' '

b31

b32

* ...

*

b(n+l)l b3(n+1)

bZ(n+l)

c12

C13

c22

C23

... ..

c31

c32

c33

.*.

c(n+1)1

c(n+1)2

c(n+1)3

0 0 ... 0

1 - a11ts at 0 . . . 0

*

c21

+ ((1 -

S

0

c11

Clearly, all b,, E I for i tsall)

* ' '

bl(n+l)

0 1 ... 0 ... ... . . . ., 0 0 ... 1

*

' ' '

C(n+l)(n+l)

# j. Furthermore, b22 = ((1- a l l t s ) a l l + alltazl)(l -

a11ts)ulz

+

al1tazz)sall.

Since

all, u22, s t

E 1

+ I and a12 E I ,

298

+

+

we deduce that b22 = (1 - t s ) ( l - t s ) t s = 1 (mod I ) . Hence b22 E 1 I . Using Lemma 6 again, (*) is right T-reducible if and only if this is SO for the row with elements 1 0 0 ... 0

. .

0

* * ... *

1 0 0 ..’ 0

. . .

*

0 0 ...

C13

c21

c22

C23

c 31

c32

c33

... ...

c(n+1)3

’ ‘’

c(n+1)1 c(n+1)2

0 0 ...

...

c12

c11

0 0 ‘.’

s

C(n+l)(n+l)

Thus, we assume that all = 1,ali = 0 = ail for i = 2, ... , n Moreover, we may assume that (*) is in the following form:

1

+ 1 in

(*).

0

+

+

where D E diag(1,1,. . . , 1) M , ( I ) . Clearly, we have DB22 CZZ= I,. By the induction hypothesis, Mn(I)is a T-stable ideal of M,(R). Thus we can find a Z2 E M n ( R ) such that D CzzZ2 E T(M,(R)); hence, we pass t o the right unimodular row with elements

+

So it suffices to show that the right unimodular with elements ‘12’2

is T-reducible. Inasmuch as D

)

and

(

‘11

‘12)

c21

c22

+ C Z Z ZE~ T(M,(R)),( 0

T(Mn+l(R)). By induction, we complete the proof.

D

+

C2222

0

Corollary 8. Let R be a regular ring, I an ideal of R. If I is a T-stable ideal of R, then for any A E M,(I) there exist E = E2 E M n ( I ) , W E T(M,(R)) such that A = EW.

Proof. Because I is a T-stable ideal of R, from Theorem 7, M,(I) is a Tstable ideal of Mn(R). Given any A 6 M,(I), we have B E M,(R) such

299

that A = ABA and B = BAB. As AB

+ (I,

+ ( I , - AB) = I,,

we have ( A

+

+

AB)(I, - B ) = I, with A + ( I , - AB) E I , Mn(I). Thus we have Z E M,(R) such that A ( I , - A B ) ( I , ( I , - B)Z) = A + ( I , - AB) ( I , - A B ) ( I ,- B ) Z = W E T(M,(R)). Set E = AB. Then 0 E = E2 E M,(R) and A = ABA = E W , as asserted. (I, - AB))B

-

+

+

+

Corollary 9. Let R be a regular ring, I an ideal of R. If I satisfies the comparability axiom, then for any A E M n ( I ) there exist E = E2 E M,(I), left invertible U E M n ( R ) and right invertible V E M,(R) such that A = EUV.

Proof. By virtue of Theorem 4, I is a T-stable ideal of R. Thus we complete 0 the proof by Corollary 8. Lemma 10 Let R be a regular ring, I an ideal of R. I f I is a T-stable ideal of R, then a R = bR with a E I , b E R implies that a = bw f o r a w E T ( R ) .

Proof. Suppose that a R = bR with a E I , b E R. Then we have x , y E R such that a = bx and b = ay. Clearly, ayx = a and bxy = b. Since ( x + ( 1 - x y ) ) y + ( 1 - x y ) ( l - y) = 1 with x + ( 1 - x y ) E 1 + I , we can find a z E R such that x + ( 1 - x y ) + ( l - x y ) ( l - y ) z = w E T ( R ) . That is, x + ( l - x y ) ( l + ( l - y ) z ) = w . Therefore a = bx = b ( x

+ (1 - xy)(l+ (1

-

0

y ) i ) ) = bw, as asserted.

Lemma 11 Let R be a regular ring, I an ideal of R. If I is a T-stable ideal of R, then a R bR with a E I , b E R implies b = wlaw2 for some w1, w2 E T ( R ) .

Proof. Suppose that $ : a R 3 bR. Because of the regularity of R, R a = R$(a) and $(a)R= bR by [12, Lemma 11. As a E I , we claim that $ ( a ) E Ra C I ; so b E $ ( a ) R C I . By Lemma 10, we have w1,w2 E T ( R ) such that $ ( a ) = w l a 0 and b = $(a)w2. Consequently, b = w1aw2 with w1, w2 E T ( R ) . Theorem 12. Let R be a regular ring, I an ideal of R. If I is a T-stable ideal of R, then every square matrix over I admits a diagonal reduction b y one-sided invertible matrices.

Proof. Suppose that A E M n ( I ) . Since R is regular, there exist E = E2 E M,(R) such that AM,(R) = EM,(R). Clearly, we have idempotents e l , ' . ' , en E Rsuch that ER" E e l R @ . . . @ e , R " diag(el,." ,en)Rn as right R-modules, so we claim that E R n X 1E diag(el,... , e n ) Rnxl,where

R n X 1= {

(::i ;

I

. . , x, E R } is a right R-module and a left M,(R)-module. Let R l X n= { ( X I , . . , 5,) 1 X I , . . . , x, E R}. Then Rlxn is a left R-module and a right

XI,.

300 M,(R)-module.

Therefore ( E R n X 1@) Rlxn E diag(el,. R

. . , en)Rnxl @ RIXn. R

One easily checks t h a t Rnxl@ Rlxn% Mn(R) as right Mn(R)-modules. T h u s EMn(R) d i a g ( e l , . . . ,en)Mn(R). Applying Lemma 11, we have U,V E 0 T(Mn(R)) such t h a t UAV = diag(el,. . . , en), as desired.

Corollary 13. Let R be a regular ring, I a n ideal of R. If I satisfies the comparability axiom, then every square matrix over I admits a diagonal reduction by one-sided invertible matrices. Proof. According to Theorem 4, I is a T-stable ideal of R. Therefore t h e 0 result follows by Theorem 12.

REFERENCES [l] P. Ara, K.R. Goodearl, K.C. O’Meara and E.Pardo, Diagonalization of matrices

over regular rings,linear Algebra Appl., 265( 1997), 147-163. [2] P. Ara, K.R. Goodearl, K.C. O’Meara and E.Pardo, Separative cancellation for projective modules over exchange rings, Israel J. Math., 105(1998), 105-137. [3] P. Ara, G.K. Pedersen, F. Perera, An infinite analogue of rings with stable range one, J . Algebra, 230(2000), 608-655. [4] M.J. Canfell, Completion of diagrams by automorphisms and Bass’ first stable range condition, J. Algebra, 176(1995), 480-503. [5] H. Chen, Elements in one-sided unit regular rings, Comm. Algebra, 2 5 (1997), 2517-2529. [6] H. Chen, Rings with stable range conditions, Comm. Algebra, 26(1998), 36533668. [7] H. Chen, Related comparability over exchange rings, Comm. Algebra, 2 7 (1999), 4209-42 16. [8] H. Chen, On generalized stable rings, Comm. Algebra, 28(2000), 1907-1917. [9] H. Chen, Morita contexts with many units, Comm. Algebra, 30(2002), 26992713. [lo] H. Chen and F. Li, Exchange rings having ideal-stable range one, Science in China(Series A), 44(2001), 580-586. [Ill K.R. Goodearl, Von Neumann Regular Rings, Pitman, London, San Francisco, Melbourne, 1979; second ed., Krieger, Malabar, Fl., 1991. [12] R.E. Hartwig and J. Luh, A note on the group structure of unit regular ring elements, Pacific J. Math., 71(1977) 449-461. [13] H.P. Yu, Stable range one for rings with many idempotents, Trans. Amer. Math. SOC.,347 (1995), 3141-3147.

Simple groups which are product of the linear fractional group with the alternating or the symmetric group M. R. Darafsheh Department of Mathematics and Computer Science Faculty of Science, University of Tehran, Tehran, Iran. E-mail: Daraf Qkhayam.ut .ac .ir

Abstract We will find the structure of finite simple groups G with proper subgroups A and B such that G = AB and A isomorphic to the linear fractional group PSL2(q), q 2 11, and B isomorphic to the symmetric group on n 2 5 letters.

1

Introduction

Let G be a group with subgroups A and B such that G = AB = {abla E A, b E B } . Then we say that G is a factorizable group and A and B are called factors of this factorization. Of course if any of A or B is not a proper subgroup of G, then obviously we have the factorization G = AB which is called the trivial factorization of G. Therefore by a factorization of G we mean a factorization with proper factors. It is an interesting problem to know which groups are factorizable. An infinite group all of whose proper subgroups are finite is not factorizable. By [11] if L is an exceptional group of Lie type except G2(q),q = 3n, F4(q),q = 2n or G2(4), then L does not have a factorization. Also the Mathieu group M22 of order 27.32.5.7.11is not factorizable. We assume G is a finite group and will mention some research works towards factorization of G with specified factors. In [5] and [6] factorizations of sporadic simple groups and simple groups of Lie type with rank 1 or 2 as the product of two simple subgroups are obtained respectively. In [lo] groups G with factorization G = AB, where A % B A5 are classified and in [14] groups G = AB where one

2000 AMS Subject Classification: 20D40 Key words: Factorization, Linear fractional group, Symmetric group.

30 1

302 of the factors is a non-abelian simple group and the other one is isomorphic to A:, are found. In [15] G.Walls started the study of factorizations when one factor is simple and the other is almost simple. In [2] we classified all finite groups G with factorization G = AB where A A6 and B s S n , n 2 5 , and in [4] we determined all groups G with factorization G = AB where A is a simple group and B 2 s6. Motivated by the above works we are interested to classify all finite groups G with subgroups

A and B such that G = AB when A 2 PSLz(q),q 2 11. and B Sn,n 2 5 . The isomorphisms PSL2(4) 2 A:, PsL2(5), and PSLz(9) A6 are well know. As and B 2 S n , n 2 5, are determined. But in [15] groups G = AB where A A6 and B 2 S,,n 2 5, are studied. Also in In [3] groups G = AB where A [2] we investigated finite groups G = AB, where A PSL2(7) or PSLz(8) and B Sn, n, 2 5. Therefore in this paper we will assume y 2 11.

2

Preliminary Results

In our consideration we need to know about the degrees of the transitive permutation representations of PSLz(y). This is equivalent to knowing the indices of all subgroups of PSLZ(q). By [7] page 213 we know all subgroups of PSLz(q), and therefore we can find the indices of all subgroups which are listed in the following Lemma.

Lemma 1. The degrees of all transitive permutation representations of the groups PSL2(q),q = pf, are as follows where k = (pf - 1,2). ( a ) k p f - y p 2 f - l),0 5 i 5 f . ( b ) tpf(pff l), ti*, (c) i t p f ( p f f 1).t1*. (d) &pf(p2f - l),if p is odd, & p f ( p 2 f - l), if p = 2 and 21f . (e) & p f ( p 2 f - I), if 161p2f - 1.

(f)& P f ( P 2 f

- I), if P = 5 , &pf(p2f - l), i f 51p2f - 1. (9)&pf-"(p2,2f - l), tip" - 1 and tl# - 1.

( h ) %pf-"

(G)where rnlf

(&)

and k' = (p" - 1 , 2 ) ,

ipf-" where 2rnl f . We use the above numbers in our considerations. We call these indices 1-numbers.

303

We also need to know about k-transitive and k-homogeneous actions of the group PSLz(q) where k 2 2. Note that the group PSL2(q) has a natural action on the q 1 points of the projective line. It is clear that a.ny k-transitive group is k-homogeneous. According to [13] any permutation group which is k-homogeneous on n points, n 2 2k > 2 is (k - 1)-transitive, and for k 2 5 it is even k-transitive. If a permutation group is k-homogeneous but not k-transitive, then it is called of type k*, and so by [13] we must have k E {1,2,3,4}. Groups of type k', k 2 2, are determined in [8] and we list the groups PSL2(q) in the following Lemma.

+

Lemma 2. The only groups of type k* occuring among PSL2(q) are the followings: (u) PSLz(8) o n 9 points is a group of type 4'. ( b ) if q = 3(mod 4 ) , then PSL2(q) o n q 1 points is of type 3'. The groups PSLz(q) which have k-transitive action, k 2 2, are given in the following Lemma.

+

+ +

Lemma 3. In general the group PSL2(q) acts 2-transitively o n the q 1 points of the projective line. If q is even then PSL2(q) acts 3-transitively o n q 1 points. The group PSL2(11) acts 2-transitiwlg o n 11 points. PSL2(5) acts 3-transitively on 5 points and PSL2(9) acts 4-transitively o n 6 points. If G is a permutation group, then knowing about permut.ation representations of G and using the following Lemma we can obtain factorizations with one of the factors isomorphic to G. Lemma 4. Let H be a k-homogeneous permutation group on a set R. Let G be a k-homogeneous subgroup of R, 1 1. k 2 101. T h e n H = H(A)G where A is a k-subset of R and H(A) is its global stabilizer. Now using Lemmas 1, 2, 3 and 4 together with Theorem D of [ l l ]we can find how alternating or symmetric group can be decomposed as product of PSLz(q)with alternating or symmetric group. Lemma 5. Let G, = A, or S,. If G, = Lz(q)B, where B is a proper subgroup of

G, and q 2 11, Then: (i) G, = PSL2(q)Gn-1 where n is a n 1-number. (ii)G, = PSLz(q)B where n = q 1 and B = Gn-l, Gn-2 or G,-2.2. (iii)GI1 = PSL2(11)(G,.2) = PSL2(11)Gg = PSL2(11)Glo. ( i w ) G, = PSL2(q)B where n = q+ 1 and B is a subgroup of G, with one of the following properties: Gla-31. B 5 Gn-3.S3, Gn-2 5 B 5 Gn-2.S2 or B = G,-1. ( v ) G, = PSL2(q)B where n = q + 1 and q = 3(m,od 4) a n d G,-3 < B 5 Gn-3.S3, Gn-2 5 B 5 Gn-2.S2 or B = G,-1.

+

304

Proof. Let G, = PSLz(q)B where G, A, or S,. Then by Theorem D of [ l l ] either n= 6 , 8, 10 or one of the PSLz(q) or B is k-homogeneous on n letters, 1 5 k 5 5 . If n= 6, 8, 10, then by [ l l ]none of decompositions of Gn involve PSL2(q),so we will assume the second possibility. Let G, = PSL2(q)B and PSL2(q) be k-homogeneous on n letters, 1 5 k 5 5 and An-k a B 9 Sn-k x s k . If k = 1, then by Lemma 1, n is an 1-number and B = G,-l which is case (i) of the Lemma. If k = 2, then we use Lemma 3. In general we have G, = PSLz(q)Bwhere n = q 1, B = G,-1 or B = Gn-2.2 and this is case (ii) of the Lemma. Since PSL2(11) acts 2-transitively on 11 points case (iii) also occurs. If lc = 3, then PSL2(q) acts 3-transitively on q + 1 projective points if q is even and case (iv) occurs. In this case Lemma 2 is also applicable. Since in this case PSL2(q) on q + 1 points is 3-homogeneous but not 3-transitive, so we have G, = PSL2(q)B where Gn-3 < B I: Gn-3.S3, or Gn-2 5 B 5 Gn-2.S2 or B = G,-l, and this the case (v). If B is k-homogeneous on n points and An-,+ PSL2(q) 9 Sn-k x &, then by simplicity of PSLz(q) we get n - k = 1 or 2. Since 1 5 k I: 5, we obtain n 5 7 and obviously no such decomposition of G, occurs.0

+

a

3

Main Results

This section is devoted to our main results. Here we deal with the factorizations of finite simple groups.

Lemma 6. Let G be a sporadic finite simple group. Then it is impossible to write G as G = PSLz(q)B where q 2 11 and B is isomorphic to an alternating or symmetric group.

Proof. If G is a sporadic finite simple group and G = PSL2(q)B where q 2 11 and B S A,, n 2 5 , then as both factors are simple groups, so by [5] we don’t get a possibility. Therefore we will assume G = PSL2(q)B,where q L 11 and B Z S,, n 2 5 . By [ l l ]factorizable sporadic simple groups are MII,h f 1 2 , M22, M2 3 , h f 2 4 , J2, H S , H e , Ru, Suz, Fizz, or Col. By [ l ]we know the subgroup structure of the above groups. Therefore the maximum n such that S, is a subgroup of the above groups is obtained as follows. If Sn 5 J2, then n = 4; if S , 5 A411,M12, or Suz, then n = 5; if S, is a subgroup of one of MZJ,M23,Ilf24,H e , Ru, then n = 6; if S, I. H S or Co1, then n = 8, and if S, 5 Fizz, then n = 10. Now by order consideration it is enough to prove G = PSL:,(q)B,B S,, q 2 11, is impossible for

305

a maximum n. We will deal with G = Co1, the other cases may be treated similarly. First we will show that if PSL2(q) 5 Co1, q 2 11, then q = 11,13,23,25,27. We know JPSL2(q)J = ,$q(q2 - l), where q = pm, p prime, and d = ( q - 1,2). Since IPSL2(q)lIIColl = 22'.39.54.72.11.13.23,hence from q = p"llCo1l we will obtain p = 2 , 3 , 5 ,7 ,1 1 ,1 3or 23. If p = 11, 13 or 23, then obviously q = 11, 13 or 23, and by [l]the group Col has subgroups isomorphic to PSLz(q) for q = 11, 13 or 23. To treat the other primes we use this fact that if q is odd, then PSL2(q) has elements of order If p = 7, then only q = 72 is possible. But this would imply that Col must have an element of order 25 which is impossible by [l].If p = 5 , then q = 5 2 ,53 and 54 are possible. Since Col does not, contain elements of order 63, and 313 and Col has a subgroup isomorphic to PsL2(25), so q = 5'. If p = 3, then q = 3', 3 5 i 5 9. Now again considering elements of order we see that only q = 33 is possible. Therefore we have proved that if PSLa(q),q2 11 is a subgroup of Co1, then y =11, 13, 23, 25 or 27. Now to prove that Col = PSL*(q)B,B Sn, is impossible it is enough to prove that the decomposition Col = Ps&(q)Ss is impossible in the cases q =11, 13. 23, 25 or 27. But this is easily done by order consideration and the Lemma is proved.

q.

9

0

Lemma 7 . Decomposition of the projective special linear group as product of PSL2(q),q 2 11, with A, or S,, n 2 5, is impossible. Proof. Let PSL,(q') = PSL;!(q)B,where B E A, or S,. We will prove the Theorem in the case of B Z A,. The case B % S, is similar. We will consider two cases:

(i). n 2 9. In this case by [9] the minimum degree of a projective modular representration of A , is n, - 2 and therefore m 2 n - 2 which implies n 5 m 2. Since n 2 9 we = qrk-l . .. 1 2 will obtain m 2 7. For any natural number k we have Case

+

2k-1

+ +

+ . . . + 1 = 2k - 1 2 k + 2, hence q I k - 1 2 ( k + 2)(q' - 1) 2 ( k + 2)q'i. But

where d' = ( m ,q'

- 1). Therefore using above inequa1it.ywe will obtain:

Since we have assumed PSL,(q') = PSLz(q)An we get IPsL,(q')l I IPSL2(q)llAnl and since n 5 m+ 2 this implies IPSL,(q')I 5 w q ( q 2 - 1) where d = (2,q - 1).

306 ,,b2 - 1

Combining the inequalities obtained so far yields & ( m , + 2 ) ! q ’ T 5 w q ( q -2 1) which implies q ’ e 5 yq(q2 - 1). But d = ( 2 , q - 1) = 1 or 2 and so

> 1,

therefore q“ > Tq(q2 - 1) 2 q’*, hence q 2 q ’ e . Since q and q’ are prime powers we may assume that q = pf and q‘ = ptf’ where p and p’ are prime numbers and f and f’ a.re natural numbers. We will consider two subcases.

Subcase 1. p‘ # p . Since PSL,(q’) = PSL2(q)A,, hence PSL2(q) is embeded in PSL,(q’) and by a. result in (121 we get m 2 Therefore q - 1 _< 7r~dwhich implies q 5 m d + 1.

9.

&-1

m2-1

But. we obtained q 2 4 ’ 7 .It is easy to prove 4 ’ 7 >

q.

(*)

q’

+1 2

+1 =

+

Hence < md 1, so m2 - 4m.d - 1 < 0. Since d = 1 or 2 from this last inequality we get m 5 8 which contradicts our original assumption if m > 8. The case m = 7 and 8 are treated below. If m, = 7, then n 5 9 and hence n = 9. If m = 8, then n 5 10 therefore n = 10 or9. F r o m m ~ ~ w e o b t a . i n q ~ m d + 1 ~ 7 d + 1 < 1 5 o r q ~ 8 d + 1 < 1 7 the respective cases m = 7 or 8. Therefore q = 11, 13 or 16. Now we must. treat the cases PSLT(q’) = PSLZ(11)Ag or PSLz(13)Ag and PS&(q‘) = PS&(q)A, for q = 11: 13 or 16 and contradiction.

76

= 9 or 10. Now order consideration simply gives a

Subcase 2. p = p’. m2-1 Since q = pf a.nd q’ = p’f‘ = pf‘, hence from q 2 q‘” we will obtain f 2 If p“f‘ - 1 2 p 2 f - 1, then m.f‘ 2 2f and using the previous inequality we will obtain m2 - 4m, - 1 5 0 contradicting m > 8. Therefore p”’r - 1 < p2f - 1 implying m.f’ < 2f. Now consider p 2 f - 1. If f # 1 and (p, 2f ) # (2,6) then by a result of Zsigmondy quoted in [ll]page 37 there is a prime r dividing p2f - 1 such that T Apz - 1 for all 1 5 i 5 2f. But from PSLa(pf) 5 PSL,(pf‘) we obtain p2f - l/(p“f’ - 1).. . (p2f’ - 1) and t.hereforerlpkf’ - 1 for some k , 2 5 k 5 m. But from mf’ < 2 f we have kf‘ < 2f for all 2 I k 5 m,.Hence we get a contradiction with respect to the Zsigmondy prime T . If f = 1, then from mf’< 2f = 2 we obtain mf’ = 1 implying m.= 1 which is a contradiction. If (p,2f) = (2,6), then p = 2 and f = 3. Therefore mf’ < 2f = 6 contradicting m 2 7. Now by proving subcase 2 the Lemma is proved if n 2 9.

w.

Case (ii). n < 9. In this case we must consider decompositions PSL,,(q’) = PSLz(q)A, with n = 5 , 6 , 7 and 8 with q 2 11. If n = 5 , then by [14] the only case is PSLZ(9) =

307

PSLz(4)As = PSLz(5)As which is not the case. If n = 6, then by [4]the only which is not the case. Now we are left with possibility is PSL4(2) = PSJ!&(~)AG n = 7 and 8 and both cases can be treated by order consideration ‘and no possibility arises. The proof is complete now. 0 Lemma 8. If PSP2m(qf)= PSLZ(q)G,, where Gn = An or Sn, q 2 11, n 2 5, then m = 2, q’ = 4, n = 6 and the factorization PSP4(4) = P s L z ( 1 6 ) s ~ is possible. Proof. Our proof is more or less the same as the proof of Lemma 7. We will consider two cases.

Case(i). n 3 9 In this case again by [9] we have 2 m 2 n - 2 and hence n 5 2 m 2 , so mi 2 4. We set d = ( 2 , q - 1 ) and d‘ = 1 or 2 and compute with the order of PSPzm(q’). Using the same inequality as in case (i) of Lemma 7 we get

+

From the other hand 1

1

lPSPzm(q’)I = IPSLZ(q)GnI I ;iq(q2 - l)n! 5 -q(q2 - 1 ) ( 2 m d 111

+ 2)!.

am+i)

2 y q ( q 2 - 1 ) _< q4 for q 2 11. Hence q 2 4 ’ 8 .Again we will assume q = p f and q‘ = p’f‘ and Therefore by combining the two inequalities we will obtain q’+ m(2 m i-1)

consider two subcases.

Subcase (1). p’ # p . Since PSLz(q) is embeded in PSPznL(q)and consequently in PSLz,(q) by [12] we get 2m, 2 hence q 5 2 m d + 1. Now we calculate

9

qfrrA(2~+1)

,m ( 2 m + 1) q ’ + l l m ( 2 m + 1) + 1 = 2 m 2 + m + 4 4

8

Therefore 2 m d

+1 2 q 2 q

’

v

2

2m’$m+4

implying d = 2 and m < 7. In this rn(lrn+l

case m = 6, 5 or 4. If m = 6, then q 5 2m.d+ 1 = 25 or q 5 25. From q 2 q+‘ we get q 2 qm hence q” 5 25 which is a contradiction. The cases m = 5 or 4 yield contradictions as well.

Subcase (2). p’ = p. In this case we obtain f 2 f ‘ m ( ~ m f l ) .If m f ‘ 2 f , then combining the two inequalities yields a contradiction. If mf ‘ < f , then using Zsigmondy primes we will obtain a contradiction. Case (ii). n < 9.

308 By [4] and [15] we may suppose n = 7 or 8. From case (i) we see that without any further assumption we have

If p # p’, then q

I 2 m d + 1 5 4m + 1. We also have

Therefore

1 + m + 1)(2m+ 2)! 5 -(4m + 1)(16m2+ 8m)8!. d

1 -(2m2

Gd‘

Now considering d , d’

I 2. from the above inequality we obtain m < 3. Hence m = 1

or 2 implying q 5 5 or 9 respectively contradicting the fact that q 2 11. Now we will assume p = p’. In this case d’ = 1 or 2 according to p even or odd

respectively, and therefore d = (2,p- 1) = d’. Since PSL2(q) L PSP2m(q’)we must I ’ n 2 ( q / 2 7 n - 1 ) . . . (qr2- 1) where q = pf and q‘ = pf’. Let f # 1 have i q ( q 2 - 1)l;irq or (p: 2 f ) # ( 2 , 6 ) , then there is a Zsigmondy prime r dividing q2 - 1 = p2f - 1 such that r dpk - 1: 1 I k < 2f . Suppose mf’< f . Since ?-/IPSP2m(q’)Itherefore we must have rJq’2t- 1 = p2tf’- 1. But 2t f’ 5 2mf’, so 2tf’ choice of r. Therefore mf‘ 2 f. I f f = 1, then

1

f’rn(2.nL+l 2

GP

I 2f

contradicting the

1 ’ ( 2 m + 2)! 5 2p(p2 - 1)8! I 8!p4,

from which we obtain m 5 1 giving no possibilities. The case p = 2, f = 3 gives no possibility as well. Now we consider the inequality

f(znL-sy

+

Using d = d’ and q2 - 1 < q2 and mf’2 f we obtain p 2 (2m a)! 5 6n!. But p 2 2 and n 5 8!, therefore 2-(2m 2)! < 241920 from which we obtain rn 5 3. The case m = 1 is impossible as P S q ( q ’ ) S PSLz(q), therefore we will consider m = 2 and m = 3. Let m = 3. By subst,ituting in the above inequality we get p2i < $. If n = 7, i then pz < which is not impossible. Therefore n = 8, hence pf < 36. Putting into the original inequality yields &q’$8! I 536(3S2 - 1)8! implying f’ = 1. Now

+

8

f I mf‘ = 3f’ implying f 5 3. So from PsPs(p) = PSLz(pf)G,, G 2 A8 s8, we have obtained pg(p6 - l)(p4 - l)(p2 - 1) 5 pf(p2f - 1)8! <_ 8!pg and

or

309

therefore (p6- l)(p4 - l)(p2 - 1) 5 8!, implying p = 2. But P S P , ( 2 ) has no such decoinposit ion. If m = 2 , then the decomposition PSPd(q’) = PSL2(q)An is impossible by [6]. Therefore we will assume PSP4(qf) = PSLz(q)S,. If n = 5, then by [15] no possibilities arise. If n = 6, then by [4] we obtain PSP4(4) = PSLa(l6)&. Therefore we will consider n = 7 and 8. From PSP4(q‘) = PSL2(q)S,, we obta.in p4”(p4f’ - 1)(p2f’- 1) = pf($f - l)[Sn : PSLz(q)n S ~ I . We set D = [S, : PSLz(q)flSn] and recalling f 5 m.f’ = 2f’ we obtain p4f’-f (p4f’l)(p2f’- 1) = (p2f - 1)D. Since DI7!or 8! and p is a prime divisor of the left hand side of the above equality, hence p4f’-f17! or 8!. If p = 5 or 7, then 4f’- f 5 1 which contradicts f 5 2f’. If p = 3, then 4f’ - f 5 2 and we obtain f‘ = 1, f = 1 or 2. But 7 ]IPSP4(3)(and a contradiction is reached. If p = 2 , then 4f’ - f 5 3 or 6 in the cases n = 7 or 8 respectively. If 4f’- f 5 3, then f’ = 1, f = 1 or 2 , hence q 5 4 a cont.radiction. If 4f’ - f 5 6, then f’ = 1, f = 1 or 2; f’ = 2 , 1 5 f 5 4; f’ = 3, 1 5 f 5 6: and again order consideration yields a contradiction. The proof is complete now.0

Lemma 9.

Decomposition of the unitary group o r the orthogonal groups as the 2 A, or S,, is impossible.

product PSL2(q)Gn,n 2 5 , q 2 11, G,

Proof. Our proof is the mine as the proof of Lemma’s 7 and 8, and here for the unitary and orthogonal groups we describe the proof briefly. We have I,L m-1

- (-1y). . . (q’2 - l ) ,

~Psu,(q’2)1 = $y’+(q‘““

+

+

where d’ = (m,q 1). Again we have qlk 2 (k 2)q’i and the same consideration
1

IPfi2m+1(qf)I = 2 4

tm?

(Q

t2m

- 1 ) . . . ( 4’2 - I>,

where d‘ = (2,q’ - 1). Since IPS22,+l(q’)l = IPSP,(q’)I, hence the same argument as used in the proof of Lemma 8 is applicable. In this case nothing new is obtained. Finally we will consider the orthogonal groups in even dimensions Pfig,(y‘), m 2 4, E = f.We have: IPflLn(q’)I = 214m t m - l ) ( q m

where d‘ = ( 4 , q f m IPQ57n(Qf)I3 QY

+ E).

,+

+

E ) ( q t ~ m - 2-

Since rn >_ 4: we have

1 ) . . . ( q’2 - l),

dm + E

>_ ( 2 m + a)&,

hence

( 2 m + 2 ) ! . Now t.he same argument as in the proof of

310

Lemma 7 is applica.ble. The proof of the Lemma is complete now. 0

Theorem 10. Let G be afinite non-abelian simple group such that G = PSLz(q)B where B is a subgroup of G isomorphic to an alternating or a symmetric group, and q 1 1 1 , then ( a ) G = A, = PSL2(q)An_1,where n is an l-number. (b) G = A, = PSL2(q)An-1 = PSL2(q)Sn-2 where n = q 1. (c) G = All = PSL2(ll)Ag= PSL2(11)Alo= PSLz(q)Sg. ( d ) G = A, = PSL2(q)An-3 = PSL2(q)An-2 = PSL2(q)A,-1 = PSL2(q)Sn-3 = PSL2(q)Sn-2 where n = q 1. ( e ) G = A, = PSL2(q)Sn-3 = PSL2(q)A,,-z = PSL2(q)Sn-2 = PSLz(q)A,-i where n = q 1 and q = 3(m.od 4). (f) G = PSP4(4) = PSL2(16)Ss.

+

+

+

Proof. By classification Theorem for finite simple groups every non-abelian finite simple group is an alternating group, a finite group of Lie type or one of the sporadic simple groups. Now let G be a finite non-abelian simple group such that G = PSLz(q)B where B is an alternat,ing or a symmetric group. If G is an alternating group, then by Lemma 5 all the cases (a)-(e) occur. If G is a sporadic group, then by Leinma 6 the decomposition G = PSL2(q)B,q 2 11, B S,, is impossible. Now we consider groups of Lie type. If G is an exceptional simple group of Lie type, then by [ l l ]table 5 all decompositions like G = A B are know and we see non of them contains decompositions like the one mentioned in the Theorem. Therefore we a.re left with classical groups, PSL,(q), PSPZ,(q), PSV,(q2) and the orthogonal groups. But: in the Lemmas 7, 8 and 9 we have dedt with these groups and the only possibility is case (f) of the Theorern. The proof is complete now.O

Acknowledgement I would like to thank the Bureau of the International affairs and also trhe research deputy of the Universit,y of Tehran for their support which made possible my participation in the ICM satellite Conference in Algebra and related topics, 1417 August 2002, Chinese Universky of Hong Kong.

References [I] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R. Wilson, Atlas of finite groups, Oxford University Press, 1985.

[2] M.R. Darafsheh, Finite groups which are product of L2(8) or L2(7) with a symmetric group, Submitkd.

311

[3] M.R. Darafsheh and G.R. Rezaeezadeh, Factorization of groups involving symmetric and alternating groups, Int. J. Math. and Math. Science, 27:3, 2000, 161-167. [4] M.R. Darafsheh, G.R. Rezaeezadeh, G.L. Walls, Groups which are the product of st3 and a simple group, to appear in Alg.Colloq.

[5] T.R. Gentchev, Factorizations of the sporadic simple groups, Arch. Math. 47, 1986, 97-102. [6] T.R. Gentchev, Factorizations of the groups of Lie type of Lie rank 1 or 2,

Arch. Math. 47, 1986, 493-499. [7] B. Huppert, Endlich gruppen I, Springer-Verlag, 1983. [8] W.M. Kantor, k-homogeneous groups, Math. Z. 124, 1972, 261-265. [9] P. Kleidmcm and M. Liebeck, The subgroup structure of the finite classical groups, Cambridge University Press, 1990.

[lo] 0. Kegel and H. Luneberg, Uber die kleine reidemeister bedingungen, Arch. Math. 14, (Basel) 1963, 7-10.

[ll] M.W. Liebeck, C.E. Praeger and J. Sax1 , The maximal factorizations of the finite simple groups and their automorphism groups, AMS 86, No. 432 1990. [12] V. Landazuri and G.M. Seitz, On the minimal degrees ofprojective representations of the finite Chevalley groups, J Algebra 32(1974), 418-443. [13] D. Livingst;one and A. Wagner, Transitivity of finite permutation groups on unordered sets, Math. Z. 90(1965), 393-403. [14] W.R. Scott, Products of 165-171.

A5

and a finite sample group, J. Algebra 37, 1975,

1151 G.L. Walls, Products of simple groups and symmetric groups, Arch. Math. 58, 1992, 313-321.

HIGH-DENSITY CLOSE-CLOSED LOOP BURST ERROR DETECTING CODES Bal Kishan Dass Department of Mathematics, University of Delhi, Delhi 110 007, India Sapna Jain Department of Mathematics, Miranda House, University of Delhi, Delhi 110 007, India Abstract. In this paper, we study cyclic codes detecting a subclass of close-closed loop bursts viz. high-density close-closed loop bursts. A subclass of CT close-closed loop bursts called CT high-density close-closed loop bursts is also studied. A comparative study of the results obtained in this paper has also been made. 1. Introduction. Burst errors are the most common type of errors that occur in several communication channels. Codes developed to detect and correct such errors have been studied extensively by many authors. The most successful early burst error correcting codes were due to Fire(1959). Fire in his report gave the idea of open and closed loop bursts defined as follows:

Definition 1. An open loop burst of burst of length b is a vector all of whose nonzero components are confined to some b consecutive components, the first and the last of which is nonzero. Definition 2. A closed loop burst of length b is a vector all of whose nonzero components are confined to some b consecutive components, the first and the last of which is nonzero and the number of positions from where the burst can start is n (i.e. it is possible to come back cyclically at the first position after the last position for enumeration of the length of the burst). Definiton 2 of closed loop burst can also be formulated mathematically on-the lines of Campopiano (1962) as follows: I

Definition 2a. Let V n ( q )be the set of all ordered n-tuples with components belonging t o GF(q). Let X = ( a g , a l , ..., u n - l ) be a vector in V n ( q ) .Then

312

313

X is called a closed loop bursts of length b, 2 5 b 5 n, if 3 an i, 0 5 i 5 n - 1 such that ai.aj # 0 where j = (i b - 1) modulo n

+

and

{

= ... - ai-1 = 0 a0 = a1 = ... = ai-1 = aj+l = aj+2 = ... = an-l = O Uj+l = aj+2

ifi>j, ifi < j

There is yet another definition of a burst due to Chien and Tang (1965) which runs as follows: Definition 3. A CT burst of length b is a vector all of whose nonzero components are confined to some b consecutive components, the first of which is nonzero. Based on these definitions, Dass and Jain(2001) defined close-closed loop bursts, open-closed loop bursts; CT close-closed loop bursts, and CT open-closed loop bursts and proved results for close-clsoed loop bursts and CT close-closed loop bursts. The definitions and the results proved by Dass and Jain (2001) are as follows: Definition 4. Let X = ( a o , a l , ..., a n - l ) be a vector in V n ( q ) ai , E GF(q) and let 2 5 b 5 n.Then X is called a close-closed loop bursts of length b, if 3 an i, 1 5 i 5 b - 1 such that a,-b+i.ai-1

# 0, ai

= aa+l =

... = an-b+i-l

= 0.

Definition 5.The class of open loop bursts as considered in definition 1 may be termed as open-closed loop bursts. Definition 6. Let X = ( a o , a l ,..., an-1) be a vector in V n ( q ) ai , E GF(q) and let 2 5 b 5 n.Then X is called a CT close-closed loop bursts of length b, if 3 an i, 1 5 i 5 b - 1 such that an-b+i # 0 ; atleast one of a o , a l , ..., ai-1 is nonzero and ai = ai+l = ai+2 = ... = an-b+i-l = 0. Definition .'7 The class of CT open loop bursts as considered in Defintion 3 may be termed as CT open-closed loop bursts. Theorem A An (n,k ) cyclic code can not detect any close-closed loop burst of length b where 2 5 b 5 k 1.

+

314

Theorem B. The fraction of close-closed loop bursts of length b(2 5 b 5 k 1) that goes undetected to the total number of close-closed loop bursts in any ( n ,k ) cyclic code is

+

Theorem C. An ( n ,k ) cyclic code can not detect any CT close-closed loop burst of length b where 2 5 b 5 k 1.

+

Theorem D. The fraction of CT close-closed loop bursts of length b (2 5 b 5 k 1) that goes undetected to the total number of CT close-closed loop bursts in any ( n ,k) cyclic code is

+

q(k-b+l) -

qb-l((b

b 1

( Q - - 1) - 1)q - b ) ) 1

+

There are of course many situations in which errors occur in the form of bursts but not all digits inside the burst get corrupted so that the weight of the burst is w or more (w 5 length of the burst). Such bursts are known as high-density bursts. High-density bursts with respect to close-closed loop bursts may be termed as high-density close-closed loop bursts defined as follows:

Definition 8. A close-closed loop burst of length b with weight w or more (w 5 b) is called a high-density close-closed loop burst. The development of codes which detect/correct high-density closeclosed loop bursts can economize in the number of parity check digits required, suitably reducing the redundancy of the code or in the other words, suitably increasing the efficiency of transmission. In the second section of this paper, we obtain results similar to Theorems A and B for high-density close-closed loop bursts whereas in the third section, we obtain results similar to Theorems C and D for CT high-density close-closed loop bursts. The last section viz. Section 4 gives a comparison of the results obtained in Section 2 and Section 3. In what follows, an (n,k) cyclic code over GF(q) is taken as an ideal in the algebra of polynomials modulo the polynomial X" - 1.

31 5

2. High-Density Close-Closed Loop Burst Error Detection In this section, we obtain results of Theorem A and B for high-density close-closed loop burst.

Theorem I . An ( n , k ) cyclic code can not detect any high-density closeclosed loop burst of length b with weight w or more (w b) where 2

<

<

b
Proof. There is no deviation in the final conclusion of this theorem from that of Theorem A because the proof is based on the length of the burst giving rise to a polynomial which is of the same degree even when the weight consideration over burst is considered. Hence the proof is ommitted.

Theorem 2. The fraction of high-density close-closed loop bursts of length b(2 5 b k 1) with weight w or more (w 5 b) that goes undetected to the total number of high-density close-closed loop bursts in any ( n ,k ) cyclic code is

< +

q k (1 - q - b + l )

-

r2 - 1 where (w - r1, 1) = max.{w - r1, l}

Proof. Let r ( X ) denote a high-density close-closed loop burst of length b(2 L: b 5 k 1) with weight w cIr more (w 5 b). Let g ( X ) denote a generator polynomial of the code of degree n - k. Now r ( X ) will be of the form r ( X ) = X n-b+i (un-b+i Un-b+i+lX + ... + u ~ - ~ x ~ - ~ - Z ) (a0 -k UlX -t- U2X2 + ... ai-1Xi-'); 1 5 i 5 b - l , U n - b + i , ai-1 # 0 and at least w number of u!,s including Un - b + i , ai-1 are nonzero. = Xn-b+irl(X) + Q ( x ) say , Un-b+i+lX + ... + Un-1Xb-'-2 where r 1 ( X ) = an-b+i and r 2 ( X ) = a0 + a l X + a 2 X 2+ ... + ui-1Xi-l.

+

+

+

+

+

316

Let r1 be the number of nonzero coefficientsin r1 ( X I and 7-2 be the number of nonzero coefficients in r z ( X ) , where (for a fixed value of i), ( 1 5 i 5 b - 1)

15rlLb-i and 1 5 r2 5 i

+

such that w 5 r1 r2 5 b. For any fixed value of i, let us give different values to (i) Let

r1

(.:w

= 1. Then

5 r1

+

r2

r1.

< w - 1 , 1 > 5 7-2 5 b - 1

5 b + w - r1 5 7-2 5 b - r1, also 7-2 2 1

:. < w - r1,l > 5 r2 5 b - q )where (w - r l , 1) = max.{w

- r1,l)

we have then

# of polynomials of type r 1 ( x >= ( q - 1) (b-,l)

( q - 110

b- 1

C

# of polynomials of type r 2 ( X ) =

( q - 1)

r z = ( w - 1,l)

:.# of polynomials of type r ( X ) b-i-1

( 4 - 1)rz-l rz=(w- 1,l)

@)For r1 = 2. we get

< w - 2 , l > 5 7-2 5 b - 2

Then

# of polynomials of type q(x)= ( q - 1 ) ( b - i - 1 ) (4- 1 ) b-2

# of polynomials of type r

2 ( ~= )

C

1'2 =(

(4 - 1 )

(q - 1 y - l

w - 2,l)

:.# of polynomials of type r ( X )

=(

c b-2

b-i-1 1

)(4-1)3

1'2 =(w

-2,l)

Continuing the computation for various values of have r1=

b-i

* <w-b+i,l>

5r2

Ii

r1 =

3,4,..., we finally

317

:.# of polynomials of type r ( X )

So, for a fixed value of i,

# of polynomials of type r ( X )

Summing over i, we get Total # of polynomials of type r ( X )

where (w1 - T I , 1) = max.{wl - T I ,1) Again r ( X ) will go undetected if g(X) divides r ( X )

+ r ( X ) = g(X)Q(X) for some polynomials Q(X) +

Xn-b+i ri(X) + r 2 ( X ) = g(X)Q(X)

Now,

# of polynomials of type Q(X) = q'(1-

q P b f 1 (ref431)

318

:. Ratio of high-density close-closed loop bursts that goes undetected to the total number of high-density close-closed loop bursts is qk

r1 - 1

(1- q - b + l )

r2 =(w-rl

1)

,l)

where (w - T I , 1) = max.{w

-

r1,l)

Hence the proof.

0

Special Case. For b = w = 2 , the ratio obtained in the preceding theorem reduces to the ratio given in Theorem B for b = 2 and the ratio in each k-l case becomes q(q-1). ~

3. CT High-Density Close-Closed Loop Burst Error Detection In this section we extend the studies made in Section 2 for CT highdensity close-closed loop bursts. Firstly, we obtain the following result, the proof of which is omitted.

Theorem 3. An (n,k ) cyclic code can not detect any CT high-density close-closed loop burst of length b(2 5 b 5 k 1) with weight w or more (w 5 b).

+

We now prove the following result.

Theorem 4. The fraction of CT high-density close-closed loop bursts of length b(2 5 b 5 k + l ) with weight w or more (w 5 b) that goes undetected to the total number of CT high-density close-closed loop bursts in any (n,k ) cyclic code is

-

-

Qk(1 - 4-bfl) b-rl

7-2 = ( w -r1,1)

where (w - T I , 1) = max.{w

-TI,

1)

319

Proof. Let r ( X ) denote a CT high-dnesity close-closed loop burst of length b(2 5 b 5 Ic 1) with weight w or more (w 5 b). Let g ( X ) denote the geneartor polynomial of the code of degree n - k.

+

Now r ( X ) will be of the form

+ an-b+i+lX + ... + an-1Xb-'-2 1 + (a0 + a l X + a2X2 + ... + ai-1XZ-l); 1 5 i 5 b - 1,an-b+i # o and

r ( X ) = Xn-bfi(an-b+i

at least w number of ais including an-b+i are nonzero and atleast one amongst ao, a l , ..., ai-1 is nonzero.

+ r2 ( X ), say + Un-b+i+lX + ... + and r 2 ( X ) = a0 + a l X + a2X2 + ... + a 2.- 1 X i - l . - xn-b+i

TI ( X )

where r 1 ( X ) = an-b+i

Let r1 be the number of nonzero coefficients in r1 ( X ) and 7-2 be the number of nonzero coefficients in r2 ( X ) , where (for a fixed value of i), (1 5 i

15r

l 5

and 1 5

5 b - 1)

b-i

5i such that w 5 r1 + r2 5 b. r2

For any fixed value of i, let us give different values to (i) Let

rl =

1. Then

rl.

< w - 1,1> 5 r2 5 b - 1

# of polynomials of type r

1 ( ~=) ( q -

1)

('-:-')( q - 110

c () b- 1

# of polynomials of type r 2 ( ~ =>

rz=(w-1,l)

(q - 1)"

7-2

:. # of polynomials of type r ( X ) b- 1

=

("-;-I)

(q - 1)

c

?-2=(w-

(ii)For r1 = 2. we get

1,l)

< w - 2 , l > 5 r2 5 b - 2

Then

# of polynomials of type r l ( X >= ( q - 1) ( b - t - ' )

( q - 1)

320 b-2

:. # of polynomials of type

c ()

r2(x) =

2

rz=(w-2,1)

7-2

( q - 1)"

# of polynomials of type r(X) =

(

b-i-1 1

) (4r z = ( w -2,1)

Continuing the computation for various values of r1 = 3,4,..., we finally have r1= b - i

+ <w -b + i , l >

5 7-2 5 i

and # of polynomials of type rl(x>= ( q - 1)

# of polynomials of type

(!I:-:)

( 4 - 1)b-i-l

=

r2(X)

r z =(w -b+i, 1 )

:. # of polynomials of type r(X) i

= (b-2-1) b-i-1

( 4 - 1)b-i r z = ( w -b+i, 1)

So, for a fixed value of i, :. # of polynomials of type r(X)

Summing over i, we get Total # of polynomials of type r(X) b-rl

where (w - r1,l) = max.{w

-

r1,1>

32 1

Again, r ( X ) will go undetected if g ( X ) divides r ( X )

+ r ( X ) = g ( X ) Q ( X )for some polynomials Q ( X ) Xn-b+i

r i ( X )+ r 2 ( X ) = g ( X ) Q ( X ) Now, # of polynomials of type Q ( X ) = q'(1 - q P b + l ) (refer[3]) :. Ratio of CT high-density close-closed loop bursts that goes undetected to the total number of CT high-density close-closed loop bursts is + ,

Hence the proof.

0

Special Case. For b = w = 2, the ratio obtained in the preceding theorem reduces to the ratio given in Theorem D for b = 2 and the ratio in each q"-l case becomes (q-1)

*

4. Comparative Study

In this section, we present the comparison of the results obtained in Section 2 and Section 3 viz. Theorem 2 and Theorem 4. The comparison has been presented in the form of a table by taking specific values of b; b and w in the binary case. For b = w = 2, both definitions viz. of highdensity close-closed loop burst and of CT high-density close-closed loop burst coincide. Therefore, we start comparing the results for b = 3, and onwards.

322

Table CT High-Density CloseClosed Loop Bursts (Theorem 4)

High-Density Close-Closed Loop Bursts (Theorem 2)

Iq = 2,b = 3 =+-w = 2 , 3 (

k=2 k=3 k=4 k=5 k=6

0.75 1.50 3.30 6.00 12.00 Iw = 31

k=2 k=3 k=4 k=5 k=6

1.50 3.00 6.00 12.00 24.00

1.50 3.00 6.00 12.00 24.00 19 = 2,b = 4

+ w = 2.3.41

Iw = 21

k=3 k=4 k=5 k=6 k=7

0.63 1.27 2.54 5.09 10.18

IW k=3 k=4 k=5 k=6 k=7

0.77 1.55 3.11 6.22 12.44

= 31

0.63 1.27 2.54 5.09 10.18 (Table contd.)

323

k=3 k=4 k=5 k=6 k=7

2.33 4.66 9.33 18.66 37.33

2.33 4.66 9.33 18.66 37.33

0.93 1.87 3.75 7.50 15.00

0.78 1.57 3.15 6.31 12.63

k=4 k=5 k=6 k=7 k=8

k=4 k=5 k=6 k=7 k=8

k=4 k=5 k=6 k=7 k=8

(W =

k=4 k=5 k=6 k=7 k=8

3.75 7.50 15.00 30.00 60.00

51 3.75 7.50 15.00 30.00 60.00

Note: The fractions have been calculated upto 2 decimal places.

324

Acknowledgement. The second author wishes to thank University Grants Commission for providing grant (vide ref. No. F 13-3/99 (SR-I)) under Manor Research Project to carry out this research work. References 1. C.N. Campopiano (1962), Bounds on Burst Error Correcting Codes, IRE Trans., IT-8, pp. 257-259. 2. R.T. Chien and D.T. Tang(1965), On Definitions of a Burst, IBM J. Res. & Develop., July, pp.292-293. 3. B.K. Dass and Sapna Jain (2001), On a Class of Closed Loop Burst Error Detecting Codes, International Journal of Nonlinear Sciences and Numerical Simulations, Vol 2., pp. 305-306. 4. P.Fire( 1959), A Class of Multiple-Error-Correcting Binary Codes for Non-Independent Errors, Sylvania Report RSL-E-2,SylvaniaReLon.Sys. Lab., Mountain View, California. 5. W.W.Peterson and E.J.Weldon,Jr.(1972), Error-Correcting Codes, Cambridge, Mass. The M.I.T. Press.

M-solid Pseudovarieties and Galois Connections K. Denecke, B. Pibaljommee Abstract

The concept of a solid pseudovariety, indroduced in [Gra-P-V;97],can be easily generalized to M-solid pseudovarieties. Using identity filters we will give a characterization of M-solid pseudovarieties by Galois connections. Finally we prove that the lattice of all M-solid pseudovarieties forms a complete sublattice of the lattice of all pseudovarieties. Key words: Pseudovariety, identity filter, M-solid pseudovariety AMS Subject Classification: 08B15, 08A30

1

Introduction

A pseudovariety of type T is a class of finite algebras of type T closed under taking of subalgebras, homomorphic images and finite direct products. ClearlL the subclass Algri,(T) A/g(T)of all algebras of type T = (2) is a pseudovariety. If V is an arbitrary variety of type T , then its finite part, i.e. the class V’i, of all finite algebras contained in V , is a pseudovariety. Such pseudovarieties will be called equational pseudovarieties. Not every pseudovariety is equational, for example, the class of all finite groups regarded as algebras of type T is a pseudovariety and the class of all finite nilpotent semigroups, i.e. satisfying the identities 5 1 . . .z,x,+l w z,+1z1.. .x, M z1.. .x, for some n, is also a pseudovariety. In both cases the smallest equational pseudovariety containing them, consists of all finite semigroups. This shows that we cannot always define pseudovarieties by identities. A logical description of pseudovarieties was given by Reiterman and Banashewski ([Rei;82],[Ban;83]) using the concept of a pseudoidentity. We will not follow this approach. Instead of this we will use the description which was given by Eilenberg and Schiitzenberger ([Eil-S;76]) and which uses the concept of an identity filter. ; n) is the power set lattice of A , then a filter A is a sublattice If A is a set and if ( P ( A ) U, of ( P ( A ) U, ; n) with the additional property that for any C E A and C’ E P ( A ) we have C U C’ E A. This can be equivalently characterized by the following two conditions:

325

326 (i) A is closed under finite intersections, (ii) A is closed under supersets. If 3 C P ( A ) ,then by (F) we denote the filter generated by 3 and if .F is one-element, 3 = {C}, we speak of the principal filter generated by C. In this case we will skip one pair of brackets and will simply write (C). We remember of the following fact:

Proposition 1.1 A is the filter generated by the set F,i.e. A = ( F ) ,iff A is a filter and for e v e y C E A there exists an integer n 2 2 and sets CO, . . . ,CnA1E 7 such that C>C~~...~C,-,.

w

An easy, but useful consequence is the following Lemma:

Lemma 1.2 If A = (3)is the filter generated by the subset 3 C P ( A ) and i f 3 is closed under finite intersections, then for every C E A there exists a set C' E 3 such that C C'. w

>

Definition 1.3 Let (fi)iGI be a sequence of operation symbols of type 7 where f i is ni ary and let X be a set of variables. By W T ( X )we denote the set of all terms of type T . An identity filter A of type 7 is a filter in the power set lattice P ( W T ( X ) 2of) W T ( X ) 2 , By I F ( T ) we denote the class of all identity filters of type T . Definition 1.4 We say that an algebra A := ( A ;(j))iE1)of type 7 ultimately satisfies an identity filter A E I F ( 7 ) if there exists a set C E A such that A satisfies each equation from C as identity, i.e. A a:= X E A ( A ~ c ) . U.S.

Using this concept of ultimately satisfaction, Schiitzenberger characterized pseudovarieties as follows:

Proposition 1.5 (see [Alm;94]) For each identity filter A, the class of all finite algebras which ultimately satisfy A, is a pseudovariety. Conversely, for any pseudovariety V there exists an identity filter A such that V consists of exactly all finite algebras which w ultimately satisfy A.

327

We remark that for the case of 7 being a finite type, an algebra A ultimately satisfies the identity filter A iff there is an identity sequence \I, such that A ultimately satisfies A if and only if it satisfies all but finitely many identities of \I,, In this case we say that A ultimately satisfies \I,. For more background on the definition of pseudovarieties via identity filters see [Alm;94]. We will now give another characterization of pseudovarieties using the Galois connection which is induced by the relation

I= C Algjin(7) x I F ( 7 ) . U.S.

For a class K

C A1gfin(7) and for a set F C I F ( 7 ) we define PMod F

:= { A E

Algfin(7)I VA E F ( A

I=

A)}

u.9.

and

Filt K := {A E I F ( 7 ) I VA E K ( A

A)}. U.S.

Clearly, the pair (PMod,Filt) is a Galois connection between Algfi,(T) and I F ( 7 ) , i.e. it satisfies for all 3 1 , 3 2 C I F ( 7 ) and for all K1, Kz C Algfin(7)the conditions

Fi G Fz + K1 C Kz E 3 K

*

PMOdFi 2 PMod F2 Filt K1 _> Filt Kz FiltPMod F PModFilt K .

As usual for a Galois connection, we have two closure operators PModFilt and FiltPMod and their sets of fix points, i.e. the sets

{F C I F ( 7 ) I FiltPMod F = F} and {K C A l g f i n ( 7 ) I PModFilt K

=K

)

form complete sublattices of the power set lattice of I F ( 7 ) and of the power set lattice of Algjin(7),respectively.

By definition, PMod 3 is the intersection of all PMod A for A E PMod A.

PMod 3 = A€F

F , i.e.

328

It is well-known that the class of all pseudovarieties of the same type T forms a complete lattice with the intersection as meet operation. Therefore PMod 3 is a pseudovariety. Then by Proposition 1.5 there must be a filter A’ such that PMod 3 = PMod A’ and one can ask how t o construct such filter A‘. By definition of PMod 3,for every A E 3 there is a set C i of equations such that for every A E PMod F we have A Xi.Then U Xi), i.e. the principal filter generated it is not difficult t o show that A’ = ( AE3,dEPModF

U

satisfies PMod 3 = PMod A‘.

by A € 3 , d € P M c d 3 By P ( Twe ) denote the complete lattice of all pseudovarieties of type

T.

Ultimately M-hypersatisfaction

2

To define M-solid pseudovarieties we need the concept of a hypersubstitution. Hypersubstitutions of type T are mappings from the set of all operation symbols of type T into the set of all terms of type T which preserve the arities. This means, to each n,-ary operation symbol of type T we assign an ni-ary term from W T ( X ) .Hypersubstitutions can be extended t o mappings ci : W T ( X )+ W T ( X )which are defined on the set W T ( X ) of all terms of type 7 by the followings inductive definition:

(i) +[x] := x for every variable x E X . (ii)

ci[fi(tl,.

. . ,tn,)]:=

a(fi)(ci[tl], . . . ,ci[tn,])for composite terms fi(tl,.. . , tn,).

Let H y p ( ~ be ) the set of all hypersubstitutions of type T . On the set H y p ( ~ we ) may define a binary operation oh by 01 oh 02 := $1 o 02. Let (Tid be the hypersubstitution which maps each operation symbol fi to the term fi(x1, ..., xn,).Then ( H y p ( ~ oh, ) ; uid) is a monoid. Hypersubstitutions can be applied to equations s x t of type T . This gives new equations of the form 3[s] x ci[t].For arbitrary sets C C W 7 ( X ) 2C , # 0 and for a submonoid M C H y p ( ~we ) define

&[C] :=

u u 6[s] ci[t]. M

s=tEC u € M

It is not difficult t o prove that xg : P(WT(X)’) + P ( W T ( X ) 2 )is a closure operator which is called completely additive because of its definition as union of singleton sets. To apply hypersubstitutions to algebras we consider so-called derived algebras. If

329

ft)iEl)

A = ( A ;( is an algebra of type T , then u(A) = ( A ;(a(f i ) A ) i E I ) is called algebra derived from A by u. Here u(fi)d is the term operation induced on the algebra A by the term a(f i ) . For the class K A l g ( T ) and for a submonoid M C H y p ( ~ we ) define

x$

is also a completely additive closure operator. Between both operators there is a close interconnection given by

x$

and

xg

* x$[A] s t. Because of this property we call (x&, xE)a conjugate pair of operators. For more backA

t=

&[s

M

t]

M

ground on conjugate pairs of additive closure operators see [Den-W;OO]and [Den-W;02]. - solid if x$[V] V. To find the right definition of an M-solid pseudovariety, we request that the finite part of an M-solid variety should be an M-solid pseudovariety. This is satisfied if we define:

A variety V is called M

Definition 2.1 A pseudovariety V of algebras of type T is called M - solid if x$[V] (i.e. if we have x$[V] = V).

V

Our aim is to characterize M-solid pseudovarieties by a Galois connection. We start with 5 Algfin(T) x I F ( 7 ) which was defined first in [Gra-P-V;97]. the following relation

t==

u.Mh. s.

Definition 2.2 Let A E I F ( T ) be an identity filter of type T and let A E A l g f t n ( T ) be an algebra of type T . Then we say that A ultimately M-hypersatisfies A if for every hypersubstitution u in M there exists a set C, of equations in A such that A satisfies .(Cu):

A

+

u.Mh.s.

A :e V a E C 3Cu E A ( A

+ .(En)).

The characterization of solid pseudovarieties given in [Gra-P-V;97] can be easily generalized to M - solid pseudovarieties.

Theorem 2.3 For a class V of finite algebras the following conditions are equivalent:

330 (a) V is an M

-

solid pseudovariety.

(ii) V is a pseudovariety and for each identity filter A E I F ( r ) , V ultimately M hypersatisfies A whenever V ultimately satisfies A, i.e.

V

t=

A*V+A.

u.Mh.s.

u.3.

(iii) There exists an identity filter A E I F ( r ) such that V consists precisely of all finite algebras which ultimately M-hypersatisfy A, in this case we write, V = HMPMod A.

3

A Second Galois Connection

For a class K

c A1gfin(r)and for a set 3 HMFilt K

:=

I F ( r ) we define

{ A € IF(r) I V A E K (A

A)} u.Mh.3.

Then the pair (HMPMod,HMFilt) forms a Galois connection. It follows that the products HMPModHMFilt and HMFiltHMPMod are closure operators and that the sets

of their fixed points form complete lattices which are dually isomorphic Proposition 3.1 Let 3 C I F ( r ) . Then every finite class of algebras of the form HMPMod F is an M-solid pseudovariety and conversely, for every M-solid pseudovariety V there i s a set 7 C I F ( r ) offilters such that V = HMPMod F.

33 1

Proof. By Proposition 1.5 for every M-solid pseudovariety there is an identity filter A E I F ( T ) such that V = HMPMod A. Therefore with F := {A} we have our set of filters. For the converse we apply also Proposition 1.5 and show that the intersection of two M-solid pseudovarieties is an M-solid pseudovariety. Indeed if Vl,Vz are M-solid pseudovarieties, then x$(V,) = V,, i = 1,2. From K n V, & V,, i = 1 , 2 , by the monotonicity of the operator x$ we get x$[Vl n V,] 2 x$[V,] & V,, i = 1 , 2 , and therefore x$[& nVz] 2 & nvz.The inclusion & nV, C x$[Vl nV,] is clear and therefore x$[K n VZ] = V, n & and this means that V, n V, is M-solid. (Here we used that by Definition 2.1 every M-solid pseudovariety is a pseudovariety and that the intersection of two pseudovarieties gives a pseudovariety too). Clearly, this can be generalized to the intersection of arbitrary families of M-solid pseudovarieties. By definition of HMPMod F we have HMPModF = n HMPMod A. A€?=

Now using Theorem 2.3 we know that each class HMPMod A is an A4-solid pseudovariety and also their intersection is an M-solid pseudovariety. Therefore HMPMod F is an M-solid pseudovariety and then again by Theorem 2.3 there is an identity filter A' such rn that HMPModF = HMPMod A'. This proof guarantees the existence of a filter A' with HMPMod 7 = HMPMod A' but does not construct this filter. Such a filter A' can be constructed in the following way: A and this means that for If A E HMPMod F then for every A E 7 we have A u.Mh.8.

every A E F and for every u E A4 there exists a set C',: For every u E M we form the principal filter J, := (

E A such that A

U

CT(E:>~).

U Ct,d) generated by

AEHMPMod 3 A E 3

U

IJ C:id.

Using these filters we form the filter A' := ( U J,) generated by

AEHMPMod 3 A E 3

u€M

the union of all J,. Then it is not difficult to show that HMPModF = HMPMod A'. As a corollary we have:

Corollary 3.2 A class V of finite algebras v = HMPModHMFilt v .

of

type T forms an M-solid pseudovariety iff

Proof. If v = HMPModHMFiltV then F := HMFaltv is a set of identity filters of type T and then by Proposition 3.1 = HMPModHMFiltV is an M-solid pseudovariety. If conversely V is an M-solid pseudovariety of type T , then again by Proposition 3.1, there exists an identity filter A E I F ( T ) such that V = HMPMod A. But then HMPModHMFiltV = HMPModHMFdtHMPMod A = HMPMod A = by a property of Galois connections. rn

v

v

As fix points of the closure operator HMPModHMFilt the M-solid pseudovarieties form a complete lattice SL'(T). If M = H y p ( ~ )we , simply write S"(T).

332 If V is a certain pseudovariety, then by Lps(V)we denote the lattice of all subpseudovarieties of V and SGs(V) := Lps(V)nSGS(T) is the lattice of all M-solid subpseudovarieties of V . Our question is whether for every pseudovariety V the lattice SLS(V)is a complete sublattice of the lattice Lps(V). Using filters we get one more characterization of M-solid pseudovarieties:

Proposition 3.3 Let V be a class of finite algebras of type tions are equivalent:

T.

Then the following condi-

(i) V is an M-solid pseudovariety. (ii) Filt V Proof.

= HMFilt

V.

(i)I (ii).Assume that V is an M-solid pseudovariety. Then we have:

VA E V (d

A E Filt V

A)

V d E V (A

A)

A

E HMFilt

V

u.Mh.s.

U .S.

by Theorem 2 . 3 . (ii)I ( i ) .Because of Filt V = HMFilt V , for every filter A E I F ( T ) and for every algebra A E V we have

A

A U A € F i ~ ~ V U A € ~ M F i l t V U /=A A u.Mh.s.

U.S.

and therefore for every filter A E I F ( T ) we have

V b A e V U.S.

A. u.Mh.s.

Application of Theorem 2.3 shows that V is an M-solid pseudovariety.

4

Galois-closed subrelations

Now we know that the set of all M-solid pseudovarieties of type T forms a complete lattice which is contained in the complete lattice of all pseudovarieties of type T . Our next aim is t o show that the lattice of all M-solid pseudovarieties forms even a complete sublattice of the set of all pseudovarieties of the same type. For the proof we will apply the characterization of complete sublattices via so-called Galois closed subrelations.

Definition 4.1 Let R and R‘ be relations between sets A and B . Let ( p , L ) and ( p ’ ,L’) be the Galois connections between A and B induced by R and R’, respectively. The relation R’ is called a Galois-closed subrelation of R if,

333 (i) R'C R, and (ii) V T

E A , V S C B ( p ' ( T )= S

A L'(S)= T )

+ ( p ( T )= S

A L(S)= T ) .

From this definition we can prove the following characterization of complete sublattices of a complete lattice.

Theorem 4.2 Let R connection (p,1 ) . Let (i)

CA x

B be a relation between sets A and B , with induced Galois be the corresponding lattice of closed subsets of A .

A x B is a Galois-closed subrelation of R , then the class UE, := l-lL!p,is a complete sublattice of El@.

If R'

(ii) If U is a complete sublattice of X l M ,then the relation

Ru

:= U { T x

p(T)IT E U }

is a Galois-closed subrelation of R. (iii) For any Galois-closed subrelation R' of R and any complete sublattice U of have U b = ZA and &, = R'.

n,, we

For the proof and more background see [Den-W;02].

Now we apply Theorem 4.2 to our situation and set A := A l g f i n ( T ) , B

, R':=

R:= U.S.

:= I F ( r ) ,

, u.Mh.5.

( p ,L ) := ( F i l t ,P M o d ) , (p', L ' ) := (HMFilt,HMPMod), zLp := Lps(T),zh!@r :=

sLs(T).

Proposition 4.3 For every monoid

M

of hypersubstitutions the relation u.Mh.8.

A l g f i n ( T ) x I F ( 7 ) is a Galois closed subrelation of the relation

C

C Algfin(T)x I F ( T ) . U.S.

Proof.

For a finite algebra d E Algfi,(T) and for an identity filter A E I F ( T )we have:

I=

s.Mh.s.

k

u.Mh.s.

A

334 Further we get

+

A

a +

V ~ E M ~ (CA E + ~~( c ) )

u.Mh.3.

+ +

A

+ C E A with o

= aid

A + A U.S.

* Ma) E

k U.S.

.

&

Thisshows that u.Mh.s.

u.3.

Let 3 C I F ( I - )be a set of identity filters of type I- and let K C Algfin(I-)be a class of finite algebras of type I- such that HMFih! K = F and HMPMod F = K . The last equation means by Proposition 3.1 that K is an M-solid pseudovariety and then by Proposition 3.3 we have Filt K = HMFilt K = 3 and thus Filt K = 3. It is left to show that PMod 7 = K . We have PModF = PModHMFilt K = PModFilt K if we use again that K = HMPModF is an M-solid pseudovariety and therefore HMFilt K = Filt K . But every M-solid pseudovariety is a pseudovariety by Theorem 2.3 and by section 1 we is a Galois closed subrelation have PMod 3 = PModFilt K = K . This show that

+

u.Mh.s.

of

k

H

U.S

Now we apply Theorem 4.2 and obtain:

Theorem 4.4 For every monoid M E H y p ( ~ -of) hypersubstitutions the lattice S&'(T) is a complete sublattice of the lattice L"(I-) of all pseudovarieties of type I-. H It is very natural t o ask for the relationship between the complete lattices S;;(I-) and S&:(I-)when MI and Mz are both submonoids of H y p ( ~ -and ) MIis a submonoid of Mz. Assume that V E S ~ : ( I -then ) , x&[V]= V and this means that for every 'a E Mz and every A E V the derived algebra 'a(A)belongs also to V . Because of A41 C M2 this is also true for every hypersubstitution from MIand thus x&[V]= V and then V E SGf(r). This means that the set S&:(r) is a subset of S;~(I-). To show that S;:(I-) forms even a complete sublattice of the lattice S ~ : ( I -we ) prove that forms a Galois-closed subrelation of .

+

u.Mzh.s.

u.Mlh.3.

Proposition 4.5 For any two submonoids MI,Mz of the monoid H y p ( r ) of all hypersubstitutions of type I- with MIC M2 the relation is a Galois-closed subrelation

+

u.Mzh.s.

.

of the relation u.Mi h.s.

335 Proof.

For every finite algebra A of type T and for every identity filter A E I F ( T ) we

have:

k

(A,A) E

A

k

+A

u.Mzh.s.

A. Further

u.Mz h.s.

A =+ V O E M ~ ~ C E ( AA~ u ( C ) ) u.Mzh.8.

*

* This shows

VU E Mi 3C E A ( A k u(C))

I=

(A,A)E

u.Mlh.8.

k

k

u.Mzh.s.

u.Mih.8.

.

Let 3 I F ( T ) and let K C_ A l g f i , ( T ) such that H M z F a l t K = 3 and H M z P M o d 3 = K we want t o show that H M , P M o d F = K and H ~ ~ F iKl t= F. If A E H M a F i l t K = 3 then for all A in K we have A k A and then also for all A u.Mz h.3.

in K there follows A

k u.M1 h.s.

A, i.e. A

E HM,Fdt

K = 3.

This show that 3 = H M a F i l t K E H M , F i l t K. Further, we have H M I F i l t K C Filt K = H M a F i l t K = 3 by Proposition 3.3 since HMzPMod3 = K is Mz-solid by Proposition 3.1. Altogether we have H M , F d t K = 3. Clearly, H M , P M o d 3 C PMod 3 = P M o d H M 2 F d t K = PModFilt K = K since K is an M2-solid pseudovariety and therefore a pseudovariety. It is left to show that K C H M ~ P M3. o~ A and then for all If A E H M z P M o d F then for all A in 3 there holds A u.Mzh.s.

A E 3 we have A HM,

t= %.Mih.s.

A, thus A

EHM,PMod

PMod 3 and altogether we have K

3.This shows K

= H M , PMod

=HM2PMod

3.This shows that

k

3 C_ is a

u.Mzh.s.

Galois closed subrelation of

k u.Mih.s

Corollary 4.6 For any two submonoids M l , M 2 of the monoid H y p ( ~ of ) all hypersubstitutions of type T with M I C_ M 2 the lattice of all M 2 - s o l i d pseudovarieties is a complete sublattice of the lattice of all Ml -solid pseudovarieties. Let H y p ( ~be ) the monoid of all hypersubstitutions of type T and let LPs(7-)be the lattice of all pseudovarieties of type T . Then we define a relation R H y p ( ~x) LPs(7)by (c7,

V ) E R :e.(V)

c v,

336 i.e. all algebras derived by u from algebras in V belong t o V . This relation defines a Galois-connection and for any set S of hypersubstitutions and for any set U of pseudovarieties of type r we have

c

E Lps(r)and Vu E S ( a ( V ) V ) } LR(U) := .I.{ E H y p ( r ) and VW E U ( u ( W ) W ) } . The sets ~~(24) are submonoids of H y p ( ~ and ) the sets p R ( S ) are complete sublattices. Therefore we have a Galois-connection between submonoids of Hyp(.r) and complete sublattices of Lps(7-).

~ R ( S := ) {VlV

5

Ultimately strong M-hypersatisfaction

In [Gra-P-V;97] a second concept of M-hypersatisfaction was defined:

Definition 5.1 Let A E I F ( T ) be an identity filter of type T , let M C H y p ( r ) be a monoid of hypersubstitutions of type 7 , and let A A l g f i n ( 7 ) be an algebra of type T , then A ultimately strongly M-hypersatisfies A if there exists a set C of equations in A such that for every u E M the algebra A satisfies u(C),i.e.

k

A

A

:H 3C E

+

AVu E M (d .(C)).

u.s.Mh.3.

Using the operator

x;

introduced in section 2 we can also write

A

A :*3C E A ( A

x;(C)).

u.s.Mh.h.s.

Clearly, if A ultimately strongly M-hypersatisfies A then A also ultimately Mhypersatisfies A, i.e. the relation k is a subrelation of the relation . For u.s.Mh.s.

u.Mh.s.

finite algebras of finite type in [Grac-P-V,97] was proved:

Proposition 5.2 ([Grac-P-V,97]) Let A = ( A ;(ft)iEl) be a finite algebra of finite type (a,.. I is finite) and let M E H y p ( r ) be a monoid of hypersubstitutions. T h e n A ultimately strongly M-hypersatisfies a n identityfilter A E I F ( T )iff A ultimately satisfies A. The relation

k

Algfi,(T) x I F ( T )defines a Galois connection. For a class K

u.s.Mh.s.

Algfin(r)and for a set

F

c I F ( T ) we define

c

337 S H M P M o d 3 := { A I A E Algfi,(T) and V A E 3 ( A

k

u.s.Mh.s.

A))

and

SHMFzltK := { A \ A E I F ( 7 ) and VA E K ( A u.s.Mh.s.

A)}.

Then the pair (SHMPMod,SHMFilt) is a Galois connection and the products SHMPModSHMFilt, SHMFiltSHMPMod are closure operators. If we consider only fiand nite algebras of finite type, then by Proposition 5.1 the two relations u.s.Mh.s.

u.Mh.5.

are equal and this means,

K

= SHMPModSHMFilt

K

H

K

= HMPMOdHMFdt

K

and

F = S H M F i l t S H M P M o dF H F = HMFiEtHMPMod3 for K a class of finite algebras of finite type, 3 C I F ( 7 ) , and M H y p ( 7 ) . Therefore, the fix points with respect to the closure operator SHMPModSHMFilt are exactly the M-solid pseudovarieties of finite type. We want now t o consider the general case and define at first an operator : I F ( T )4 P(P(W,(X)')) by A H {&(C) I C E A}. In general, &(A) is not a filter, but we can extend our definition of ultimately satisfaction to sets of the form x Z ( A ) . The operator x$ is monotone and idempotent, but in general not extensive.

xg

Lemma 5.3 Let A E I F ( 7 ) , then SHMPMod A is a psevdovariety.

Proof.

By definition, SHMPMod A

=

{ A E Algfin(7) I A

A } .By the remark u.5.Mh.s.

following Proposition 5.2 we have

Here x g ( A ) is defined t o be the set x g ( A ) = { x g ( C ) I C E A } . Since x&(A) is not a filter, we form A' := (x$(A)) = ({&(C) 1 C E A}). To show that SHMPMod A = PModA' we assume that A E SHMPModA. Then

AESHMPModA

j

A

k u.s.Mh.s.

A

338 This shows that SHMPMod A

C PModA’.

For the converse inclusion assume that A E PModA’. Then A E PModA‘ + A k A‘= (&(A)) u.9.

+

3 2 E A’(A

C).

Since &(A) is closed under finite intersections by Lemma 1.2 there exists a set C E A such that x E ( C ) C’. Therefore we have A k C ’ =+ A F & ( C ) . + A A.

+

u.s.Mh.s

This shows that SHMPhfod A 2 PModA‘. Altogether, SHMPhfod(A) is a pseudovariety.

Then in a similar way as in Proposition 3.1 we can prove Corollary 5.4 Let F

w

I F ( T ) , then SHMPModF is a pseudovariety.

Proposition 5.5 Let A E I F ( r ) and M be a monoid of hypersubstitutions Then SHMPModA is an M-solid pseudovariety.

of

type r .

Proof. We have to find an identity filter A‘ E I F ( r ) such that SHMPModA = HMPModA‘. By definition we have SHMPModA = { A E Algfin(r)lA k A } and u.s.Mh.s.

A

k

u s . Mh.s.

A H 3C E A ( A t= & ( C ) ) .

Let Jd be defined by Jd = {C E AlA k x g ( C ) } . We form the filter A’ generated by U Jd, i.e. A‘ = ( U J d ) . Now we have dESHMPModA

A E SHMPModA

+

d € S H M PModA

3C E A ( A k & ( C ) ) + VCTE M3C E A ( A k CT(C)) =+ A A’since C E A’

+

u.Mh.3.

A E HMPModA’. Therefore, SHMPModA C HMPModA’. Conversely, we assume that A E HMPModA’. Then for every CT E M there exists a set C’ E A’ such that A k o(C’).Now we show that the set U Jd is closed under finite intersections. Indeed, let &,& E

d € S H M PModA

U

AESHMPModA

Jd. Then there exist algebras

339 E SHMPModA such that for all u E M we have A1 .(El) and But then also b u(C1n Cz) for all u E M and this means C1 n Cz E

A2

b u(C,).

U

AESHMPModA

Now by Lemma 1.2 there is a set C E

Jd.

Ja such that C’ 2 C. Therefore,

IJ

dESHMPModA

x g ( C ) and A E SHMPModA and thus there exists a set C E A such that A HMPModA’ C SHMPModA. Altogether, we have HMPModA’ = SHMPModA and obtain that SHMPModA is an M-solid pseudovariety. Clearly, if 3

IF(T),then S H M P M o d F =

n

SHMPModA. Since the intersection

A€F

of M-solid pseudovarieties is M-solid, we obtain that SHMPModF is an M-solid pseudovariety. By Proposition 5.1, for the case that the type is finite, we have one more possibility to characterize M-solid pseudovarieties. In this case the following conditions are equivalent:

(i) V is an M-solid pseudovariety (ii) SHMPkfodsHMFiltv = v.

In the arbitrary case by Theorem 5.3 we have the implication (ii)+(i). We introduce the following denotation := {vlv is a class of finite algebras and SHMPModSHMFzltV = v for a monoid M of hypersubstitutions.}

sCPs(‘r)

Clearly, SCPs(7)is a complete lattice and by Theorem 5.3 we have SCp”(~) C S;’(T).

6

Proper Hypersubstitutions

If we want to check whether a pseudovariety is M-solid, we have not to consider all hypersubstitutions from M .

Definition 6.1 Let V be a pseudovariety of type T . A hypersubstitution u of type called a V-proper hypersubstitution if for all A E IF(T)we have A E FiZtV

(.(A))

E FiltV.

By P(V) we denote the set of all V-proper hyperubstitutions.

T

is

340 Definition 6.2 Let V be pseudovariety of type r . Two hypersubstitutions u1, u2 of type r are called V-equivalent if ( { u l ( f i % ) u2(fi)}) E FiltV for all i E I . In this case we write u1 -I/

u2.

Clearly, -v is an equivalence relation on H y p ( r ) and the restrictions of -V are equivalence relations on its submonoids. Then we have

Proposition 6.3 Let V be pseudovariety of type r and let following propositions are equivalent: (i)

01

u1,uz

E H y p ( r ) . Then the

-v 02.

(ii) For all t E W T ( X )we have ({61[t]x & [ t ] }E) FiltV. (iii) For all algebras A E V we have u1(A)= 4 A ) . Proof.

(i) + (ii): If u1-V

u2

1. If t = z is a variable, then

then for all i E I we have

@1[z] = 2 = 62[z]and

({z

({ul(fi) M ~ ( f i ) }E) M z})

FiltV.

E FiltV.

2. If t = f i ( t 1 , . . . ,tn,) and if ({61[tj]M &&I}) E FiltV for all j = 1,.. . ,ni then by Lemma 1.2 we have A u l ( f i ) % u2(fi) for all i E I and A d l [ t j ]M & 2 [ t j ] for all j E (1,.. . ,ni}. From this it follows A u1(fi(&1[tl],.. . ,i+l[tni]) M ? 1 [ t ] % &[t] and this means A u:!(fi(&2[tl], . . . ,62[tni]),i.e. A ({g1[t]M U.S.

& [ t ] }for ) arbitrary terms t and any algebra A

E V. But this means, ({C1[t] M

& [ t ] } )E F i l t V .

+

= f i ( z l , .. . ,zni) we have also ( { & l [ f i ( z l ,... ,zn,)] M FiltV and then again by Lemma 1.2, ul(fi) M u 2 ( f i ) E IdV for every i E I . But this means, ul(A)= u 2 ( A ) . (iii) + (i): This follows immediately from the definition of a derived algebra.

(ii)

(iii): For t

& [ f i ( z 1 ,... , z n i ) ] } ) E

Proposition 6.4 Let V be pseudovariety satisfied. (a)

of

type r . Then the following propositions are

For all ul, u2 E H y p ( r ) , if u1 -V u2, then u1 is V-proper i f f u2 i s V-proper. (Therefore P ( V ) is a union of full equivalence classes with respect to -v.)

(ii) For all terms , t E W T ( X )and for all

u1,uz

E H y p ( r ) with

u1 N V u2

({61[s]M &l[t]})FiltV H ({&[s]M & [ t ] } )E FiltV.

we have

341 Proof. (i): Let u 1 , u ~E Hyp(.r) with u1 -V u2 and assume that u1 E P ( V ) . By definition this means that for all filters A E FiltV we also have (ul(A)) E FiltV and then V (ul(A)).Therefore there is a set C’ E (ul(A)) such that V C’. By Lemma U.S.

1.2 there exists a set C E A such that C’ 2 ul(C) and then also V b ul(C). Using the definition of the equivalence 0-1 -v u2 we obtain V uz(C) ([Plo;94]) and then V k (uz(A)) and (uz(A)) E FiltV. The opposite implication is clear. U.S.

(ii): Assume that u1 -V u2 and that ({61[s] x 6 l [ t ] } E ) FzltV. Using the last proposition we have ({61[t]M 62[t]}) E FiltV and ( { C ? ~ [ S ] x & [ s ] } )E FiltV. Then Lemma 1.2 gives V 6l[s] x &[s]. From this we get C ? ~ [ S ] M 61[t],Vk 61[t]x 62[t] and V V &[s] N 62[t]and then ( { C ~ [ Sx] & [ t ] } )E FiltV. The second part follows in the same way. m

7

Example

Let A be the principal filter of type T = (2) generated by the set C := {x: M zl, ~ 1 ( ~ 2 ~x3 (xlx2)53, ) 51x2 x q}, i.e. a = ({xy = 2 1 , xl(5253) = (21x2)x3, zlzz x xl}). Let f be a binary operation symbol. We consider the pseudovariety PModA. If A E PModA, then A k { x 2 M x, q ( x 2 x 3 ) M (51x2)x3, ~ 1 x 2x q},i.e. A is a finite left-zero-semigroup and then PModA C LZ where LZ is the variety of all left-zerosemigroups. If conversely A is a finite algebra from L Z , then A k {xf N 5 1 , xl(x2Z3) M ( q x 2 ) ~ ~, 1 x 2M XI}, i.e. A and thus A E PModA. This shows that PModA = LZfi,. Let L e f t be the monoid of all hypersubstitutions ut with u(f)= t and leftmost(t)= xl, i.e. where the first variable occurring in t is 2 1 . If t is a binary term term starting with zl, then because of ( ~ 1 x 2M xl} E {x: x zl, q(Z223) x ( ~ 1 ~ 2 ) ~xlxg 3 , x XI} we have ui N p M o d A uxl with ux,(j)= x1 and if t starts with x2 we get ut N P M ~ Au x 2 .Thereand [ u x z ] N P MFrom odA. fore L e f t / - p M o d A consists of exactly two classes [uZ1]NPModA [Oxl]NPMod we A choose the identity hypersubstitution u,,,,. Since the type is finite, we ask whether A E P M o d A strongly Le f t-hypersatisfies A. By Proposition 6.4 we have only to check with uXlx2and with ax,. Clearly, we can restrict our checking to ux2and consider uxz(C) = {xl M xl}. Every algebra A E PModA satisfies x1 M x1 and then A A.

+

u.s.Mh.s.

This shows that PModA is a Left-solid pseudovariety.

References [Alm;94] J. Almeida, Finite Semigroup and Universal Algebra, World Scientiic 1995.

342 [Ban;83] B. Banahewski, T h e Birkh,off Theorem for varieties of finite algebras, Algebra Universalis 17 (1983) 360-368. [Den-W;OO] K. Denecke, S. L. Wismath, Hyperidentities and Clones, Gordon and Breach Science Publishers, 2000. [Den-W;OP] K. Denecke, S. L. Wismath, Universal Algebra and Applications in Theoretical Computer Science , CRC/Chapman and Hall 2000. [Eil-S;66] S. Eilenberg, M. P. Schiitzenberger, O n pseudovarieties, Adv. Math. 19 (1976), 413-418. [Gra-P-V;97] E. Graczyriska, R. Poschel, M. Volkov, Solid Pseudouarieties, in: General Algebra and Applications in Discrete Mathematics, Proceedings of the Conference on General Algebra and Discrete Mathematics, Potsdam 1997, pp. 93-110. [Plo;94] J. Plonka, Proper and inner hypersubstitutions of varieties, Proceedings of the International Confeence Summer School on General Algebra and Ordered Sets, Olomouc 1994, 106-116. [Rei;82] J. Reiterman, T h e Birkhoff theorem f o r finite algebras, Algebra Universalis 14 (1982), 1-10,

Bundit Pibaljommee University of Potsdam Institute of Mathematics Am Neuen Palais 14415 Potsdam Germany e-mail: [email protected] Klaus Denecke University of Potsdam Institute of Mathematics Am Neuen Palais 14415 Potsdam Germany e-mail: [email protected]

On Regular Ternary Semirings T.K.Dutta and S.Kar* Department of Pure Mathematics University of Calcutta

35, Ballygunge Circular Road, Kolkata-700019

INDIA e-mail address: karsukhendu6yahoo.co.in

Abstract

In this paper, we introduce the notions of ternary semiring and idempotent pair in a ternary semiring. We particularly investigated regular ternary semiring and k-regular ternary semiring. AMS Mathematics Subject Classification (2000): 16Y30.

Keywords: Ternary semiring, Idempotent pair in a ternary semiring, Regular ternary semiring, k-regular ternary semiring.

1 Introduction The notion of semiring was firstly introduced by Vandiver [4] dated back to 1934. In fact, semiring is a generalization of ring. In 1971 Lister [2] introduced ternary ring and regular ternary rings were studied by Vasile in [5]. We now introduce the notion of ternary semiring which is a generalization of the ternary ring introduced by Lister. In fact, ternary semiring is an algebraic system consisting of a set S together with a binary operation, called addition and a ternary multiplication, denoted by juxtaposition, which forms a commutative semigroup *Thesecond author is thankful to CSIR, India for financial assistance.

343

344 relative to addition, a ternary semigroup relative to multiplication and the left, right, lateral distributive laws hold. Ternary semiring arises naturally as follows of integers 2. The subset

consider the ring

~

Z+ of all positive integers of Z is an additive semigroup which

is closed under the ring product. Such algebraic system is called a semiring. Now if we consider the subset Z- of all negative integers of

Z then we see that Z- is an additive

semigroup which is closed under the triple ring product; however, 2- is not closed under the binary ring product, that is 2- forms a ternary semiring. More generally, in an ordered ring, we can see that its positive cone forms a semiring where as its negative cone forms a ternary semiring. Thus a ternary semiring may be considered as a counterpart of semiring in an ordered ring. We will also introduce the notions of left, right, lateral ideals and also characterize the regular ternary semiring and k-regular ternary semiring. Our main purpose of this paper is to generalize the concept of regularity in semiring to arbitrary ternary semirings. In

5 2, we give some preliminary definitions and examples.

In

3, we will introduce the notions of idempotent pair in a ternary semiring. In § 4, we will

study regular ternary semirings and some of their properties. In § 5, we will characterize k-regular ternary semiring.

2

Some Basic Definitions and Examples

Definition 2.1 A non-empty set S together with a binary operation, called addition and a ternary multiplication, denoted by juxtaposition, is said to be a ternary semiring if S is an additive commutative semigroup satisfying the following conditions: (i) abc 6 S , (ii) (abc)de = a(bcd)e = ab(cde), (iii) ( a + b)cd = acd abd

+ bcd, (iv) a(b + c)d =

+ acd, (v) ab(c + d ) = abc + abd for all a, b, c, d , e E S .

Example 2.2 Let Z-be the set of all negetive integers. Then,with the usual binary addition and ternary multiplication,

Z- forms a ternary semiring..

Example 2.3 Let S1 = { r f i

: r 6

Z} and

5’2

= {ri : r E

Z,i=

a}. Then both

Sl and Sz together with the usual binary addition and ternary multiplication form ternary semirings.

Definition 2.4 A ternary semiring S is said to be

345 (i) commutative if abc = bac = bca for all a, b, c E S. (ii) laterally commutative if abc = cba for all a, 6, c E S. We note that if S is commutative, then also abc 5 acb = cba = cab for all a, b, c E S.

Definition 2.5 Let S be a ternary semiring. If there exists an element 0 E S such that 0 + z = z and Ozy = soy = sy0 = 0 for all z,y E

S then

‘0’ is called the zero element or

simply the zero of the ternary semiring S. In this case we say that

S is a ternary

semiring

with zero. Throughout this paper S is always a ternary semiring with zero. Let A, B,C are three subsets of S. Then by ABC, we mean the set of all finite sums of the form

C aibici, with ai E A , bi

E

B,ci E C.

Definition 2.6 An additive subsemigroup T of S is called a ternary subsemiring if t 1 t 2 t 3 E T for all tl, t z , t 3 E T .

Definition 2.7 An additive subsemigroup 1 of S is called a left (right, lateral) ideal of S if slszi (respectively i s l s 2 , slisz) E I for all s1, s2 E S and i E I . If I is a left, a right, a lateral ideal of S then I is called an ideal of S.

Definition 2.8 A ternary semiring S is called a simple ternary semiring if it contains no non-zero proper ideal of S.

Example 2.9 Let M 2 ( Z - ) be a ternary semiring of the set of all 2 x 2 square matrices over 2-, the set of all negative integers.

{ (::)

: a , b ~ Z - } , h=

{ (1 t)

Let I1 =

:a,b,c,dE22-};Ii =

Then I I is a left ideal of M 2 ( Z - ) but not a right ideal of

M 2 ( 2 - ) but not a left ideal of M z ( 2 - ) .

4 is a

{(::)

:aiZ-}.

Mz(Z-).I, is a right ideal of

lateral ideal of M z ( Z - ) . In fact, I, is an

ideal of M z ( Z - ) and 14 is a ternary subsemiring of M z ( Z - ) but not an ideal of Mz(Z-).

Definition 2.10 Let S be a ternary semiring and a E S. Then the principal

+ na/ri, si E S;n E Z+}. {C arisi + na/ri, si E S;n E Z+}

(i) left ideal generated by a is given by < a > I ={Crisis (ii) right ideal generated by a is given by < a

>r=

346 (iii) two-sided ideal generated by a is given by


{Cp,qzu

+ C ar,s, + Cp;qkar;sk +

z+>

na/p,,qz,r,,s,,p;,q;,r;,s; E

(iv) lateral ideal generated by a is given by

< a >m= {Cr,as,+Cp,q,ar,s,+na/p,,

q j ,rZ,3, E

S ; n E Z+}

< a >= { x p Z q , u + Car,s, + CukaVk + XpiqiaTiS: + s;n E Z+},where denotes the finite sum and 2’ IS

(v) ideal generated by a is given by na/p,, qz,r,, s,,

u k , v k , p { ,q;, r;, si

E

the set of all positive integers.

Definition 2.11 An ideal I of S is called a k-ideal if x + y E I , z E S , y E I imply that

x E I.

S.Then the k-closure of A, denoted by 2, is defined by = { a E S : a + b = c for some b, c E A}. We can easily show that A 5 A, A E B j 2 E B and A= We note that an ideal A of S is a k-ideal if and only if A = x. Let A be an ideal of

z.

We also note that the intersection of any set of k-ideals of a ternary semiring S is a k-ideal of S.

Definition 2.12 An equivalence relation p on S is said to be a congruence relation or simply a congruence of

S if the following conditions hold:

(i) apa’ and bpb‘

+ ( a + b)p(a’ + b’) as well as (ii) apa’, bpb‘ and cpc‘ + (abc)p(a’b’c‘) for all

a, a’, b, b‘, c, c’ E S. The condition (ii) of the above definition is equivalent to the following condition: (ii)’ apa’

(abc)p(a’bc), (bca)p(bca’) and (bac)p ( ba’c). Definition 2.13 Let I be a proper ideal of S. Then the congruence on S , denoted by and defined by sp~s’if and only if s ternary congruence on

+ a1 = s‘ + a2 for some a l , a2 E I , is called the Bourne

S defined by the ideal I.

We denote the Bourne ternary congruence

(PI)

class of an element r of S by r / p r or

simply by r / I and denote the set of all such congruence classes of

S by S/pI or

simply by

S/I. It should be noted that for any s E S and for any proper ideal I of S , s / I is not necessarily equal to s + I = { s + a : a E I} but surely contains it.

+

347 Definition 2.14 For any proper ideal I of S if the Bourne congruence PI , defined by I , is proper i.e. O/I

all

+ b/I

=

(a

# S then we define the addition and ternary multiplication on S / I by

+ b ) / I and ( a / I ) ( b / I ) ( c / I =) (abc)/I for all a, b, c E

S . With these two

operations S / I forms a ternary semiring and is called the Bourne factor ternary semiring or simply the factor ternary semiring.

Definition 2.15 A ternary semiring S is said to be multiplicatively

+ x = y, (ii) right cancellative if xab = yab + x = y, (iii) laterally cancellative if axb = yb + x = y, (i) left cancellative if abx = aby

(iv) cancellative if S is left, right and laterally cancellative, for all a, b, z, y E

S.

Throughout this paper, cancellative means multiplicatively cancellative.

Definition 2.16 An element a in a ternary semiring S is called an idempotent element if a3 = a. A ternary semiring S is called an idernpotent ternary semiring if every element of S is idempotent. An ideal I of S is called idempotent if I 3 = I .

Definition 2.17 An element e in a ternary semiring S is called a bi-unital element if eex = xee = z for all x E S . For any bi-unital element e E S, we have xlxz ...............xmeex,+3 ...............xzn+1 = X l X Z ...............x,xm+3

...............X z n + 1 , xi E

s.

A bi-unital element is, in particular, an idempotent element. But the converse is in general not true. We will show latter on that the converse is true for cancellative ternary semirings.

Definition 2.18 Let S and T be two ternary semirings and f be a mapping from S into T . Then f is called a homomorphism of S into T if (i) f ( a -t b) = f ( a ) + f ( b ) and (ii) f(abc) = f ( a ) f ( b ) f ( c )for all a , b, c E S .

348

Idempotent pairs and cancellativity in a Ternary Semir

3

ing Definition 3.1 A pair ( a ,b) of elements in a ternary semiring S is called an idempotent pair if ab(abx) = abx and (xab)ab= xab for all x E S ,

Definition 3.2 A pair ( a , b) is called a natural idempotent pair if aba

=a

and bab = b.

Definition 3.3 Two idempotent pairs ( a ,b) and (c,d ) are said to be commute if ab(cdx) = cd(abx) and (zab)cd = (xcd)ab for all x E S . Definition 3.4 Two idempotent pairs ( a ,b) and (c, d ) are equivalent i.e. (a, b)

N

(c, d ) if

abx = cdx and xab = xcd, for all x E S. Clearly, the relation defined above is an equivalence relation. The equivalence class

-

containing the idempotent pair ( a ,b) is denoted by ( a ,b).

Let E ( S ) denote the set of all equivalence classes of idempotent pairs in S.

Lemma 3.5 Every idempotent pair (a,b) in S is equivalent to a natural idempotent pair

( c , d ) in S . Proof: Suppose c = aba and d = bab. Then cdc = (aba)(bab)(aba) = aba = c and

dcd = (bab)(aba)(bab)= bab = d, since ( a ,b) is an idempotent pair in S. Again, cdx = (aba)(bab)x= ab(aba)bx = ab(abx) = abx and xcd = x(aba)(bab)= xa(bab)ab = (xab)ab = xab. Hence the lemma. Now, assume that S is a ternary semiring in which all the idempotent pairs commute mutually. If ( a ,b) and ( c ,d ) are two idempotent pairs in S , then (abc,d ) is also an idempotent pair in S , because (abcd)(abcdx) = (abcd)(cdabx) = ab(cdabz) = ab(abcdx) = abcdx for all x E S and (xabcd)(abcd) = (xcdab)(abcd) = (xcdab)cd = (xabcd)cd = xabcd for all z E

S . Similarly, we can show that ( a ,bcd) is also an idempotent pair in S. Moreover,

(abc,d )

N

( a ,bcd).

Definition 3.6 A commutative idempotent semigroup is called a semi-lattice Theorem 3.7 Let S be a ternary semiring in which all the idempotent pairs commute mutu-~ ally. Then E ( S ) is a semi-lattice under the (binary) product defined by ( a , b).(c, d ) = (abc,d ) .

349 Proof: We first show that the above definition of product in E ( S ) is well-defined. Suppose - _ _

- _ _

that (a,b) = (a’, b’) and ( c , d ) = (C‘,d‘) in E ( S ) . Then abz = a’b’z, xab = xa’b’ and

cdx = c’d’z, xcd = xdd’ for all z E S. Hence (abc)dx = (a’b’c)dx = a’b’(cdz) = a’b’(c’d’z) + (abc)dz = (a’b’c’)d’z and similarly, z(abc)d = z(a’b’c’)d‘ for all 2 E S, proving that (abc,d )

N

-~

(a’b‘c’, d’) i.e. (abc, d ) = (a’b’c‘, d’). Thus the product in E ( S ) is well-defined. Clearly, the associative property holds in E ( S ) . Hence E ( S ) is a semigroup. Since by hypothesis, the -__ idempotent pairs in S commute, E ( S ) is commutative. Further, (a,b).(a,b) = (aba,b) =

(a,b), since (aba,b)

N

( a ,b). This shows that every element of E ( S ) is idempotent. Thus

E ( S ) is a semi-lattice. -

Remark 3.8 The associated partial ordering in the semi-lattice E ( S ) is given by ( a , b) 5

- -- __ (c,d ) if ( a ,b).(c, d ) = ( a ,b), and is called the natural partial ordering of E ( S ) .

Theorem 3.9 A ternary semiring S is cancellative if any one of the following conditions holds:

(i) S is left and right cancellative, (ii) S is laterally cancellative, (iii) For any a

E S,

the equation axa = aya implies that x = y .

Proof: We shall prove this theorem by showing that all the three conditions are equivalent. First we show that (i) + (ii). Let axb = ayb. Then c(azb)d = c(ayb)d 3 ca(xbd) =

ca(ybd) + zbd = ybd, ( by left cancellation)+

3:

= y, ( by right cancellation).

Clearly, (ii)+ (iii). To show that (iii)+ ( i ) ,we First let abz = aby. Then (abx)ba = (aby)ba + a(bzb)a =

a(byb)a + bxb = byb, by using (iii), we obtain + x = y . This shows that the left cancellation law holds in S . Similarly, we can show that the right cancellation law holds in S as well. This completes the proof.

Lemma 3.10 A n y idempotent element of a cancellative ternary semiring S is a bi-unital element. Proof: If e E S is an idempotent element, then ee(eez) = (eee)ex = eex for all x

E

S. By

using the left cancellation law, we get eex = z. Similarly, we can show that z e e = z. Hence

e is a bi-unital element.

350

Lemma 3.11 The idempotent pairs of elements of a cancellative ternary semiring S (if they exist) are all equivalent. Proof: Let ( a ,b) be an idempotent pair in a cancellative ternary semiring S . Then it follows from the equation ab(abx) = abx that abx = x for all x E S , by left cancellation. Hence, for any two idempotent pairs ( a ,b) and (c,d ) , we have abx = x = cdx for all x E S . Similarly,

xab = x = xcd for all z E S . Consequently, (a,b)

4

N

(c,d ) .

Regular Ternary Semirings

Definition 4.1 An element a in a ternary semiring S is called regular if there exists an element x in S such that axa = a. A ternary semiring is called regular if all of its elements are regular. If axa = a, then it is not in general true that xax = x . But if we take b = xax, then

aba

= axaxa = axa = a

and bab = xaxaxax = xaxax = xax = 6.

We note that if a E S is regular with aba = a and bab = b, then both ( a ,b) and (6, a ) are idempotent pairs. On the other hand, if ( a ,b) is an idempotent pair, then the element aba is regular. Hence, regular elements give rise to idempotent pairs and conversely, idernpotent pairs give rise to regular elements. We note that every left and right ideal of a regular ternary semiring may not be a regular ternary semiring; however, for a lateral ideal of a regular ternary semiring, we have the following result :

Lemma 4.2 Every lateral ideal of a regular ternary semiring S is a regular ternary semiring. Proof: Let L be a lateral ideal of a regular ternary semiring S . Then for each a E L , there exists x E S such that a = axa. Now, a = axa = axaxa = a(xax)a = aba, where b = xax

E

L. this implies that L is a regular ternary semiring.

Theorem 4.3 Let f be a homomorphism from a regular ternary semiring S onto a ternary semiring T . Then T is a regular ternary semiring that is, the homomorphic image of a regular ternary semiring is still a regular ternary semiring.

351 Proof: Suppose b E T . Since f is onto, there exists a E S such that f ( a ) = b. Again,

since S is regular, there exists x E

S such that a = axa. Therefore, b = f ( a ) = f(axa)=

f ( a )f ( x ) f ( a )= byb, where y = f (x)E T . This shows that T is a regular ternary semiring.

Corollary 4.4 Let S be a regular ternary semiring and I be a k-ideal of S . Then the factor ternary semiring

SII is regular.

Proof: The proof follows from the above Theorem 4.3,since S/I is a homomorphic image of

s.

Theorem 4.5 The following conditions on a ternary semiring S are equivalent:

(i) S is regular (ii) A n B = ASB for every right ideal A and every left ideal B of S

(iii) F o r a , b E S; < a

>r

(iv) For a E S ; < a >,.

n < b >1=<

a >,. S < b >1

n < a >1=,. S < b > l .

Proof: We first show that (i) + (ii). Let A be a right ideal and B be a left ideal of S . Then ASB

C ASS

A and ASB

C S S B c B.

Hence, ASB

a E A n B , a = axa for some x E S, by using

(2).

CA nB

(1). Again, for

This implies that a = axa = axaxa E ASB.

(2). From (1) and (2), it follows that A n B = ASB.

Thus A n B C_ ASB

Clearly, (ii) + (iii) and (iii)

+ (iv).

To complete the proof it remains to show that (iw)+ ( 2 ) . For this purpose, we let a E < a >,.

rna)tj(Crkska+na)

n < a >l=<

a >,. S < a

>l.

+

Then a = C(Capiqz

= C[(Capiqi)tj(C rkw)+(Capiqi)tj(na)+(ma)tj(Crkska)+(ma)tj(na)l E

aSa. Thus a E a S a , so that there exists an element x of S such that a = axa. Consequently, a is regular. Since a is arbitrary, S is regular. This completes the proof.

Theorem 4.6 A commutative ternary semiring S is regular i f and only if every ideal of S is idempotent.

Proof: Let S be a regular ternary semiring and I be any ideal of S. Then I3= 111 c

SSI C I. Let a

E

I. Then there exists x

E

S such that a = axa = axaxa. Since I is an

ideal and a E I, xax E I. Hence, we find that a = axa = axaxa E 13. Consequently, I so that I3= 111= I i.e.

c 13,

I is idempotent.

Conversely, suppose that every ideal of S is idempotent. Let A and B be two ideals of

S . Then ASB

c ASS C A and ASB c S S B c B. This implies that ASB C A n B. Also,

352

(Af l B)(Af l B)(Af l B)C ASB.Again, since A fl B is an ideal of S,( A nB)(A(l B)(A B)= A n B. Thus we have shown that A n B ASB and hence A n B = ASB. Therefore, by Theorem 4.4, S is a regular ternary semiring. The proof is completed. Definition 4.7 A regular ternary semiring in which all the idempotent pairs are mutually commute is called a strongly regular ternary semiring.

Definition 4.8 A left (respectively right) ideal L (respectively R ) of a ternary semiring S is said to have an idempotent representation if there exists an idempotent pair ( a , b) such that L = Sab (respectively R = abS). The left ideal L (respectively the right ideal R) is said to have a unique idempotent representation if for any two idempotent pairs ( a , b) and (c,d ) such that L = Sab = Scd (respectively R = abS = cdS), we have ( a ,b)

Lemma 4.9 An element a E

N

(c,d ) .

S is regular if and only if the principle left [respectively right)

ideal of S generated by 'a' has an idempotent representation. Proof: If a is regular then a = axa for some x E S . Now if b = x a x then we have aba = a

C< a

and bab = b. In this case ( b , a ) is an idempotent pair and Sba ideal of S generated by a. Again, if r E < a

(Csitia + na)ba E Sba.

>1,then

= C sitia

r

Consequently, < a >iE Sba and hence

Conversely, suppose that < a

>i

>l,

the principal left

+ n a = C sitiaba + naba =

< a >1= Sba.

has an idempotent representation i.e. < a

>I=

Sdc

for some idempotent pair ( d , c ) . Since a E < a >I= Sdc, a = tdc for some t E S and

adc a

>i,

=

(tdc)dc = tdc = a. Therefore, a = adc there exist ei, fi E

S such

=

(adc)dc = ad(cdc). Since, cdc E S d c =<

that cdc = C e i f i a

+ n a for some n E Z+.Therefore,

a = ad(&) = a d ( C ei fia+na) = a ( C dei fi+nd)a =+ a = axa for some x =

deifi+nd E S.

Consequently, a is a regular element of S.

Theorem 4.10 Let S be a strongly regular ternary semiring. Then every princapal left (respectively right) ideal of S has a unique idempotent representation. Proof: By our Lemma 4.9, we see immediately that every principal left ideal of S has at least one idempotent representation. Therefore, < a >I= Sba, where (b, a ) is an idempotent pair such that a = aba. Now suppose that


>i=

Sba = S d c for some idempotent pair

( d , c). Then for any 2 E S there exist y, z E S such that xba = ydc and z d c = zba. Therefore,

353 (xba)dc = (ydc)dc = ydc = xba -(ii).

(i) and (xdc)ba = (zba)ba = zba = xdc-

Since, by hypothesis, S is strongly regular, all the idempotent pairs commute

mutually. Thus, the elements on the left side of ( i ) and (ii) are equal. Hence for all

x

E

S, we have xba = zd-(iii).

Again, since S is strongly regular, for all x E S,

ba(dcx) = dc(bax) + b(adc)x = d(cba)x. Now from (iii), we have aba = adc and cba = cdc. (iv). From (iii) and (iv),

Thus, b(aba)x = d(cdc)x. This implies that bax = dcx we have (b, a )

5

-

( d ,c). Hence the theorem is proved.

k-Regular Ternary Semirings

Definition 5.1 Let S be a ternary semiring and a E S. Then a is called k-regular if there exist x , y E S such that a+aya = a m . If all the elements of S are k-regular then S is called a k-regular ternary semiring .

Remark 5.2 Suppose that S is a regular ternary semiring. Then for any a E S there exists x E S such that a = axa. By adding aya to both sides of a = axa, we have, a axa + aya

= a(z

+ y ) a = aza, where z = ( x + y ) E S .

+ aya =

Thus, it follows that each regular

ternary semiring is also a k-regular ternary semiring; however, the converse is not necessarily true general. Definition 5.3 Let S be a ternary semiring and a E S. Then we call a is weak k-regular if there exist r, s, x , y E S such that a

+ arysa = arxsa.

A ternary semiring S is called weak

k-regular if all the elements of S are weak k-regular.

Remark 5.4 A weak k-regular ternary semiring is a k-regular ternary semiring but the converse is not generally true. Note 5.5 If S happens to be a ternary ring, then all the concepts defined above coincide. Theorem 5.6 Let S be a k-regular ternary semiring. Then A n B = ASB holds for each right k-ideal A and each left k-ideal B of S. Proof: Since A is a right k-ideal, we have ASB

2 ASS

c A and consequently, we have

ASB C 2 = A . Again, since B is a left k-ideal of S, we have ASB C S S B C B and hence

ASB C B = B. Thus, we obtain that ASB

C_

A nB

(i). To show the converse

354 inclusion, we let a E A n B . Since S is k-regular, a + aya = a z a , for some x,y E S. Now

aza E ASB, for each z E S. Therefore, by the definition of k-closure, we get a E ASB and hence A

B

C ASB

(ii). From (i) and (ii) it follows that A n B =

m.

Theorem 5.7 Let S be a ternary semiring and K ( S ) the set of all k-ideals of S . Then ~

( K ( S ) , C )is a complete lattice. This lattice is distributive if A n B = ASB holds for all k-ideals A =

and B = B of S i.e. in particular i f the ternary semiring S is k-regular.

Proof: The intersection of any set of k-ideals of a ternary semiring S is known to be a

, is a complete k-ideal of S , and S itself is also a k-ideal of S . This yields that ( K ( S ) C) lattice, where A A B = A n B is the infimum and A V B = A

A , B E K ( S ) . To prove the next part, suppose A n B

+B

is the supremum of

ASB

=

each lattice, we always have ( A A B )V ( A A C ) C A A ( B V C )

(i) holds. In (ii) holds for all

A, B,C E K ( S ) . Therefore, it remains to show the converse inclusion. Now, A I \ ( B V C ) =

A n ( B + C ) = A S ( B ) ,by using (i). Suppose that z E A S ( B

+ C ) . Then x + u = v

for some u,w E A S ( B + C ) . Hence, u = C a i t i y i and w = C a j t j y j , where

C

the finite sum and ai, aj E A; ti, tj E S and yi, y j E B + C. Since, yi, y j E B

+ C , yi +

denotes

+ ci = b: + c: and y j + bj + cj = b> + c> for some bi, b:, b j , b> E B and q,c:, cj, c> E C . Thus aitiyi + aitibi + a&ci = aitibi + aitici + x a i t i y i + Caitibi + Caitici = Caitib: + C aitic: + u + C aitibi + C a& = C aitib: + C aitid u E ASB + ASC. Similarly, we can show that u E ASB + ASC. Now z + u = w implies that z E ( A S B t- ASC) = ASB + ASC, since ASB + ASC is k-closed. So we find that A S ( B ) C bi

~

j

A A ( B V C ) C A S B + A S C . Again, A S B + A S C g ( AA B ) V ( AA C ) . Hence A A @ V C )

m+m= ( A n B ) + ( A n C )=

C ( AA B ) V ( AA C )

(iii). From (ii) and

(iii) it follows that A A(B V C ) = ( AA B ) V ( AA C ) . This shows that the lattice ( K ( S ) C) , is indeed a distributive lattice. The proof is completed.

References [l]Adhikari, M.R.; Sen, M.K. and Weinert, H.J.; O n k-regular semirings; Bull. Calcutta

Math. SOC.88 (1996), no.2, 141-144. [2] Lister, W.G.; Ternary rings; Trans. Amer. Math. SOC.154 (1971), 37-55.

355 [3] Sioson, F.M.; Ideal theory in ternary semigroups; Math. Japonica.; Vol.10 (1965), 63-84. [4] Vandiver, H.S.; Note o n a simple type of Algebra in which the cancellation law of addition

does not hold; Bull. Am. Math. SOC.40, 920 (1934).

[5] Vasile, Tama.; Regular ternary rings; An. Stiin. Univ. Al. I. Cuza. Ia si Sec. Ia Mat.33 (1987), no.2, 89-92.

Indecomposable decompositions of CS-modules * Josh L. G6mez Pardo Departamento de Alxebra, Universidade de Santiago 15782 Santiago de Compostela, Spain Email: [email protected] Pedro A. Guil Asensio Departamento de Matema'ticas, Universidad de Murcia 30100 Espinardo, Murcia, Spain E-mail: [email protected]

1

Introduction.

The common early ancestor of the results that we are going to study here, is the fact that all divisible abelian groups are direct sums of groups isomorphic to Q or to Z(p"), where p is prime. Thus divisible abelian groups are direct sums of indecomposable groups and their study reduces, in some sense, to the study of the latter. This is one of the sources of an important idea in module theory: since the determination of the structure of a module is hopelessly difficult, one can try to reduce it to the case of indecomposable modules. For this setup to work, it is necessary that the module one wants to study has an indecomposable decomposition, which is not always the case. The result mentioned at the beginning gives a clue on how to find modules with the desired property and has an important moduletheoretic generalization in which divisible groups are replaced by injective modules. This is the M a t h - P a p p theorem, according to which a ring R is right noetherian if and only if every injective right R-module has an indecomposable decomposition. Over a right noetherian ring, every injective right R-module E is C-injective, in the sense that any direct sum of copies of E is injective (actually, this is a necessary and sufficient condition for R to be right noetherian) and this leads in a natural way to a result, due to 'Work partially supported by the DGI (BFM2000-0346, Spain). The second author was also partially supported by the Fundaci6n S6neca (PL76/00515/FS/Ol).

356

357

A. Cailleau [3], which can be regarded as the module-theoretic version of Matlis-Papp: a n injective module E is C-injective i f and only if E i s a direct s u m of indecomposable C-injective modules. This result reveals a large class of injective modules with indecomposable decompositions over rings which are not necessarily noetherian. For example, all the countable injective modules are C-injective by a result of Megibben [16]. Cailleau’s result was soon extended to quasi-injective modules by Cailleau and Renault [4] (recall that M is quasi-injective when it has the injectivity property relative to monomorphisms ending in M ) : every C-quasi-injective module has a n indecomposable decomposition. The passage from injective to quasi-injective modules is not a big jump from the conceptual point of view but has the advantage of posing the problem in a purely module-theoretic context, free from any explicit reference to the ring. In recent years, it has been observed that many of the characteristic properties of (quasi-) injective modules still hold for the much more general class of CS-modules (the term comes from Chatters and Hajarnavis [5]; they are also called extending modules in the terminology introduced by Harada). A module Ad is called CS whenever every essentially closed submodule of M (i.e., any submodule which has no proper essential extensions within M ) is a direct summand of M or, equivalently, every submodule of M is essential in a direct summand of M IS]. The generalization of the Matlis-Papp theorem to CS-modules was proved by Okado in [18]: Every CS right R-module has a n indecomposable decomposition i f and only if R is a right noetherian ring. The proof was not difficult and suggested that the decomposition theory of CS-modules could be very similar to the one of injective and quasi-injective modules but, as we are going to see, things do not work in such a straightforward way as one might be tempted to conclude from this result. For example, after Okado’s result it was still unknown whether a ring R such that every CS right R-module is C-CS (i.e., any direct sum of copies of the module is CS) was necessarily right noetherian (this was a question of Dung [7]). Even more intriguing and important was the following problem, which was posed in 1994 in [8, p. 1041 : Does every C - CS-module have a n indecomposable decomposition? Until very recently, the known results about this question were restricted to modules satisfying some additional condition like nonsingularity, or some form of projectivity. In the latter case these results were often inspired by Oshiro’s theorems on right C-CS rings which show, among other things, that these rings are, in fact, two-sided artinian, and are also characterized by the property that the class of projective right modules is closed under essential

358

extensions [19, 201. Another remarkable result not directly related to CCS-modules was the celebrated Osofsky-Smith theorem [23, 81, according to which every cyclic module whose cyclic subfactors (submodules of factor modules) are CS, is a (finite) direct sum of uniform modules. In this paper we are going to review some more recent results related to preceding questions which, in particular, provide a complete solution to the last problem mentioned but, at the same time, give rise to new and difficult questions that remain unsolved so far. At the end of the paper, we also obtain some new results related to these questions. Throughout this paper, all rings R will be associative and with identity, and by a module we will usually mean a right; R-module. The cardinality We refer to [2, 17, 251 for all undefined of a set X will be denoted by notions used in the text.

1x1.

2

Indecomposable decompositions

We begin with a simple remark: the indecomposable CS-modules are precisely the uniform modules, and so asking whether a CS-module M has an indecomposable decomposition amounts to asking if M is a direct sum of uniform modules. Thus an indecomposable decomposition of a CS-module need not have local endomorphism rings, but in the case of C-CS-modules we have the following important fact, which was first observed by Al-attas and Vanaja in [l]:

Proposition 2.1. An indecomposable module i s C-CS if and only i f it is C - quasi-injective.

As a consequence of this result, if a C-CS-module M has an indecomposable decomposition M = @ I Mi, then the Mi have local endomorphism rings by [2, Lemma 25.41 or [25, 19.91. An important consequence is that the indecomposable decompositions of C-CS-modules are unique (in the sense that they are all “equivalent”, cf. [a]) by the Krull-Remak-Schmidt-Azumaya theorem [2, Theorem 12.61. Recall that a decomposition M = @ I M is ~ said to complement direct summands (cf. [2, $121) when, for every direct summand N of M , there exists a subset J I such that M = N @ ( @ J M ~A ) . module M is said to have the exchange property if, for any index set I , whenever M @ N = @iErXi for modules N and X i , then M @ N = M @ ( @ i E ~ Yfor , ) submodules Y, C_ X i . From Proposition 2.1 and [7, Corollary 3.51 it follows that, if a C-CSmodule M has an indecomposable decomposition, then this decomposition complements direct summands and, moreover, M has the exchange property.

359

We see from the preceding remarks that if a C-CS-module has an indecomposable decomposition, then this decomposition has most of the desirable properties one might expect. This made even more important the problem of finding whether these decompositions must exist. The answer is yes and the proof was given in [12]:

Theorem 2.2. Every C-CS-module is a direct sum of uniform modules The original proof of this result used set-theoretic counting arguments but, in [14], a proof was obtained that avoids counting, except for the use of the theorem of Kaplansky that asserts that if c is an infinite cardinal and M a module which is a direct summand of a direct sum of c-generated modules, then M itself is also a direct sum of c-generated modules [2, Theorem 26.11. This new proof can be summarized as follows. Recall that, if A4 is a module, the quasi-injective hull of M is precisely the injective hull of M in the category a[M]whose objects are all the submodules of M-generated modules (where a module is M-generated when it is a quotient of a direct sum of copies of M ) [25, 17.111. First we can reduce the problem to the quasiinjective hull of M by means of the following lemma [14, Lemma 2.21:

Lemma 2.3. If M is a GS-module whose quasi-injective hull is a direct sum of uniform modules, then M itself is a direct sum of uniform modules. Now, to prove the theorem, it will suffice to show that if M is a C-CSmodule and Q its quasi-injective hull, then Q is C-quasi-injective because, as we have already remarked, Q is then a direct sum of indecomposable modules and we can apply Lemma 2.3. If I is a set and E denotes the quasi-injective hull of Ad'),then it is well-known that E is an M-generated module and so we have an epimorphism p : A d J ) E , for some set J . Since M is a submodule of E , we may assume that the canonical inclusion j of M into E factors through p , i.e., there exists q : M + A d J ) such that p o q = j . Thus p ( q ( M ) )= Im(p o q ) is an essential submodule of E and, since q ( M )nKerp = 0, it follows from [14, Lemma 2.11 that Kerp is a direct showing that E belongs to the class Add M of the direct summand of summands of direct sums of copies of M . Then, using Kaplansky's theorem [2, 26.11, it follows that E is a direct sum of c-generated modules, where c = max(N0, / M I ) . Now, using [8, 2.41, we see that Q is a C-M-injective module (and a C-quasi-injective module) and so the theorem follows. From Theorem 2.2 and the preceding remarks we know that C-CSmodules have very well behaved indecomposable decompositions. This fact can also be appreciated in the following characterization of C-CS-modules --f

360

among CS-modules [12], which is a direct consequence of the theorem, together with several previous results of [l]:

Corollary 2.4. Let M be a module. Then the following conditions are equivalent:

(i) M is C-CS. (ii) M is CS, the quasi-injective hull ofM is C-quasi-injective, and M is a direct s u m of indecomposable quasi-injective modules. (iii) M is countably C-CS, and a direct sum of modules unth local endomorphism rings. Also, from Theorem 2.2, the following result, measuring up the gap between C-CS and C-quasi-injective modules, is obtained [12]:

Corollary 2.5. A module M is C-quasi-injective i f and only af it is C-CS and quasi-continuous. We have already mentioned Dung’s question [7] asking whether a ring R such that every right CS-module is C-CS is a right noetherian ring. It follows from Theorem 2.2, together with Okado’s result previously mentioned, that the answer is affirmative. In fact, a stronger result was obtained by Huynh, Jain and L6pez-Permouth in [15, Theorem 21 where, using again Theorem 2.2, the following was proved:

Theorem 2.6. If R is a ring such that every CS right R-module is C-CS, then R is right artinian. In relation with this result, it should be mentioned that the converse is far from being true. In fact, there are (left and right) artinian right serial rings for which not all the right CS-modules are C-CS [l,Examples 5.131. The preceding results and, in particular, Theorem 2.2 and some of its consequences, might give the impression that the decomposition theory for C-CS-modules behaves exactly in the same way as the one for C-(quasi)injective modules, but this conclusion would be completely wrong. For example, we have already mentioned the fact that there exists (two-sided) artinian rings such that not every right CS-module is C-CS, in contrast with what happens if one replaces “CS” with “injective” (all injective right modules over a right noetherian ring are C-injective by a well-known result of C. Faith [2, Theorem 25.11). This is just one of the ways in which both

36 1

theories differ and we are now going to explore another direction in which deep differences between them are also readily apparent. As is well known, it is enough to assume that a module M is countably C-injective -meaning that each countable direct sum of copies of M is injective- to conclude that M is already a C-injective module [2, Theorem 25.11. But a countably CCS-module need not be C-CS and need not even have an indecomposable decomposition, as the following easy example shows [9, 81:

Example 2.7. Let F be a field and V an infinite dimensional (right) vector space over F . Then the endomorphism ring R = EndF(V) is a right selfinjective Von Neumann regular ring which is not left self-injective. Then

RR is countably C-CS but is not C-CS. In fact, it is easy to check that RR does not have an indecomposable decomposition (as this would imply that R is semisimple). The phenomenon just mentioned suggests that one could study N-C-CSmodules (i.e., modules M such that every direct sum of N copies of M is a CS-module) and the possible existence of indecomposable decompositions for such modules. An obvious question in this direction, which was tackled in [13], is the following:

Given a n arbitrary module M , is there a cardinal N (possibly depending o n M ) such that if M is N-C-CS, then M is already c-CS?

It turns out that the answer to the preceding question is yes, as the following result [13, Theorem 2.41 shows. We denote by N+ the successor cardinal of N. Theorem 2.8. Let M be an N-generated module with N 2 No. If M is N+-C-CS, then M is a C-CS-module and hence a direct sum of uniform modules. This result has a drawback: the cardinal N+ that ensures the existence of an indecomposable decomposition depends on the size of M . This is the motivation for the next questions, which were originally asked in [13] in a more restrictive way:

Q1 Is there a cardinal N such that each N-C-CS-module is already C-CS? Q2 Is there a cardinal N such that each N-C-CS-module has an indecomposable decomposition?

362

Observe that an affirmative answer to the first question -which is still open- would imply an affirmative answer to the second one, but the converse is not necessarily true. If the answer to either question is yes, then the cardinal N must be, at least, N1 and in [13] it was specifically asked if every N1-C-CS-module is a direct sum of uniforms. The answer to this problem (and hence to question Q2) is, again, affirmative 1141: Theorem 2.9. Every N1-C-CS-module is a direct s u m of uniform modules.

The proof of this theorem relies on infinitary counting arguments that go back to Tarski and are inspired in what we will refer to as “Tarski’s lemma” [24, Thkorkme 71. Tarski’s lemma was also used by Osofsky [22] in her proof of the finiteness of the socle of an injective cogenerator ring (among other places). The following lemma [12, Lemma 2.11, obtained with the help of a variant of Tarski’s lemma (cf. [ll,Lemma 3.1]), is a crucial step of our proof Lemma 2.10. Let A4 be a CS-module and {Li(iE I } a local direct summand of M such that 111 = No. Then there exist subsets d,K C 2’ such that:

(i) A is a partition o f 1 with [dl= No and IAl = No for every A E A. (ii) d g K , IKl = N1 and IKl = No for each K E K .

(iii) K n K’ is a finite set for all K , K’ E K such that K

# K’.

(iv) If LK denotes an essential closure of @ K Lin~ M for each K E Ic, and K , Kl, . . . ,K, are pairwise distinct elements of K , then LK n ( L K + ~ . . . + LK,) = @Kn(K1u..uKT)Li. In fact, this lemma was also used in the original proof of Theorem 2.2 given in [la], which was later superseded by the more conceptual proof that we have sketched in the first part of these notes. It would be interesting to obtain a simpler proof of Theorem 2.9 along these lines but this seems difficult as the cardinal numbers must somehow enter the picture explicitly. Since the proof of Theorem 2.9 is rather involved, we will not sketch it here and, for the details, we refer to [14]instead. This result leaves open the question whether every N1-C-CS-module is already C-CS, which would answer question Q1 above. The following corollary of Theorem 2.9 ([14, Corollary 2.71) shows that N1-C-CS-modules are, in some sense, pretty close to being C-CS, but our proof of Theorem 2.9 highlights the differences between both concepts and suggests that maybe they are not the same.

363

Corollary 2.11. The quasi-injective hull of an N1-C-CS-module is a Cquasi-injective module. As another consequence of Theorem 2.9 we obtain the following result, which shows that in order to obtain an affirmative answer to question Q1, it is enough to show that every uniform N1-C-CS-module has local endomorphism ring.

Corollary 2.12. Let M be a right R-module. Then the following conditions are equivalent: (a) M is a C-CS-module. (ii) M is an N1-C-CS-module such that every uniform direct summand of M has local endomorphism ring. The preceding corollary will allow us to give an affirmative answer to Q1 in a few interesting particular cases. To provide some background, we mention that it was shown by Clark and Dung [6], drawing on Osofsky’s ideas [21], that a right nonsingular right quasi-continuous right N1-C-CS ring is already right C C S and hence a semisimple ring. More generally we see, as a consequence of Theorem 2.8, that any countably generated N1-CCS-module is already C-CS and so, in particular, every right N1-C-CS ring is a right C-CS ring. This is also a consequence of the next result which answers Q1 for projective modules.

Corollary 2.13. Every projective N1-C-CS-module is a C-CS-module. Proof. By Corollary 2.12 it suffices to show that every uniform projective N1C-CS-module has local endomorphism ring. If M is such a module, then M is countably generated by Kaplansky’s theorem [2, 26.11. Then M is (C)quasi-injective by [13, Proposition 2.11 and so it has local endomorphism 0 ring. Recall that, if M is a module, M is said to be non-M-singular [8] whenever, for every M-subgenerated module N (i.e., a module N which is isomorphic to a submodule of a factor module of a direct sum of copies of M ) , there are no nonzero homomorphisms f : N -+M with essential kernel (so that , in particular, nonsingular modules M are non-M-singular).

Corollary 2.14. Every non-M-singular N1-C-CS-module M is a C-CSmodule.

364

Proof. Every uniform direct summand of M is countably C-CS and nonM-singular, and hence is C-CS (and C-quasi-injective) by [l,Lemma 2.71. 0 Then the result follows using Corollary 2.12. We end these notes with a result that shows that Q1 also has an affirmative answer if we fix the underlying ring R.

Corollary 2.15. L e t R be a ring. T h e n there exists a cardinal N such that each N-C-CS right R-module is already a C-CS-module.

Proof. Since the isomorphism classes of cyclic right R-modules form a set, there exists a cardinal c such that the cardinal of the injective hull of every cyclic right R-module is 5 c. Let now N = max(N1, c} and let U be an N-CCS uniform right R-module with quasi-injective hull Q. Since Q is contained in the injective hull of any cyclic submodule of U , we have t h a t IQI 5 c and hence Q is a c-generated module. But Q is also a U-generated module, and so there exists a n epimorphism q : U @ )+ Q such that Ker q is not essential in U"). Then, using [l,Lemma 2.21, we obtain that U is quasi-injective. Thus U has local endomorphism ring and it follows from Corollary 2.12 that every N-C-CS right R-module is C-CS. 0

References [l]A.O. Al-attas and N. Vanaja, O n C-extending modules, Comm. Algebra 25 (1997), 2365-2393. [2] F.W. Anderson and K.R. Fuller, Rings and categories of modules, SpringerVerlag, Berlin and New York, 1974.

[3] A. Cailleau, Une caractdrisation des modules E-inject+, C.R. Acad. Sci. Paris, Ser. A-B, 269 (1969), 997-999. [4] A. Cailleau and G. Renault, Etude des modules Sigma-quasi-injectifs, C.R. Acad. Sci. Paris, Ser. A-B, 270 (1970), 1391-1394. [5] A.W. Chatters and C.R. Hajamavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math 28 (1977), 61-80. [S] J. Clark and N.V. Dung, O n the decomposition of nonsingular CS modules, Canadian Math. Bull. 39 (1996), 257-265.

[7] Nguyen V. Dung, O n indecomposable decompositions of CS-modules 11,J. Pure Appl. algebra 119 (1997), 139-153.

365

[8] Nguyen V. Dung, Dinh V. Huynh, P.F. Smith, and R. Wisbauer, Extending modules, Longman, Harlow, 1994.

[9] Nguyen V. Dung and P.F. Smith, C-CS-modules, Commun. Algebra 22 (1994), 83-93.

[lo] C. Faith, Algebra 11Ring Theory, Springer-Verlag, Berlin and New York, 1976.

[I11 J.L. G6mez Pardo and P.A. Guil Asensio, On the Goldie dimension of injective modules, Proc. Edinburgh Math. SOC.41 (1998), 265-275.

[12] J.L. G6mez Pardo and P.A. Guil Asensio, Every C-CS-module has an indecomposable decomposition, Proc. Amer. Math. SOC.,129 (2001), 947-954.

[13] J.L. G6mez Pardo and P.A. Guil Asensio, Indecomposable decompositions of N-C-CS-modules, Contemporary Mathematics, 259 (2000), 467-473.

[14] J.L. G6mez Pardo and P.A. Guil Asensio, Indecomposable decompositions of modules whose direct sums are CS, J. Algebra, to appear.

[15] Dinh V. Huynh, S.K. Jain and S.R. L6pez-Permouth, Rings characterized by direct sums of CS modules, Commun. Algebra 28 (2000), 4219-4222.

[16] C. Megibben, Countable injectives are C-injective, Proc. Amer. Math. SOC.84 (1982), 8-10.

[17] S.H. Mohamed and B.J. Miiller, Continuous and discrete modules, Cambridge University Press, Cambridge, 1990.

[18] M. Okado, On the decomposition of extending modules, Math. Japonica 29 (1984), 939-941.

[19] K. Oshiro, Lifting modules, extending modules and their applications to QFrings, Hokkaido Math. J. 13 (1984), 31G338. [20] K. Oshiro, On Harada rings, I, 11, 111, Math. J. Okayama Univ. 31, 161-178, 179-188 (1989) and 32, 111-118 (1990). [21] B.L. Osofsky, Rings all of whosefinitely generated modules are injective, Pacific J. Math. 14 (1964), 645-650.

[22] B.L. Osofsky, A generalization of Quasi-Frobenius rings, J. Algebra 4 (1966), 373-387. [23] B. Osofsky and P.F. Smith, Cyclic modules whose quotients have all complement submodules direct summands, J. Algebra 139 (1991), 342-354.

366 [24] A. Tarski, Sur la dtkomposition des ensembles en sous-ensembles presque disjoints, Fund. Math. 12 (1928), 188-205.

[25] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.

HEREDITARY RINGS, QF2 RINGS AND RINGS OF FINITE REPRESENTATION TYPE C.R.HAJAKNAVIS Mathematics Institute, University of Warwick, Coventry CV4 7AL, England E-mail: [email protected]

A well-known result of Eisenbud and Griffith states that if R is a Noetherian prime hereditary ring then the ring R/I is an Artinian serial ring for all non-zero ideals I of R . There are several partial converses t o this theorem. We review the literature in this area and reinterpret some of these results in terms of QF2 rings. We also describe a recent extension of this theory to Noetherian polynomial identity rings where the Artinian factor rings are of finite representation type.

1. Introduction It is well-known that a quasi-Frobenius (QF) ring R is a direct sum of uniform one-sided ideals [ 171. Highlighting the significance of this property, such rings have been named QF2 rings in the literature on Artinian rings and obviously in this theory the Artinian chain condition is assumed to be part of the definition. However, as shown in [4], interesting results can be obtained using weaker chain conditions or even no chain assumptions at ail. So following the precedent set in [4] we define a ring to be QF2 if it is a direct sum of uniform one-sided ideals. Thus for example, hereditary Noetherian prime rings [l 11 and serial rings are QF2 under this definition. It is shown in [ 6 ] that proper factor rings of a prime hereditary Noetherian ring are Artinian serial rings. Several authors have obtained partial converses by considering proper factor rings which are QF rings, principal right and left ideal rings or Artinian serial rings, some of these results being successive generalisations. All these theorems need some boundedness assumption which is always fulfilled by Noetherian rings satisfying a polynomial identity (PI). In this line, we present some new work of Villani where rings whose proper factor rings are of finite representation type are considered. In some sense Villani’s results can be viewed as a completion of the above project. 2. Preliminaries and Background All rings will be assumed to have an identity. J will always denote the Jacobson radical of the ring.

367

368

When the ring R contains no infinite direct sum of right ideals we have R = e,R 0 e2R 0 ... 0 e,R = Rel 0 Re2 0 ... 0 Re, where each eiR (Rei) is a primitive right (left) ideal. This will be our standard notation. 2. I Definitions: A ring R will be called a QF2 ring if it is expressible as a direct sum of uniform right ideals as well as uniform left ideals. By [S, Lemma 7.11, in the Artinian case, it is enough to assume that R is an ordinary sum of uniform right (left) ideals. Further, we will say that the ring R is a fully QF2 ring if all non-zero factor rings of R (including R itself) are QF 2 rings. A module M is called (ungserial if whenever A and B are submodules of M , we have either A G B or B c A . The ring R is said to be a serial ring if each eiR (Rei) is a serial module. By [29 or 5, Ch. 61 if R is a serial ring then the ring WJ is semi-simple Artinian and if R is also Noetherian then the chain eiR 3 eiJ 2 eiJ2 3 ... 2 0 gives the complete list of submodules of eiR . It is clear that a serial ring is fully QF2 but obviously the converse cannot hold even if the ring is Noetherian. However, in the Artinian case we do have a characterisation. 2.2 Theorem (30, Theorem I]: Let R be an Artinian ring. Then R is a serial ring if and only if R is a fully QF2 ring. In order to appreciate the hierarchy of the results given below we draw the reader’s attention to the following which comes from [2, 15 and 301. 2.3 Theorem: Let R be an Artinian ring. Then the following are equivalent: (i) All factor rings of R (including R itself) are QF rings. (ii) R is a principal right and left ideal ring. (iii) R is a direct sum of primary rings, each being a serial ring. The results below describe a relationship between hereditary and serial rings. 2.4 Theorem (6, Corollary 3.21: Every proper factor ring of a prime Noetherian hereditary ring is an Artinian serial ring.

369

2.5 Theorem (29, Theorems 5.11, 5.141: A Noetherian serial ring with zero Artinian radical (i.e possessing no non-zero Artinian one-sided ideal) is hereditary. For Commutative rings a converse of Theorem 2.4 was proved in [18] as can be seen noting the equivalence stated in Theorem 2.3. The results stated below all need some sort of boundedness hypothesis. The most natural class fulfilling these conditions arises from rings satisfying a polynomial identity.

2.6 Definition: A ring R is said to satisfy a polynomial identity (PI) if there is a non-trivial polynomial f in the fiee algebra Z<X1, X2, ... , X,> such that f(r,, r2, ... , r,) = 0 for all choices of ri E R . We now mention the results which have been obtained as converses to Theorem 2.4. We present them here in their most natural form. The enthusiastic reader seeking the greatest generality should consult the original papers. Recall that a Noetherian Dedekind prime ring is a hereditary Noetherian prime ring which contains no non-zero idempotent ideal. 2.7 Theorem (12, Theorem 3.5; 31, Theorem 101:

Let R be a Noetherian prime PI ring. If every proper factor ring of R is a QF ring then R is a Dedekind prime ring. Note that proper factor rings of a Dedekind prime ring (bounded or not) are always QF rings. This follows from Theorem 2.3 and [22, Theorem 3.51. The hypothesis in Theorem 2.7 assumes a priori that proper factor rings are Artinian. This requirement is eliminated in the next formulation. Recall that a ring R is called an ipri ring if every ideal of R is principal as a right ideal. 2.8 Theorem (13, Theorem 5.41:

Let R be a Noetherian prime PI ring. If every proper factor ring of R is an ipri ring then R is a Dedekind prime ring. A one-sided version of the above and a different proof are given in [26, Theorem 61. See also [31, Proposition 161.

The next result characterises hereditary rings which need not be Dedekind.

370

2.9 Theorem (19, Theorem; 25p.8831:

Let R be a semi-prime Noetherian PI ring such that EUI is a serial ring for each right essential (two-sided) ideal I of R . Then R is a hereditary ring. 3 Relation between Hereditary and Fully QF2 Rings 3.1 Lemma I l l , Theorem 7.71:

A prime Noetherian hereditary ring is QF2. Combining 2.4 and 3.1 we obtain: 3.2 Theorem:

A prime Noetherian hereditary ring is a fully QF2 ring. Now a well-known result of Chatters [5, Theorem 5.41 states that a Noetherian hereditary ring is a direct sum of prime rings and Artinian hereditary rings. While the prime part is QF2, the Artinian part need not be as can be seen from : 3.3 Theorem 14, Corollary 2, p. 70; 14; 211:

A hereditary Artinian QF2 ring is serial. Remark: The above, in fact holds for hereditary serni-prirnary QF3 rings. More explicitly we can demonstrate the following: 3.4 Example:

Let A be the Kronecker algebra

(i ') k:

where k is a field. Then by

[20, Proposition 5.11, A is an Artinian hereditary ring which is clearly not serial. In fact, it can be shown that A does not have finite representation type either (cf. 4.2). 0

3.5 Lemma:

A Noetherian QF2 ring has an Artinian QF2 quotient ring.

371

Proof: This follows from [5, Lemma 6.14 and Theorem 2.3(c)].

0

Hence the following theorem applies to fully QF2 rings. By the Krull dimension we mean the Gabriel-Rentschler Krull dimension for modules. 3.6 Theorem [ I , Theorem 4.51: Let R be a Noetherian PI ring. Then every factor ring of R has an Artinian quotient ring if and only if R is a direct sum of an Artinian ring and prime rings of Kmll dimension 1 . The proof of the above theorem is not as straightforward as one would expect and appears to depend on the fact [23] that a Noetherian PI ring can contain only a finite number of idempotent ideals. As Example 3.9 below shows this result is not valid without the polynomial identity assumption. Nevertheless, we point out the following:

3.7 Theorem: Let R be a Noetherian ring such that every factor ring of R has an Artinian 00

quotient ring. Then

n Jn = 0 . n=l

Proof: This will follow if we can show that [16, 8.3.161 applies. Thus we need to show that every meet-irreducible ideal I of R is primary. Factoring out by I , assume that the ideal 0 is meet-irreducible. We need to show that the module RR has a unique associated prime. Let P = r(U) be an associated prime of R where U is a right ideal of R . Let Q be a maximal ideal in the set K = {r(A) I A is a non-zero ideal of R} . Then clearly, Q is a prime ideal of R . Now since 0 is a meet irreducible ideal, we have RU n A # 0 and so P = r(U) = r(RU) G r(RU n A) = Q . Since R has an Artinian quotient ring, by [5, 1.25 and 2.3(c)], Q is a minimal prime ideal and so P = Q . An argument similar to the above shows that Q is unique in K . Thus P is the unique 0 associated prime in RR . Lemma 3.5 and Theorem 3.6 have the following consequence.

3.8 Theorem: Let R be a Noetherian PI ring. Then R is a fully QF2 ring if and only if R is a direct sum of an Artinian serial ring and prime hereditary rings.

372

Proofi Suppose that R is fully QF2. By 3.5 every factor ring of R has an Artinian quotient ring. So by 3.6 R is a direct sum of an Artinian ring and prime rings of Krull dimension 1. By 2.2 the Artinian part is a serial ring. If S is a prime direct summand of R then because S has Krull dimension 1 , every proper factor ring of S is Artinian and also a QF2 ring by hypothesis. By 2.2 it follows that every proper factor ring of S is serial. Hence by 2.9, S must be a hereditary ring. 0 We note that the above theorem is not valid without the PI assumption as the example below shows. 3.9 Example:

Let S be a simple Noetherian domain which is not a division ring. Consider the ring R = S[x]/(x2) . Since the ideal generated by the image of x is the unique ideal of R , it is easily seen that R is a fully QF2 ring. However, R is an indecomposable ring which is neither prime nor Artinian. We note that results from [24] are useful for constructing more varied examples of this type. 4 Finite Representation Type Rings

The main results of this section due to Maria Luisa Villani are from [27]. Detailed proofs will appear in [28]. 4.1 Definition:

A ring R is said to be offinite representation type if R is right Artinian and has only finitely many non-isomorphic finitely generated indecomposable right modules. [7, Theorem 1.21 shows that this definition is in fact left-right symmetric. Note that if R is a ring of finite representation type then so is any factor ring of R . The next result in the Artinian case is well known. 4.2 Theorem:

An Artinian serial ring is of finite representation type.

373

Proofi Let R = elR 0 e2R 0 ... 0 enR be a decomposition of R where the eiR are uniserial modules. Let M be a finitely generated indecomposable right Rmodule, Then by [29, 1.3, 2.6 and 3.41, M 2 eiR/eiJk for some i, k 2 0 . Since each eiR is an Artinian module, it follows that up to isomorphism, there are 0 only finitely many choices for M . The converse is not true as the following example which is a standard construction shows. 4.3 Example:

Let k be a field and k(x) the field of rational functions in the indeterminate x . Define a homomorphism o:k(x) + k(x) given by o(a) = a for all a E k and o(x) = x2 . We can define a right action for the module k(x)k(x) by f g = fo(g) for f , g E k(x) . Thus Ik(x), is a k(x)-k(x)-bimodule where the right action is as above and the left action is the natural one. Now consider the trivial extension ring k(x) lk(X),

[

1.

0 " k(x) R= Here the entries in the (1,l) and (2,2) position of R are the same. The ring R is an Artinian scalar local PI ring. Using the theory of quivers, R can be seen to be a ring of finite representation type. However R is not right serial since 0

Remark: Note, however, that for scalar local Artin algebras the two concepts can be shown to be equivalent. In the interests of brevity, we make the following definition. 4.4 Definition:

The ring R is said to have property (P) if every proper Artinian factor ring of R is of finite representation type. The starting point of this investigation is the following theorem of Gustafson which deals with a classical setup. 4.5 Theorem [IO, Theorem 3.31:

Let C be a complete discrete valuation ring with field of fractions K . Let Q

be a finite dimensional separable K- algebra and R a C-order in Q . Then R is hereditary if and only if R has property (P). 4.6 Theorem (28, Theorem 0.11:

Let R be a Noetherian PI ring which is not Artinian. Suppose that R has an Artinian quotient ring. Then the following are equivalent. 1. R has property (P). 2. R is a direct sum of an Artinian ring of finite representation type and hereditary Noetherian prime rings. The above requires a highly sophisticated method of proof using ideas fkom different strands of ring theory. A key idea is to apply localisation and completion techniques to reduce the problem to a situation where Theorem 4.5 becomes applicable. We refer the reader to [9] for an exposition of clique theory. We highlight below some of the main steps required. 4.7 Proposition /28, Theorem 4.111:

Let R be an indecomposable Noetherian PI ring satisfying property (P). Then all cliques of maximal ideals of R are finite. Thus in this case the clique of a maximal ideal is localisable and the localised ring is semi-local. Another crucial idea is to show that factor rings involving certain 2x2 upper triangular matrices of a certain type cannot occur under the given assumptions. This is done by generalising a method of S.Brenner [3]. 4.8 Proposition /28, Theorem 1.101:

Let D be a scalar local prime PI principal right and left ideal ring. Then the ring

(: ):

has a factor ring which is of infinite representation type.

Finally, the next theorem extends those of Section 2 to the non-semiprime case. 4.9 Theorem [28, Theorem 0.21:

Let R be a Noetherian PI ring. Then the following are equivalent. 1. Every Artinian factor ring of R is serial. 2. R is a direct sum of an Artinian serial ring and hereditary prime rings.

375

References 1. E.P.Armendariz and C.R.Hajamavis, Noetherian rings whose factor rings are orders in Artinian rings, J. London Math. SOC. (2) 34 (1984), 12--16. 2. K.Asano, a e r Hauptidealringe mit Kettensatz, Osaka J. Math. 1 (1 949), 52-6 1. 3. S.Brenner, Large indecomposable modules over a ring of 2x2 triangular matrices, Bull. London Math. Soc, 3 (1971), 333--336. 4. A.W.Chatters and C.R.Hajamavis, Rings in which every complement right ideal is a direct summand, Quart. J. Math. Oxford (2) 28 (1977), 61-40. 5. A. W.Chatters and C.R.Hajarnavis, Rings with chain conditions, Research notes in mathematics 44, Pitman advanced publishing program, London (1980). 6. D. Eisenbud, and P. Griffth, Serial rings, J. Algebra 17 (1971), 389--400. 7. D.Eisenbud and P.Griffith, The structure of serial rings, Paclfic J. Math. 36 (19711, 109-121. 8. A.W.Goldie, Torsion-free modules and rings, J. Algebra 1 (1964), 26%287. 9. K.R.Goodear1 and R.B.Warfield, An introduction to non-commutative Noetherian rings, London Math. SOC. Student Texts 16 Cambridge University Press, Cambridge 1989. 10. W.Gustafson, Hereditary orders, Comm. Alg. 15 (1987), 219--226. 11. C.R.Hajamavis, Orders in QF and QF2 rings, J. Algebra 19 (1971), 329-343. 12. C.R.Hajamavis, Non-commutative rings whose homomorphic images are self-injective, Bull. London Math. Soc. 5 (1973), 70--74. 13. C.R.Hajamavis and N.C.Norton, The one and half generator property in Noetherian rings, Comm. AZgebra 9 (1981), 381--396. 14. M.Harada, On QF3 and semi-primary pp rings I, Osaka J. Math. 2 (1965), 357--368. 15. M.Ikeda, Some generalisations of quasi-Frobenius rings, Osaka J. Math. 3 (1951), 228--239. 16. A.V.Jategaonkar, Localisation in Noetherian rings, London Math. SOC. Lecture Notes Series 98 Cambridge University Press, Cambridge 1986. 17. H.Kupisch, Beitrage zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung, , I Reine Angew. Math. 201 (1959), 100--112. 18. L.S.Levy, Commutative rings whose homomorphic images are selfinjective, Pacific J. Math. 18 (1966), 149--153. 19. L.S.Levy and P.F.Smith, Semiprime rings whose homomorphic images are serial, Canad. J. Math. 34 (1982), 691--695. 20. J.C.McConnel1 and J.C.Robson, Non-commutative Noetherian rings, Purre and Applied Mathematics, Wiley-Interscience, New York 1987. 21. H.Y.Mochizuki, On the double commutator algebra of QF3 algebras, Nagoya J. Math. 25 (1965), 221-230. 22. J.C.Robson, Non-commutative Dedekind rings, J. Algebra 9 (1968), 249-265.

376

23. J.C.Robson and L.W.Smal1, Idempotent ideals in PI rings, J. London Math. SOC.14 (1976), 120--122. 24. R.C.Shock, Polynomial rings over finite dimensional rings, PacrJic J. Math. 42 (1972), 251--257. 25. S.Singh, Modules over hereditary Noetherian prime rings, Canad. J. Math. 27 (1975), 867--883. 26. P.F.Smith, Rings with every proper image a principal ideal ring, Proc. Amer. Math. SOC.81 (1981), 347--352. 27. M-L.Villani, Hereditary rings and rings of finite representation type. Thesis, University of Warwick, (1999). 28. M-L.Villani, A new characterisation of hereditary PI rings. To appear. Israel J. Math. 29. R.B.Warfield Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), 187-222. 30. D.W.Wal1, Characterizations of generalized uniserial algebras, Trans. Amer. Math. SOC.90 (1959), 161--170. 3 1. A.Zaks, Some rings are hereditary rings, Israel J. Math. 10 (1971), 442-450.

MODIFIED RSA CRYPTOSYSTEMS OVER BICODES

K. HASHIGUCHI, K. HASHIMOTO AND S. JIMBO Department of Information Technology Faculty of Engineering Okayama University, Tsushima Okayama, 700-0082, Japan E-mail: [email protected]

Let C be a finite nonempty alphabet, and C* the free monoid generated by C. A finite sequence ((~I,~~),(Iz,Y~),...,(z~,Y~)) (n 2 l , x ; , y i E C') of pairs of words is a bicode if the following holds : for any i l , . ,i,, j1,. . . ,j , ( p , q 2 1 , l 5 i k , j L 5 n ) , if xil " ' X i , y i , " ' y ~ l = xjl . . ' xi, yj, . . . yj, , then p = q and i k = j k for all I 5 k 5 p . In this paper, we present three families of public-key cryptosysterns which consist of bicodes and whose encryption and decryption keys depend on RSA cryptosystems. In each family, encryption and decryption of messages are done by depending on RSA cryptosystems, and sending messages is done in the form of bicodes. In each such cryptosystem, we use many units of public-key systems in the style of RSA cryptosystems.

1. Introduction There have been presented and studied many families of one-channel codes such as block codes, variable length codes, error correcting codes, etc. Onechannel coding theory is very deep and large, practical and theoretical, and is related with algebra, combinatorics and formal language theory. There exists also multi-channel coding theory. This theory concerns generally notions of information theory such as entropy, transmission rate, noise, channel capacity, distortion rate, etc. It will be very interesting if one can develop multi-channel coding theory which is related with algebra, formal language theory and combinatorics. The intention of our study is to develop such theory. In [4], we introduced a family of new codes named bicodes. Let C be a finite nonempty alphabet, and C' the free monoid generated by C. A bicode is a finite sequence of pairs of words, 2 = ( ( X I , yl), . . , (xn, yn)) ( n 2 1,xi,yi E C' ) such that for any p , q 2 1 and 1 5 i l , iz , . ' . ,i,, j~ ,j z , . . . ,j q I n, if IC;, . . . xi,yi, . . yil = xj, * . . xj,yj, . . . yjl , then it holds that p = q and

-

377

378 = j k for all 1 5 k 5 n. Any bicode can be used as a two-channel code. Over the first channel, xil . . . xi, may be sent, and over the second channel, y t . * . y e may be sent : or if xil ' . ' xi, is shorter than y t y& then over the first channel, xil ' . . xi, yip . . . yi, may be sent, and over the second channel, y t . . . may be sent so that the lengths of xil . . . xi, y i p . . . yij and y e . * .yc-l are almost the same ; when y t * . ' y e is shorter than xil . . . xi,, then the roles of words may be changed. (Here for any word w = a l . . . a , ( n 2 0,ai E C), w R is the reverse of w, w R = a,...al). Or for the purpose of cryptography, any prefix of xil . . . xi,yi, . . . yil may be sent over the first channel, and the rest subword may be sent over the second channel. In any of these cases, the reciever can detemine uniquely the sequence, (il, . . , i,). The total message xil . . . xi, yip yil may be sent over the first channel so that bicodes can be used also as one-channel codes. In [4], we also introduce the notion of two-channel codes. A two-channel code is a finite sequence of pairs of words 2 = ( ( qyl), , . . . , (x,, yn)) ( n 2 1,xi, yi E C*) such that for any p , q 2 1 and 1 5 i l , i 2 , . . . , i,, jl,j2,.. . ,j , 5 n, if xil . * * xi, = xjl . . . xjq and yil . . . yip = yjl . . . yj,, then it holds that p = q and i k = j , for all 1 5 k 5 n. In [4], the following two problems are shown t o be undecidable. zk

+

+

-

+

The binary bicode problem : when given any finite sequence of pairs of words, 2 = ( ( x l , y l ) , . . . ,( x n , y n ) )( n 2 l , x i , y i E E*), where C = { a , b } is a binary alphabet, decide whether or not Z is a bicode. The binary two channel code problem : when given any finite sequence of pairs of words, 2 = ( ( x ~ , y l ) , . . . , ( x , , y n ) )( n 2 l , x i , y i E C*), where C = { a ,b } is a binary alphabet, decide whether or not 2 is a two channel code. The RSA cryptosystem was proposed in 1977 by Rivest, Shamir and Adleman. and is one of the most well known public-key cryptosystems. In this paper, we present three families of modified RSA cryptosystems. In each family, encryption and decryption of messages are done by depending on RSA cryptosystems, and sending messages is done in the form of bicodes. In each such cryptosystem, we use many groups of public-key systems in the style of RSA cryptosystems. Because we use many keys for encryption and decryption, these proposed cryptosystems will be strong against attack of cyphering. We hope that our proposed cryptosystems or their extensions will be used over communication lines in which secrecy should be kept in

379

very high levels. 2. The RSA cryptosystem

In this section, we shall overview some basic part of the RSA cryptosystem. Throughout this paper, a natural number means a positive integer. For a finite set A , IAl denotes the cardinality of A . The empty set will be denoted by @. For a natural number n, let (Z,, +, X ) denote the ring whose operations are done under (mod n ) ,and 2, = (0,1,. . . ,n - 1). We often denote this ring simply by 2,. Then 2 : denote the set, { a E 2, 1 a is relatively prime to n}. The Euler function 4 is defined by : for any n 2 1, 4(n)= IZ;I.The following follow easily by definition.

Fact 1. Let p and q be two distinct primes. (1) 4 ( P ) = P - 1. (2) 4 ( p e ) = pe-'(p

(3)

-

1) for e

21

4 ( P . 4 ) = ( P - l ) ( q - 1).

The following theorem (Euler's theorem) and corollary are well known.

Theorem 2.1. For any relatively prime two positive integers n and a, a@(n)s 1 (mod n ). Corollary 2.1. For a prime p and a positive integer a which is prime t o p , upF13 1 (mod p ) . Definition 2.1. One unit of the RSA Cryptosystem is a quintuple p , q, e , d , n > which satisfies the following. (1) p and q are two distinct primes. (2) e and d are two distint positive integers satisfying : e (mod I c m ( p - 1,q - 1)). (3) n = p . q .

.d

G

<

1

Definition 2.2. Generally one RSA cryptosystem consists of some k 2 2 units, Ui =< p i , q i , e i , d i , n i > (1 5 i 5 k ) . Each unit Ui =< p i , qi, e i , d i , ni > satisfies the above condition as a unit. In the sequel, we consider cryptosystems over the binary alphabet C = {0,1}. A plain text w is a sequence of symbols w = 0 1 0 2 . . . a, ( r 2 1,ai E C), and consists of several blocks of length n 2 1. Thus w = blb2 . . . b,, s 2

380

1, bi E C n (1 5 i 5 s). Consider any block M of w, and let C be the encrypted word of M . Then the following hold.

Encryption key ( e , n ) and Encryption : C M e (mod n ) Decryption key ( d ,n ) and Decryption : M z Cd (mod n )

Theorem 2.2. If the triple of positive integers ( e ld , n ) satisfies the condition in Definition 2.1, then the following holds : ( M e ) d= M (mod n ) . When one has k units for the SRA cryptosystem, for the strongness of the system, one generally'needs the folloing (i)-(iii) : (i) ni # nj (1 5 i < j 5 k ) , (ii) ei should be sufficiently large, and (iii) ei # ej (1 5 i < j 5 n). For the computational complexity and the safety problem of the RSA cryptosystem, see the literature (e.g., [1][5]). 3. Bicodes

In this section, we present some part of results presented in [4] about bicodes. First we recall some results on (finite) codes. 3.1. Codes

Let C be a finite alphabet. The free monoid generated by C is denoted by C*. The null word will be denoted by A, and C+ is C* - {A}. For x,y,z E C*, x is a prefix of xy, z is a suffix of yz, and y is a factor of xyz. In this paper, one channel code means a finite code. Generally a finite code is a finite set of words. But in this paper, for notational convienience, we employ definitions as follows.

Definition 3.1. Let seq(C*) denote the set of finite (nonnull) sequences of words over C. For any X = ( X I , . . . , x,) E seq(C*), it holds that n 2 1 and for all 1 5 i 5 n, xi E C*. The length of X is n and denoted by

1x1.

Definition 3.2. A code is a sequence X = (xl,+..,xn) E seq(C*) such t h a t f o r a n y p , q > l a n d i l , . . . , i , , j l , . . . , j g ( I < i k , j l 5 n ) , i f z i , . . - x i p= xjl ...xjq, then p = q and for all 1 5 k 5 p , i k = jk. Each xi is called a code word of X . The following theorem is well known : see, e.g., [2].

Theorem 3.1. For any given sequence X = (x~,..*,z,) E seq(C*), one can eflectively decide whether or not X is a code.

38 1

3.2. Properties of bicodes

For sending mechanisms of bicodes, see Section 1. In this subsection, we shall present definitions and some properties of bicodes. Definition 3.3. Let seq(C* x C*) denote the set of finite ( nonnull) sequences of pairs of words over C. For any 2 = ( ( x ~ , y l ) , . . . , ( x n , y n )E) seq(E* x E"), it holds that n 2 1 and for all 1 5 i 5 n, x;,yi E C*. The length of Z is n, and denoted by 121. Definition 3.4. For any Z = ( ( x ~ , y ~ ) , . . . , ( x ~ ,Ey ~seq(C* )) x C"), Z(+) denote the set of words, {xil . . ' x i p y i p. . . y i l I p 2 1 and for any 1 5 k 5 p , 1 5 i k 5 n}. z(+) is called the set of (nonnull) words generated by z.For any i l , . . . , ip,j$,. . . ,jg( p , q 5 1,1 2 i k , j l 2 n ) , if w = xil ... xipyip. . . yil = xj1 . . . xjqyjq. . ' y j l , then we write w G~ (xil,.. . , xi,, yip,. . . ,yil) -z ( x j l , . ' ' , xi,,yj,, . * . ,yjl ). If Z is clear from the context, then instead of -=, we write =. We put Z(*) = Z(+) U {A}. The following proposition is well known in formal language theory.

Proposition 3.1. For any Z E seq(C* x C*), Z(+) is a linear context-free language.

Definition

Example 3.1. Let C = {a,b,c},Z1 = ( ( a b , a ) , ( c , a ) , ( a , a )(bc,a)) , and 2, = ( ( a , a ) ( a b , a ) (, a b c , ~ ) ) .Then 21 is not a bicode because abcaa = (ab,c, a , a ) = ( a ,bc, a , a ) . But

2 2

is a bicode which can be seen easily.

Definition 3.6. The binary bicode problem. When given any Z = ((xl,yI), . . . , (x,,y,) E seq(C* x C"), where C = { a , b}, decide whether or not 2 is a bicode. The following theorem was proved in [4] by reducing the Post correspondence problem to the binary bicode problem.

Theorem 3.2. The binary bicode problem is undecidable.

382 4. Public key cryptosystems over bicodes

In this section, we propose three families of public key cryptosystems whose encryption and decryption depend on RSA cryptosystems, and whose sending mechanism depends on bicodes. As stated in Section 1, sending can be done both over one channel and over two channels.

Remark 4.1. In the following, in each cryptosystem, we use several RSA cryptosytems, i.e., (n~,el,d1),(n2,e2,d2),..., (nm,em,d,) ( m 2 2) where for 1 5 i 5 m, (ni,ei,di) is a unit of RSA cryptosystem as in Theorem 2.2. If n1 < 722 < ... < n,, then one can define a new cryptosystem (nl, el,d l , n2,e2, d2,.. ' ,n,, e m , d,) such that one plain text w is such that 1wI < n1, and encryption and decryption of w are done by creating the following words, v1 , v2 , . . ' ,v,, and w1,w2,. . ' ,w, : (1) (i) vo = w and (ii) for each 1 5 i 5 m, vi = vzell (mod ni) ; ( 2 ) (i) w, = 21, and for each 1 5 i 5 m, wi-1 wfi (mod ni) and (ii) wg = w

In this system, one needs the relation n1 < 722 < . ' < n,. In the following, we can use any combination of (121, el,dl), (122, e2,dz), . . . , (nm,e m , d m ) , and so from this collection of keys, we can use all m! combinations of keys. +

In the following, for any bicode 2 = ((zl,yl),... , (zn,y,))E seq(C* x C*) with (i) lzil = Izjl and (ii) lyil = 1yj(for all 1 5 i , j 5 n, we say that the X-length of 2 is 1x11. For notation, let z = a1a2. a, E C+ (ui E C) be a sufficiently long word. Then hp(z) denotes "the half prefix "of z, h f ( z ) = a1 . . . ap, and h s ( z ) denotes the half suffix of 2, h s ( x ) = ap+l . . . a,, where P = TmPl. 4.1. Cryptosystem 1 Cryptosystem 1 is the simplest among all cryptosystems proposed in this section. In this cryptosystem, we prepare two sets of keys. By the first keys, we encode X-length 1 of the bicode which will be used to send the plain text by using the second keys over the bicode whose X-length is 1. The Public Key Cryptosystem 1 consists of the following terms. ( I ) A group of z (2 2 ) persons, (Al,...,A,} . (2) For each 1 5 i 5 2 , A* has eight parameters (numbers) pil,qil ,pi2,qi2, eil,dil,ei2,diz which satisfy Enc-Dec-Method 1

383

(encryption-decryption method 1) below. For 1 5 i < j 5 z, the sets of these parameters of Ai and Aj are disjoints. (3) For any 1 5 i 5 z, Ai makes the parameters, (ei1,nil) and (ei2,ni2) be open, where nil = pi1 . qil, ni2 = pi2 . qi2. (4) Sending a message M to Ai, one uses Enc-Dec-Method 1 to get the encrypted sentence C. (5) When Ai receives the encrypted message C, he uses Enc-DecMethod 1 for decrypting c t o get M .

4.1.1. Enc-Dec-Method 1 This method consists of the following eight pa,rameters. 0 0

Four distinct prime numbers pl , 91, pz, q2. Four distinc positive integers el, dl, e2, d2 which satisfy the following. el . dl e2

. d2

1

(mod 1cm(p1 - 1, q1 - 1))

=1

(mod Icm(p2 - 1,qz - 1))

3

Over the alphabet C = (0, l}, for any block code U = ( u l , . . . ,urn), encryption and decryption will be done as follows, where we assume that ui E C+ and 1uil = min{[log2(nl - 1)1,[log2(n2 - 1)1}(1 5 i 5 m ) ,where we put pl . q1 = n1 and p2 . q 2 = 722. The method conisists of the following steps.

(1) First the sender chooses some positive integer 1 with 1 5 1 5 min{ [log(nl - I)], rlog(n2 - 1)1}- 1. (2) The sender computes 1"' (mod n l ) by the key e l , and constructs a word w E C+ whose length is rlog(n1 - 1)1 and which is a binary sequence denoting lel (mod n l ) . 1. (3) The sender wants to send a message uilui2 . . . zli, (s 2 1,1 5 ij <_ m). (4) Define a bicode Z ( U , 1 ) as follows. Z(V,1) = ~

~ ~ O , ~ O ~ ~ ~ ~ l , Y 1 ) ,. ,(( z~m2, Y , rYnZ) )~ , . ~

where (i) w = zOyo, zo = hp(w),yo = h s ( w ) , and (ii) for each 1 5 i 5 m, the following holds.

384

ziyi is a binary sequence denoting u:' (mod n2) such that its length is rlog2(n2 - 1)1 , lzil = 1 and lyil = [log,(nz - 1)1- 1. For each 1 5 i 5 m, we put X ( u i ) = zi and Y ( u i )= yi. (5) Instead of the message uilui2 . . . ui,, send the following. z0X(~i1)X('ILi*) ' * . x(~i,)y(~is)y(ui,-l) * .Y('LLi1)Y0 (6) The reciever knows 1 from zoyo by the key d l . Then by the key dz, he obtains the plain text Uil U i 2 . . . uia.

4.2. Cryptosystem 2

In this system, the integer 12 in Cryptosystem will be encrypted in several blocks. One prepares five sets of keys. The keys in the first set will be used for decrypting informtion (length 11 in 4.2.1 below) about the keys of the second, the third and the fourth sets. The keys in the second set will be used for defining the number T of dummy blocks (number r in 4.2.1 below). The keys in the third set will be used for indicating the integer b (1 5 b 5 r ) such that the inforamation about 12 will be contained in the b-th block from the left. The fourth keys will be used for determining the length 12 such that the X-length of the bicode over which encryption by the keys in the fifth set will be done. Thus the whole encrypted message consists of the message encrypting the above 11, the message encrypting the above r , the r messages encrypting the above b and the message encrypting the plain text. Cryptosystem 2 consists of the following terms. (1) A group of z (2 2) persons, { A l , . . . ,A z } . (2) For each 1 5 i 5 z, Ai has twenty parameters (numbers) pil, q i l , pi2, qi 2, pi3, qi3, pi4, qi4, p i s , q i 5 , eil, 41,ei2,di2, ei3,di3, ei4, di4 ,ei5 7 4 5 which satisfy Enc-Dec-Method 3 below. For 1 5 i < j 5 2, the sets of these parameters of Ai and Aj are mutually disjoint. (3) For any 1 5 i 5 z, Ai makes parameters, ( ( e i ~ , n i l )(ei2,72i2), , ( e i 3 , ~ )( e, i 4 , 7 4 , (ei5,72i5)) be open, where for j = 1 , 2 , . . . ,5, nij = pij . q i j . (4) In sending a message M to Ai, one uses Enc-Dec-Method 2 to create the encrypted message C. (5) When Ai receives the encryted message C , he uses Enc-Dec-Method 2 for decrypting C to get M .

385 4.2.1. Enc-Dec-Method 2 This method consists of the following twenty parameters.

f o r i = 1 , 2 , . . . ,5, ei . di = 1 (mod lcm(pi - 1,qi - 1)) Over C = (0, l}, for any block code U = (211,. . . , u r n )with ui E C+, luil = rlog2(p5q5 - 1)1 (1 5 i 5 m ) , one sends messages as follows, where for j = 1 , 2 , . . . ,5, p j . qj = nj. The method consists of the following steps.

(1) The sender chooses a positive integer 21 with 1 5 11 5 min{ [log(nj1)1 I 1 5 j 5 5) - 1. (2) The sender computes 2;' (mod 721) by the key e l , and constructs a word w1 E C+ whose length is rlog(n1- 1)1 and which is a binary sequence denoting 1:' (mod 721). (3) The sender chooses a positive integer T with 1 5 r 5 min{ rlog(nj 1)1 I 1 5 j 5 5) - 1. (4) The sender computes rez (mod 7 2 2 ) by the key e2, and constructs a word w2 E C+ whose length is rlog(n2 - 1)1 and which is a binary sequence denoting re* (mod 722). (5) The sender chooses a positive integer b with 1 5 b 5 r. (6) The sender computes b"3 (mod n3) by the key e3, and constructs a word w3 E C+ whose length is rlog(n3 - 1)1 and which is a binary sequence denoting beg (mod 723). (7) Finally the sender chooses a positive integer 12 with 1 5 12 5 min{ [log(nj - I)] I 1 5 j 5 5) - 1. (8) The sender computes 2;4 (mod n4) by the key e4, and constructs a word W b E C+ whose length is rlog(n2 - 1)1 and which is a binary sequence denoting l? (mod n4). For i = 1 , 2 , 3 , put z g i = hp(w;) and yoi = hs(wi), and put x k b = hp(wb) and Y k b = h S ( W b ) . (9) The sender constructs T - 1 binary sequences w1,.. . , Wb-1, W b + l , . . . ,wT (with length rlog(n4 - l)]), randomly, and for each 1 5 j 5 r , ( j # b ) , put x k j = hp(wj) and y k j = hs(wj). (10) The sender wants to send a message uilui2. . . uia (s 2 1,1 5 ij 5 m).

386

(11) Define a bicode Z ( U , 1) as follows.

z ( U ,1 )

((zoi, Yo1 ), ( 2 0 2 , Y02),(J:03,903)(zkl7 Ilkl) I 1

(xk,,Yk,), (21, Yl),(22,Y2)l.. . I

(Zkz1 Y k 2

1,

' ' '

( x m ,Ym))

where for each 1 5 i 5 m, the following holds. xiyi is a binary sequence denoting (mod ng) such that its length is rlog,(ng - l)],12il = 12 and Iyil = rlog2(n5 - 1)1- 12. For each 1 5 i 5 m,we put X(ui)= zi and Y(ui)= yi. (12) Instead of the message uiluiz . . . ui,, send the following. 2012022032k12kz

"

'~k,x(uil)x(Uiz). . x(Ui,)

y(uis)y(%s-l)...Y(ui~)Yk, ' ' 'YkzYklY03Y02YOI

(13) The reciever knows 11 from zoly01 by the key dl, knows T from sozyo~ by the key d2, knows b from 203y03 by the key d3, and knows 12 from X k b y k b by the key d4. Then by the key dg, he obtains the plain text uiluiz.. . uis from X(uil)X(uiz) * . * X ( ~ i , ) Y ( ~ i s ) Y ( ~ i* s * *Y(ui1). l)

4.3. Cryptosystem 3 In this cryptosystem, for encrypting a plain text over any block code consisting of m (> 1) code words, one prepares 3 3t ( t 2) sets of keys, and encrypts all code words of the plain text and their X-lengths by using these 3t sets of keys cyclically. The keys ( t keys) in the third block of sets will be used for decrypting the given plain text by the style of a bicode, and the keys in the first and the second blocks will be used for encrypting and decrypting the X-parts and the Y-parts of the bicode. Thus the whole encrypted message consists of the message encrypted by the initial three keys, the message encrypting the X-parts of messages over the bicode and the message encrypting the Y-parts of the bicode.

+

>

Cryptosystem 3 consists of the following terms. (1) A group of J: ( 2 2) persons, { A l l . . . ,A z } . (2) For each 1 5 i 5 2, A; has 4 ( 3 3 t ) parameters (numbers)

+

Pi1 7 Q i l ,

Pi2,4i21 Pi37 Qi3, Pill,Qill

Pi21 I 4i21 i

' ' '

> PiZt 7

,

I ' ' ' 7

4i2t Pi31 1 qi317

'

'

'

Pilt,4ilt

> PiJt,qi3t

~ i l ~ ~ i l ~ ~ i 2 ~ ~ i 2 ~ ~ i 3 ~ ~ i 3 ~ i l l ~ ~ i l l ~ ~ ~

387 e i z ~ , d i z l ., *

., e m , d i z t , ei31, di31, . . , *

di3t

which satisfy Enc-Dec-Method 3 below. For 1 5 i < j 5 z, the sets of these parameters of A i and Aj are mutually disjoint. (3) For any 1 5 i 5 x, A i make parameters, ((GI

7

nil

(ei2 n i z ) , (ei3 72i3), ( e i l I 1 nil 1 )

)I

(eiZ1,%21),"'

I

7

. . ., ( e i ~ tn,i ~ t ) ) ,

( e i 2 t , % 2 t ) , (ei31,%31),"'7

(ei3t,ni3t))

be open, where (i) for j = 1 , 2 , 3 , n i j = p i j . q i j and (ii) for k = 2 , 3 , 1 5 i 5 X and 1 5 j 5 t , n i k j = Pikj . Q i k j . ( 4 ) In sending a message M to Ai, one uses Enc-Dec-Method 3 below to create the encrypted message C. (5) When A; receives the encryted message C , he uses Enc-DecMethod 3 for decrypting C to get M . 4.3.1. Enc-Dec-Method 3

This method contains the following 4(3 0

2(3

+ 3t) (t 2 2) parameters.

+ 3t) (t 2 2) distinct prime numbers, Pl 7 (71, P 2 , 4 2 , P 3 , q3

Pll,qll, '

2(3

' '

,Plti q l t , ' ' ' , P 2 1 , q217

PZmr qZt, ' ' ' 7P31, q31, '

* ' '

' '

jP3tr q3t

+ 3t) distinct positive numbers, el,

4 ,-52, dz, e3, d3

. . ,elt, d l t , . . . ,e21, & I , .

ell,~ I I , .

. . ,e2t, d 2 t , .

.

e31, d 3 1 , .

. . ,e3tl d3t

which satisfy the following.

f o r j = 1,2,3,

ej

.dj

=1

(mod Icm(pj - 1, g j - 1))

for i = 1,2,3and 1 5 j

(mod

km(pij

5t

;

eij . d i j 2

1

- 1, qij - 1))

Over C = {O,l}, for any block code U = (u1, ..., urn) with ui E C+, 1ui[= min(rlog(nzj - 1)1 I 1 5 j 5 t } (1 5 i 5 m ) , one sends messages as follows, where for j = 1 , 2 , . . . ,t , p2j . q z j = n z j . The method consists of the following steps. ( 1 ) The sender chooses a positive integer min{ rlog(nlj - I)] 1 1 5 j 5 t } - 1.

11

with 1

5

11

5

388

(2) The sender computes lyl (mod nl) by the key el, and constructs a word w1 E C+ whose length is rlog(n1- 1)1 and which is a binary sequence denoting lyl (mod n l ) . (3) The sender chooses a positive integer 12 with 1 5 12 L min{ [log(nlj - 1)1 I 1 5 j 5 t } - 1. (4) The sender computes 1g2 (mod n2) by the key e2, and constructs a word w2 E C+ whose length is rlog(n2 - 1)1 and which is a binary sequence denoting 1;' (mod n2). (5) The sender chooses a positive integer l3 with 1 5 l3 5 min{ rlog(n1j - I)] I 1 5 j 5 t } - 1. (6) The sender computes lg3 (mod 723) by the key e3, and constructs a word w3 E C+ whose length is rlog(n3 - 111) and which is a binary sequence denoting lg3 (mod n3). For i = 1,2,3, put xi = hp(wi) and yi = hs(wi). (7) The sender wants to send a message M = uilui2, . . ui,(s 2 1,1 5 i j 5 m). (8) Define a bicode Z(U,13, M ) as follows.

z ( U ~ 1 3 ,=~ ((231iy31),(232,Y32)r...,(x3s,Y3s)) ) where for each 1 5 j 5 s, the following holds. x3jy3j is a binary sequence denoting ~ ; ~ 3 * 3 (mod n3j) such that (i) 1 -< kj 5 min{ij, t } and k j = ij (mod t l), (ii) its length is r1og2(n3kj - 1)1 , and (iii) 1x3jl = l3 and ly3jl = [log2n3kj] - 13. For each 1 5 j 5 s, we put X(uij) = 53j and Y ( u i j )= y3j. (9) Then for each 1 5 j 5 s, let z l j y l j be a binary sequence denoting ( ~ 3 j ) ~ l (mod ~ j n l j ) such that lxljl = 21 and lyljl = [log2nlkj] 11. (10) Now for each 1 5 j 5 s, let x2jy2j be a binary sequence denoting ( ~ ~ j ) ~ '(mod * j such that IxZjl = l2 and lyZjl = [log2n2kjl -

+

12.

(11) Instead of the message ui,u;,

. . . ui,, send the following.

Y3Y2Y1

(12) The reciever Ai knows 11 from xlyl by the key dl, knows 12 from x2y2 by the key d2, and knows l3 from x3y3 by the key d3, (13) Then he knows x l l . . . x l s y l s . . ' y l l , and for 1 5 j 5 t , he knows x3j by the key dlkj.

389 2 2 1 . . . x ~ ~ y. '2y21, ~ . and for 1 5 j 5 t , he knows by t h e key d 2 k j . (15) Finally he obtains the plain text ui,ui2 . . ui3 from X ( u i l ) X ( u i 2 )... X ( U ~ , ) Y ( ~ ~ , ) Y (. U . Y~( u~i-l )~by) .the keys,

(14) Then he knows y3j

+

d31, d32r

' ' '

d3t.

5. Conclusion In this paper, we propose new cryptosystems by combining the notions of RSA cryptosystems and bicodes. These are really new cryptosystems, and can be used both as one-channel cryptosystems and two-channel cryptosystems. From the above proposed cryptosystems, one can see that one can extend such cryptosystems into very wide and intrigued hierarchies of cryptosystems. If there exist very important communication lines whose secrecy should be kept very significantly, then we hope that our proposed cryptosystems or their new descendants will contribute t o the communication over these lines.

References 1. F.L. Bauer, Decrypted Secrets : Methods and Maxims of Cryptology, Springer,

1997 2. J. Berstel and D. Perrin, Theory of Codes, Academic Press, 1985 3. M.A, Harrison, Introduction to Formal Language Theory, Addison-Wesley,

1978 4. K. Hashiguchi and T. Mizoguchi, Introduction to bicodes, Algebraic Engi-

neering (Proceedings of The International Workshop on Formal Languages and Computer systems, Kyoto, Japan 18-21 March 1997 and The First International Conference on Semigroups and Algebraic Engineering, Aim, Japan 24-28 March 1997), C.L. Nehaniv and M. Ito eds, World Scientific, 1999 5. D.R. Stinson, Cryptography : Theory and Practice, Chapman & Hall/CRC, 2002

SOLID BURST ERROR DETECTING CYCLIC CODES Sapna Jain Department of Mathematics, Miranda House, University of Delhi, Delhi 110 007, India (e mail ID: [email protected]) Abstract. In this paper, we study cyclic codes detecting a subclass of open loop bursts viz. solid open loop bursts and a subclass of closed loop bursts viz. solid close-closed loop bursts. A comparative study of the results obtained in this paper with the earlier known results has also been made. 1. Introduction. Burst errors are the most common type of errors that occur in several communications channels. Codes developed to detect and correct such errors have been studied extensively by many auhtors. The most successful early burst error correcting codes were due to Fire (1959). Fire in this report gave the idea of open and closed loop bursts defined as follows: Definition I. An open burst of length b is a vector all of whose nonzero components are confined to some b consecutive components, the first and the last of which is nonzero.

Definition 2. A closed loop burst of length b ia s vector all of whose nonzero components are confined to some b consecutive positions components, the first and the last of which is nonzero and the number of positions from where the burst can satrt is n (i.e. it is possible to come back cyclically at the first position after the last position for enumeration of the length of the burst). Definition 2 of closed loop burst can also be formulated mathematically on the lines of Campopiano (1962) as follows: Definition 2a. Let Vn(q)be the set of all ordered n-tuples with components belonging to GF(q). Let X = (ao,a l , ..., an-l) be a vector in V n ( q ) .Then X is called a closed loop burst of length b, 2 5 b 5 n, if 3 an i , 0 5 i 5 n - 1such that ai.aj # 0 where j = (i b - 1) modulo n aj+1 = aj+2 = ... = az-1 = 0 ifi > j , and a0 = a1 = ... = ai-1 = aj+l = aj+z = ... = a,-l = 0 if i < j '

{

+

390

391

It is well known that in an ( n , k ) cyclic code a burst (according to Definition 1) of length (n- k ) or less is always detectable. However, there are several bursts of length greater than n - k which go undetected. Two known results in this direction are stated below:

Theorem A . No code vector of an ( n , k ) cyclic code is a burst of length n - Ic or less. Therefore, every (n, k ) cyclic code can detect any burst of length n - k or less. Theorem B. The fraction of bursts of length b>n-k that can be undetected by any (n,k ) cyclic code is 4-

(n- k- 1)

+1 if b > n - Ic + 1 if b = n - k

( 4 - 1)

q-(n-k)

For the proofs of Theorem A and B, one may refer to Peterson and Weldon (1972, pp. 229-230). Some implications of Theorem B have been pointed out by Brown and Peterson (1961). An application of these ideas to a practical error detection problem has been described by Fontaine and Gallagher (1960). Based on the above defintions, Dass and Jain(2001) defined closeclosed loop bursts, open-closed loop bursts and proved the results for closeclosed loop bursts. The definitions and the results proved by Dass and Jain (2001) are as follows:

Definition 3. Let X = (ao,al, ..., a,-l) be a vector in V n ( q ) ai , E GF(q) and let 2 5 b 5 n.Then X is called a close-closed loop bursts of length b, if 3 an i, 1 5 i 5 b - 1 such that an-b+i.ai-l

# 0, ai = ai+l = ... = an-b+i-l

0.

Definition 4.The class of open loop bursts as considered in Definition 1 may be termed as open-closed loop bursts. Theorem C. An ( n , k ) cyclic code can not detect any close-closed loop burst of length b where 2 I: b 5 k 1.

+

Theorem D. The fraction of close-closed loop bursts of length b (2 5 b 5 k + 1) that goes undetected to the total number of close-closed loop bursts

392

in any (n,k) cyclic code is

There is yet another kind of burst viz. solid burst in which all the digits inside the burst length gets corrupted. Such a category of burst errors may be defined as follows:

Definition 5. A solid burst of length b is an n - tuple whose all the b nonzero components are confined to b-consecutive positions. In the second section of this paper, we obtain results similar to Theorem A and B for solid open loop bursts, whereas in Section 3, we obtain a result for solid close-closed loop bursts. In the last section viz. Section 4,we make a comparative study of the results derived in this paper with the earlier known results. In what follows, an ( n ,k ) cyclic code over GF(q) is taken as an ideal in the algebra of polynomials modulo the polynomial X" - 1. 2. Solid Open Loop Burst Error Detection In this section, we obtain the results of Theorem A and Theorem B for solid open loop bursts.

Theorem I. No solid open loop burst of length n - k or less is a code vector in any ( n ,k) cyclic code.

Prooj Let r ( X ) denote a solid open loop burst of length n - k or less. Let g ( X ) be the generator polynomial of the code of degree n - k . Let the first nonzero component of the vector corresponding to r ( X ) be the coefficient of X i if we consider the correspondence as: (ao,U l , - - - , U n 4 ) e.(a0 UlX ... cL"-lX"--l) Then the polynomial r ( X ) can be written as r ( X ) = a i x i U i + l Xi+l ... U i + n - k - l Xi+"-k-l = Xi(U2 Ui+lX ... a i + n - k - l x n - k - 1 ) = X i r l ( X ) where deg r l ( X ) = n - k - 1. If g ( X ) divides r ( X ) then g ( X ) I X i r 1 ( X )

+

+ +

+ +

+ +

+ +

393

Since X iand g(X)are coprime + g(X)I q(X) which is not true .: deg q(X)< deg g(X) Thus g(X)does not divide r(X)3 r(X)is not a code vector. Hence the proof. 0 Theorem 2. The fraction of solid open loop bursts of length b > n - k that goes undetected to the total number of solid open loop bursts in any cyclic code is

I

q(b-2)-(n-k)

(q - 1)-

if b

>n -k +1

Prooj Consider r(X)to be a solid open loop burst of length b(> n - k). Let g(X)be the generator polynomial of the code of degree n - k . Then

r(X)= XZrl(X)for some non-negative integer i and deg q(X)= b - 1. Now

# of solid open loop burst of length b = # of polynomials of type T(X) = # of polynomials of type r1(X) = (q - l)b

Now r(X)will go undetected of g(X)I r(X) + g(X)must divide Xirl(X) g(X)I q(X).:Xiand g(X)are coprime. Let q(X)= g(X)Q(X) for some polynomial Q(X) Since deg q(X)= b - 1 and deg g(X)= n - k

:. deg Q(X)= (b - 1) - ( n - k). Two cases arise: Case 1. Let b = n - k + 1 Then deg Q(X)= 0 + Q(X)is a nonzero constant.

394

+ Number of possibilites of Q ( X ) = q - 1. :. The ratio of solid open loop bursts which go undetected to the total number of solid open loop bursts is

Case 2. Let b > n - k + 1 Then # of polynomials of type Q ( X ) = ( q - l)zq(b-l)-(n-k)-l. :. The ratio in this case is -

(n - 1)24 ( b - l ) - ( n - k ) - l (Q -

-

Q

(b-2)-(n-k)

( q - 1y-2 .

Hence the theorem.

0

3. Solid Close-Closed Loop Burst Error Detection

In this section, we shall calculate the ratio of undetected solid closeclosed loop bursts to the total number of solid close-closed loop bursts. Since in a solid burst all the components are nonzero,therefore, the polynomial representation of a solid close-closed loop burst will have degree n - 1 irrespective of the length of the burst which is always greater than the degree of the geneartor polynomial of the code. It is remarked that either the polynomial corresponding to a solid close-closed loop burst is an irreducible polynomial or if it is reducible and the generator polynomial of the code does not divide it, then the burst will be detected. therfore, there is no specific burst lengfth upto which solid close-closed loop burst will be detected. Now we obtain the ratio of undetected solid close-closed loop bursts to the total numebr of solid close-closed loop bursts. Theorem 3. The fraction of solid close-closed loop burst of length b that goes undetected to the total number of solid close-closed loop bursts in any ( n , k )cyclic code is -

( b - l ) ( q - l)b-2 '

395

Proof. Let r ( X ) denote a solid close-closed loop burst of length b. Let g ( X ) denote the generator polynomial of the code of degree n - k . Since r ( X ) is a close-closed loop burst of lengfth b, the first component from where the burst can start is the ( n - b 2 ) t h components.

+

If we consider the correspondence

* + a1X + a2X2 + ... + an-lxn--l

(ao,a1, u2, '", un-l) uo Then r ( X ) may be written as

+

+ +

.(X) = Xn-b(Un-b+iXi an-b+i+1Xi+' ... an-1Xb-') f a z X 2 ... ui-lXZ-'), 1 5 i 5 b - 1 and each ai # 0.

+ +

-

a2X2

xn-bfi

+ ... + a .

(an-b+i xi-1)

(a0

+ UlX +

+ an-b+i+lX + ... + an-lXb-l-i ) + (a0 + UlX +

2-1

-

xn-b+i

7-1(

X)

+

7-2 (

X) Un-b+i+lX

+ + ... + an-lXb-l-i and 7-2(X)= uo + a l X + a z X 2 + ... + a . X2-l.

where q ( X ) = Un-b+i

2-1

For a fixed value of i,

# polynomials of type r ( X ) = ( q - I ) ~ (': each coefficient in r ( X ) is nonzero and there are b coefficients in r ( X ) ) . Since i takes values from 1 to b - 1 and for each value of i, the number of polynomials of type r ( X ) is same. Therefore, Total

# polynomials of type r ( X ) = ( b - l ) ( q - l ) b .

Now r ( X ) will go undetected if g ( X ) I r ( X ) , i.e. if r ( X ) = g ( X ) Q ( X )for some polynomials Q ( X ) . or Xn-b+i r i ( X )

+ 7-2(X)= g ( X ) Q ( X ) .

(1) Now degree of L.H.S. of (1) = (n- l ) ,so should be of R.H.S. of (1). Also degree of R.H.S. of (1)= degree of g ( X )

+ degree of Q ( X ) n - 1 = (n- k ) + degree of Q ( X ) =

:.

+ degree of Q ( X )

(n- k )

+ degree of Q ( X ) = (n

-

1) - (n- k ) = k - 1.

NOW,# polynomials of type Q ( X ) = ( q - 1)2qk-2.

396

:. Ratio of solid close-closed loop bursts that go undetected to the total number of solid close-closed loop bursts is

Hence the proof.

0

4. Comparative Study

In this section we present the comparison of the results obtained in Section 2 viz. Theorem 2 with the earlier known result viz. Theorem B. The comparison has been presented in the form of a table by taking specific values of b in the binary as in the binary case. For b = 2, both defintions viz. open loop burst and solid open loop burst coincide. :. we start comparing the results for b = 3 and onwards. Let us recall once again that fraction of undetected bursts for open loop bursts is

i

q - ( n - k - 1)

b? - 1)

q-(n-k)

if b = n - k if b

+1

>n -k +1

Fraction of undetected bursts for solid open loop bursts is

I

I

( 4 - l)-(n-k) q(b-2)-(~-k)

( q - 1)b-2

if b = n - k

+1

if b > n - k:

+1

397

Table

IQ = 21 Open Loop Bursts (Theorem B)

n-k+1=3 n-k+1<3

Solid Open Loop Bursts (Theorem 2)

1/2 2-k-k)

1 21- (n-

k)

1

for k = 1

for k = 1,2,3 the above reduces to n=3

1/2

n=4 n<4 n=5 n<5

IQ n-k+1=4 n-k+1<4

= 2, b = 41

1/4 24n-k)

1 2 2 - (n-k)

for k = 1 , 2 , 3 the above reduces to n=4

1/4

for k = 1

1

n=5 n<5 n=6 n<6

1/4 2-(n-3)

1 2-(n-5)

for k = 3

(Table contd.)

398

Iq=2,b=51 n-k+1=5 n-k+1<5

118 2-(n-k)

1 23-(n-k)

for k = 1 , 2 , 3 the above reduces to

n=5

1/8

1

for k = 1

n=6 n<6 n=7 n<7

14 = 3, b = 3 )

n-k+1=3 n-k+1<3

1/6 3-(n-h)

2-(74) 31-(n-k)/2

for k = 1,2,3 the above reduces to

n=4 n<4 n=5 n<5

n-k+1=4 n-k+1<4

1/18 3-(n-k)

2-(n-k)

32-

(n- k )

14

(Table contd.)

399

for k = 1 , 2 , 3 the above reduces to

n=4

2--("-1)

for k = 1

n=5 n<5

1/18 1/18 3-("-2)

2-("-2) 3-(n-4)/4

n=6 n<6

1/18 3-("-3)

2-("-3)

1

n-k+l=5 n-k+1<5

1/54 3-("+I

for k = 2 for k = 3

3-("-5)/4

2-(n-k)

33- (n-lc) /8

for k = 1,2,3 the above reduces to

n=5

1/54

n=6 n<6

1/54 34n-2)

n=7 n<7

1/54 3-("-3)

2-(n-l) 24n-2)

3-("-5) /8

for k = 1 for k = 2

Acknowledgement. The author wishes to thank University Grants Commission for providing grant (vide Ref. No. F 13-3/99 (SR-I)) under Minor Research Project to carry out this research work.

References 1. D.T. Brown and W.W. Peterson (1961), Cyclic Codes for Error Detection, Proc. IRE, Vol. 49, pp.228-235. 2. C.N. Campopiano (1962), Bounds on Burst Error Correcting Codes, IRE Trans., IT-8, pp. 257-259.

400

3. B.K. Dass and Sapna Jain (2001), On a Class of Closed Loop Burst Error Detecting Codes, International Journal of Nonlinear Sciences and Numerical Simulations, Vol 2., pp. 305-306. 4. P.Fire( 1959), A Class of Multiple-Error-Correcting Binary Codes for Non-Independent Errors, Sylvania Report RSL-E-2, Sylvania Recon.Sys. Lab., Mountain View, California. 5. A.B. Fontaine and R.G. Gallagher (1960), Error-Statistics and Coding for Binary Transmission over Telephone Circuits, Technical Report 25G-0023, M.I.T. Lincoln Laboratory, Lexington, Mass., October. 6. W.W.Peterson and E.J.Weldon,Jr.(l972), Error-Correcting Codes, Cambridge, Mass. The M.I.T. Press.

SURVEY O F SOME RECENT RESULTS ON CS-GROUP ALGEBRAS AND OPEN QUESTIONS S.K.JAIN1, PRAMOD KANWAR, AND J. B. SRIVASTAVA

1. INTRODUCTION Let K be a ring and G be a group. Let R = KG be the group ring of the group G. We shall call R to be a group algebra if K is a field. It is well known that R is right selfinjective if and only if K is a right selfinjective ring and G is a finite group. For any ring R, we know that if M is an injective R-module then each essentially closed submodule is a direct summand of M . This property is known as property (Cl) in the literature and the modules satisfying this property are known as CIS-modules or extending modules. The subject of CS-modules has been of considerable interest to many authors and a number of papers have been written. (See for example 153, [6], [7], [S], [lo], and so on.) The question of CS-group algebras was initiated in 2000 by Jain et. al. in [8] where the group algebra of infinite dihedral group was considered. There is very little known about CS-group algebras, in general and several interesting questions are open. We will give a survey of recent results in the area and state some of the open problems. 2. PRELIMINARIES

Throughout, unless otherwise stated, K will denote a field and R = KG, the group algebra of the group G over K . The augmentation ideal w ( R ) of R is defined to be the ideal generated by (1 - g 1 g E G}. It is known that w ( R ) is a maximal ideal of R. R is a prime ring if and only if G has no finite normal subgroups. R is a semiprime ring if and only if either characteristic of K is 0 or if characteristic of K is p then G has no finite normal subgroup whose order is divisible by p . If R is local then characteristic of K is p and G is a pgroup ( [ll],Lemma 'Plenary talk at ICM Satellite Conference in Algebra, August 2002. 2000 A M S Subject Classafication. Prirnary:16834, 16R20; Secondary:16LSO, 16E60. Key words and phrases. CS group rings, finitely C - C S group rings, prime rings, semiprime rings, polycyclic-by-finite groups, locally finite groups.

401

402

1.13, p415). R is artinian (even perfect) if and only if G is finite. The characterization of semiperfect or semilocal group algebras is not easy and is known only in some special cases. A group G is called polycyclic if G has a finite subnormal series

( e ) = GO4 GI 4

f

f

.

4G, = G

such that each quotient Gi+l/Gi is cyclic. If Gi+l/Gi is either cyclic or finite then G is called polycyclic-by-finite. For example, the infinite dihedral group

D,

=

{ ( a ,b) 1 .(a) is infinite, o(b) = 2, ab = ba-l}

is a polycyclic-by-finite group as

G is called locally finite if every finitely generated subgroup of G is finite, G is called an FC-group if every element of G has finitely many conjugates. For a group G, G' will denote its commutator subgroup, Z(G), its center and G+, the set of its torsion elements. We will denote by A(G), the subgroup of all those elements of G that have finitely many conjugates, that is, A(G) = {X E G I [G : CG(Z)]< GO}. Observe that G is an FC-group if and only if G = A(G). A+(G) will denote those elements of A(G) that are of finite order, that is, A+(G) = {x E G I [G : CG(Z)]< m and

O(Z)

< m}.

For any two R-modules M and N , M is said to be N-injective if for any R-homomorphism 4 : N 4 E ( M ) , where E(A4) is the injective hull of M , 4 ( N ) c M . M is said be injective if M is N-injective for all right R-modules N . A submodule K of a right R-module M is said to be essential in M , denoted by K ceM , if for any nonzero submodule L of M , K n L # 0. M is called a CS (or extending) module if every submodule of M is essential in a direct summand of M , equivalently, if every essentially closed submodule of M is a direct summand of M . M is called finitely C-CS if direct sum of finite number of copies of A4 is CS. M is called CS with respect to uniform submodules if every uniform submodule of M is essential in a direct summand of M , equivalently, if every uniform closed submodule of M is a direct summand of M . A4 is said to satisfy condition (C3) if for any two summands M I and M2 of M with M I n M2 = 0, M I @ A42 is also a summand of 111. A CS module is called quasi-continuous if it satisfies (C3). It is known that

403

if M x N is quasi-continuous then M and N are injective relative to each other. A ring R is said to be right CS (or CS with respect to uniform right ideals) if the right R-module RR is CS (resp. CS with respect to uniform R-submodules). R is called right selfinjective if RR is injective. R is called right PP if every principal right ideal of R is projective. It is well known that all right PP rings are right nonsingular and that every right nonsingular right CS ring is right PP. For a ring R, J ( R ) will denote the Jacobson radical of R and Zr(R), the right singular ideal { r E R 1 r I = 0 for some essential right ideal I of R } of R. For an element a of a ring R, r.annR(a) will denote the right annihilator of a in R.

3. PRIME AND SEMIPRIME GROUPALGEBRAS We begin this section with a general result on nonsingular CS-group rings.

Proposition 3.1. Let R be a ring with no nontrivial idempotents and G any group such that RG is nonsingular and CS. Then for every finite subgroup H of G, o ( H ) is invertible in R. In particular, order of every torsion element in G is invertible an R. Proof. It is sufficient to prove the result for finite cyclic subgroups of G. Let H = ( h ) be a finite cyclic subgroup of G and let o ( H ) = n. Then n-1

w(RH) = ( H - l ) R H

=

C(hi- 1)RH.

Thus r.annRH(W(RH))=

i=l

n- 1

n-1

r . a n n R H ( ( h - l ) R H )= r.annRH(h-l) = ( C h i ) R H = ( C h i ) R . Since i=O

i=O

RG is nonsingular and CS, RG is PP. Consequently R H is PP. Thus r.annRH(h - 1) is generated by an idempotent. Hence there exist an n-1

idempotent e in R H such that

n- 1

(Chi)R = eRH. Let e = (Chi)-. Then i=O i=O

n-1

n-1

n-1

,

(i=OChi)'r2 = n(i=OC hi)?.

Since e is an idempotent, n( C hz)r2=

(chi)r. It follows that nr2

= r. If nr = 0 then r = nr2 = 0. Thus

e2 =

i=O

n-1

i=O

e = 0 , a contradiction as r.annRH(W(RH))# 0. Thus nr # 0. Also n2r2= nr, that is, nr is an idempotent. Since R has no nontrivial idempotents, nr = 1, as desired. I We now consider prime and semiprime noetherian group algebras which are CS. We state the following result.

404

Theorem 3.2. ([8], Theorem 3.6, L e m m a 3.8, T h e o r e m 3.9) Let R = KD,, the group algebra of the infinite dihedral group. T h e n (i) R is a right CS-ring i f and only i f char(K) # 2. (ii) T h e center C = Z ( R ) of R i s a Dedekind d o m a i n and Rc as also CS. Recall that the infinite dihedral group D , in Theorem 3.2 is a polycyclic-by-finite group. Also KD, is noetherian. Using theory of PI group algebras, it can be seen that KD, has PI (1111,Corollary 3.8, p196 and Corollary 3.10, p197). Since char(K) # 2, gl. dim(KD,) < 00 ([ll],Theorem 3.13, p450). Thus gl.dim(KD,) is equal to the Hirsch number h(D,) of D , ([ll], p450). Since h(D,) is the number of infinite cyclic quotients in any subnormal series of D,, h(D,) = 1. It follows, then, that the ring R in Theorem 3.2 is also hereditary. Thus KD, is a prime right-left CS, noetherian, hereditary ring with PI. Behn, in his Ph.D. dissertation [l], considered prime CS group algebras R of polycyclic-by-finite groups. It can be shown that the prime group algebra K G of a polycyclic-by-finite group is noetherian and has PI.

Theorem 3.3. ([2], Theorem 3.3.10) Let G be a polycyclic-by-finite group and suppose R = K G is prime. T h e n R i s CS i f and only i’G i s either torsion free or G I ID , and c h a r ( K ) # 2. Question 1. Characterize group G if R = K G is a semiprime PI noetherian CS-group algebra. (The hypothesis in Question 1 implies G is polycyclic-by-finite. (See Lemma 3.4)) As observed above the group algebra K D , is indeed hereditary. It, therefore, follows that if R = K G is a prime CS group algebra of a polycyclic-by-finite group which is not a domain then it is hereditary. By ( [5], Corollary 12.18), this is equivalent to R is finitely C-CS, that is, R” is CS as R-module for all n > 0. This leads us to raise the following question. Question 2. Is semiprime PI noetherian CS-group algebra which is not a direct sum of domains also finitely C-CS, that is, R” is CS as R-module for all n > O? We know that semiprime noetherian ring R is finitely c - C S if and only if it is right hereditary ([5], Corollary 12.18). With respect to Question 2, one may ask to characterize groups G such that R = KG is a semiprime noetherian PI hereditary ring. Hereditary group algebras have been completely characterized by Dicks ([4]). The structure of group G, in this case, is described in terms of fundamental group of connected graphs of finite groups.

405

Since the description of fundamental groups is rather too abstract, we give below a characterization of semiprime hereditary group algebras. In this special case the group G is quite easy to describe. We first give a Lemma that is of independent interest. Lemma 3.4. If KG is right (or left) noetherian and has PI then G is polycyclic- by-finite. Proof. First we note that G is noetherian because K G is noetherian. Case I: char(K) = 0. Since K G has PI, G has a normal abelian subgroup A of finite index which is finitely generated and so it is polycyclic ([ll],Corollary 3.8, p196). Since [G : A] < 00, G is polycyclic-by-finite. Case 11: char(K) = p . Since K G has PI, G has a normal pabelian subgroup A of finite index which is finitely generated ([ll],Corollary 3.10, p197). Consequently $ is finitely generated, abelian and hence polycyclic. Also as A is pabelian, A’ is finite. Since [G : A] < 00, (1) A’ d A 9 G is a finite subnormal series in which each factor is either finite or polycyclic. Thus G is polycyclic-by-finite. I

The theorem that follows gives a characterization of a semiprime noetherian hereditary group algebras with PI.

Theorem 3.5. Let K be a field with c h a r ( K ) # 2 and let K G be a semiprime, noetherian, P I group algebra. T h e n the following are equivalent. (i) K G is hereditary. (ii) Either G is finite or G has an infinite cyclic subgroup A of finite index, say, n such that in the case c h a r ( K ) = p , p does not divide n and G has no element of orderp. Proof. (i) + (ii). Since K G is hereditary, gl. dim(KG) is 0 or 1. If gl. dim(KG) = 0 then G is a finite group with no elements of order p . So let g l . dim(KG) = 1. KG, being hereditary, is nonsingular. Also K G is finitely C - C S and hence CS. We show that G has no element of order p if char(K) = p . Let, if possible, G has an element, say x , of order p . Let H = (x),the cyclic subgroup generated by 2. Since K G is nonsingular and CS, it must be PP. Thus, by ([2], Lemma 3.3.1), KH is PP. Hence KH is nonsingular. However, the right nonsingular ideal Z,(KH) of KH is w(KH) # 0 ([ll],Exercise 30, p467), a contradiction. Thus G has no element of order p , if char(K) = p . By ([ll],p450), Hirsch length h(G) of G is equal to 1. Thus by ([Ill, Lemma 2.5, p422) G has a characteristic infinite cyclic subgroup A of finite index. If char(K) = 0 then

406

we have nothing else to prove. So let us assume that char(K) = p. Since A is infinite cyclic, Aut(A) C2, the cyclic group of order 2. ) ox where I J ~: A A, Define IJ : G --f Aut(A) given by ~ ( x = ox(a) = xax-l. It can be checked that IJ is a group homomorphism and ker(o) = CG(A). Thus [G : &(A)] 5 2. Case I : [G : CG(A)]= 1. In this case G = CG(A). Thus A c Z ( G ) . Since [G : A] < 00, [G : Z(G)] < 00. Thus G' is finite and p 1 o(G'). Since for any x,g E G, g-lxg = xx-lg-lzg E xG' and G' is finite, it follows that G is an FCgroup, that is, G = A(G). Hence G+, the group of all torsion elements is finitely generated of G, is a finite normal subgroup of G and torsion free abelian ([Ill,Lemma 1.5, p116). Thus G' C G+. Consider the subnormal series

-

5

(1) a G' 9 G+ a G. Since $ is finitely generated torsion free abelian, is free abelian. G is But G+ is finite. Therefore h(g)= h(G) = 1. Therefore G+ = (xG+). Then G = G + ( x ) and an infinite cyclic group. Let p i o(G+) = [G : (x)].This completes the proof in this case. Case 11 : [G : CG(A)]= 2. Let Go = CG(A). Then CG,,(A) = Go. Note that KGo is semiprime, noetherian, PI, and hereditary because gl.dim(KG0) = h(G0) = 1 ([ll],Lemma 2.10, p426). By Case I, Go = G i (x) and p [Go : (x)]. Since [G : (x)] = 2[Go : (x)] and char(K) # 2, p i [G : (x)] and we are done in this case as well. (ii) + (i) We only need to consider the case when G is infinite. Let A be an infinite cyclic subgroup of G with [G : A] < 00. Then K A is a PID. Thus gZ.dim(KA) = 1. Since [G : A] < 00, and G has no element of order p if char(K) = p, gl. dim(KG) < 00 ([ll],Theorem 3.12, p442). Indeed gl.dim(KG) = gl.dim(KA) = 1 ([ll],p450). Thus KG is hereditary. I

+

Theorem 3.6. Suppose R = K G is a semiprime noetherian hereditary group algebra with PI. Then (i) the center Z ( R ) of R is a direct sum of Dedekind domains. (ii) R is CS as a module over its center Z(R).

Proof. (i) Since R = K G is semiprime, the classical quotient ring Qd(R) of R is semisimple artinian. Let Qcl(R) = elQd(R)@e:!Q,l(R)@ . . . @enQcl(R)where for each i, ei is a central idempotent and eiQd(R) is simple artinian. By ([ll],Exercise 32, p167), ei E R. Hence

407

e l R @ e2R @ . . . @ e,R. Since R is hereditary, e i R is hereditary. Also e i R is a prime ring satisfying a polynomial identity ([ll] Corollaries 3.8 and 3.10, p196-197). Thus e i R is a prime PI noetherian hereditary ring . So by ([9],Theorem 13.9.16, p483), the center Z ( e i R ) isaDedekinddomain. SinceZ(R) = Z ( e l R ) @ Z ( e z R ) @. . . @ Z ( e , R ) , the result follows. (ii) The proof follows from the fact that Z ( e i R ) is a Dedekind domain and e i R is CS over Z ( e i R ) . I

R

=

4. LOCALCS-GROUPALGEBRAS In this section we state recent results on local group algebras which are either CS or finitely C-CS. We begin with the following result.

Theorem 4.1. ([3], Lemma 4.1) Let K be a field and G be any group. The group algebra KG of the group G is local right CS i f and only i f c h a r ( K ) = p and G is a locally finite p-group. Lemma 4.2. 031, Lemma 3.3) Suppose R is a local right uniform ring with nil radical and S is a uniform direct summand of ( R x R)R. T h e n S = (1,b ) R f o r some b E R or ( a , l ) R f o r some a E R. Lemma 4.3. 031, Theorem 3.10) Under the conditions of Lemma 4.2, R x R has (C3), that is, if S1and S2 are direct summands of R x R with S1 n S2 = 0 then S 1 @ S2 is also a summand of R x R. Theorem 4.4. Suppose R = KG is a local group algebra. T h e n M2(R) is CS i f and only if R is selfinjective. Proof. By ([5], Lemma 12.8), ( R x R ) Ris CS. Since radical of R is nil, ( Rx R)R has (C3)(by Lemma 4.3). Thus ( Rx R)Ris quasi-continuous. Consequently R is selfinjective. (Note that in this case R is selfinjective is equivalent to saying that G is a finite pgroup.) I Theorem 4.5. Suppose R = KG is a local group algebra. T h e n M n ( R ) (n > 1) is CS if and only i f M2(R) is CS. A ring R is called semilocal if J ( R ) is semisimple artinian. For a semilocal group algebra R = KG, one can ask the following questions. Question 3. Characterize groups G such that semilocal group algebra K G is finitely c - C S . Question 4. Characterize groups G such that semilocal group algebra KG is CS. In case G is a nilpotent group, then the following result is known. Theorem 4.6. (131, Theorem 4.3) Let K be afield and G be a nilpotent group such that the group algebra R = KG is semiperfect. T h e n M n ( R ) (n > 1) is CS i f and only i f R is selfinjective.

408

REFERENCES [l] Antonio Behn, Group rings whose principal ideals are projective and group with bounded representation degree, Ph.D. dissertation, University of Wisconsin, Madison (2000). [2] Antonio Behn, Polycyclic group rings whose principal ideals are projective, J. Algebra, 232, (2000), 697-707. [3] K. I. Beidar, S. K. Jain, Pramod Kanwar, and J. B. Srivastava, CS matrix rings over local rings, Preprint. [4] Warren Dicks, Hereditary group rings, J. London Math. Soc. 20 (2), (1979), 27-38 [5] Nguyen Viet Dung, Dinh Van Huynh, Patrick F. Smith and Robert Wisbauer, Extending Modules, Pitman, London, 1994. (61 Jos6 L. G6mez Pardo and Pedro A Guil Asensio, Essential embedding of cyclic modules in projectives, Trans. Amer. Math. SOC.349 (ll), (1997), 4343-4353. [7] Jos6 L. G6mez Pardo and Pedro A Guil Asensio, Every C - C S module has an indecomposable decomposition, Proc. Amer. Math. Soc. 129, (200l), 947-954. [S] S. K . Jain, P. Kanwar, S. Malik, and J. B. Srivastava, K D , is a CS-algebra, Proc. Arner. Math. Soc., 128 (2), (2000), 397-400. [9] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, WileyInterscience, NY, 1987. [lo] B.L.Osofsky and P.F. Smith, Cyclic modules whose quotients have all complement submodules direct summands, J. Algebra, 139, (1991), 342-354. [ll] Donald S. Passman, The Algebraic Structure of Group Rings, John Wiley, NY, 1977. [la] Daniel Segal, Polycyclic Groups, Cambridge Univ. Press, Cambridge, NY, 1983. [13] S. K. Sehgal, Topics in Group Rings, Dekker, NY, 1978. [14] M. J. Tomkinson, FC-Group, Research Notes in Math 96, Pitman, London, 1984. OF MATHEMATICS, OHIO UNIVERSITY, ATHENS,OH 45701 DEPARTMENT E-mail address: jainamath.ohiou.edu

DEPARTMENT OF MATHEMATICS, OHIO UNIVERSITY - ZANESVILLE, ZANESVILLE, OH 43701 E-mail address: pkanwaramath. ohiou.edu DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY, DELHI 110016 (INDIA) E-mail address: jbsrivasamaths. iitd.ernet .in

ON E*-UNITARY CATEGORICAL INVERSE SEMIGROUPS *

ZHONGHAO JIANG Department of Mathematics, Northern Jiaotong University, Beijing, 1000&, P. R. China E-mail: zh,[email protected]. edu. cn

We describe the structure of E*-unitary categorical inverse semigroups by means of E-unitary inverse categories and actions of primitive inverse semigroups. We also obtain a P-theorem for E*-unitary categorical inverse semigroups in terms of primitive inverse semigroups acting by partial order-automorphismson a partially ordered set.

1. Introduction

An inverse semigroup with zero is called E*-unitary (also 0-E-unitary) if every element above a non-zero idempotent is also an idempotent. Lawson has pointed out in [8,9] that there are at least two reasons why E*-unitary semigroups form an important class of inverse semigroups. Firstly, there are a number of interesting inverse semigroups which are E*-unitary, for instance, the E-unitary semigroups with zero adjoined, primitive inverse semigroups, polycyclic monoids and tiling semigroups [6,9]. Secondly, Kellendonk's topological groupoid, which can be constructed from every inverse semigroup, is usually only T I , but is Hausdorff in the case of E*-unitary inverse semigroup [9]. For the results on E*-unitary inverse semigroups, the reader is refered to [l-6,8,9]. E*-unitary inverse semigroups are more complicated than E-unitary inverse semigroups. As yet, there is no structure theorem for general E*unitary inverse semigroups. In [8], Lawson discussed E*-unitary inverse monoids in terms of categories and functors. In [2], Gomes and Howie obtained a type of P-theorem for strongly categorical E*-unitary inverse semigroups. In [l], Bulman-Fleming, Fountain and Gould introduced strongly *This work is supported by SRF for ROCS, SEM and Kort Fellowship of Bar-Ilan University, Israel

409

410

E*-unitary inverse semigroups and proved that the strongly E*-unitary inverse semigroups are precisely the Rees quotients of all E-unitary inverse semigroups. Furthermore, they characterized strongly E*-unitary inverse semigroups by McAlister O-triples. But the structure theorem obtained in [I]is unsatisfactory for the reason that corresponding to a strongly E*unitary inverse semigroup, there are many McAlister O-triples. It is difficult to choose an appropriate McAlister O-triple. In this paper, we will study E*-unitary categorical inverse semigroups, a special class of strongly E*unitary inverse semigroups. The main results of this paper are Theorem 2.2 and Theorem 3.4. In [lo], Margolis and Pin proved that a group acting transitively without fixed point on a connected, locally idempotent inverse category could be used to construct an E-unitary inverse semigroup, and every E-unitary inverse semigroup arises in this way. In section 2, we generalize the approach of [lo] and prove that every E*-unitary categorical inverse semigroup can be constructed from a primitive inverse semigroup acting on an E-unitary inverse category. McAlister's P-theorem stands as one of the most significant achievements in inverse semigroup theory. The theorem refers to what have come to be called E-unitary inverse semigroups, and gives a description of such semigroups in terms of a group acting by order-automorphisms on a partially ordered set. In section 3, we obtain a P-theorem for E*-unitary categorical inverse semigroups by means of primitive inverse semigroups acting by partial order-automorphism on a partially ordered set. In order to do so, we first give a representation theorem for E*-unitary categorical inverse semigroups by using their largest O-restricted primitive homomorphic images. The tools used here are O-dual prehomomorphisms. For other undefined terms, the reader is refered t o [9][11]. We adopt the standard convention that a semigroup with zero must contain at least two elements. Recall that a semigroup with zero is said to be categorical at zero if abc = 0 implies that a b = 0 or bc = 0. An inverse semigroup with zero is E*-unitary categorical if it is both E*-unitary and categorical at zero. An inverse semigroup with zero is a primitive inverse semigroup if all its nonzero idempotents are primitive. Let E ( S ) be the set of all idempotents of a semigroup S. If S is a semigroup with zero and A S then A* = A\{O}. A mapping 0 between inverse semigroups with zero is idempotent pure if a is an idempotent whenever 0 ( a ) is an idempotent; it is @restricted if 0-'(0) = (0).

41 1

Throughout this paper, we shall make use of the following lemma, since its proof is straightforward and is hence omitted.

Lemma 1.1 Let S be a non-trivial inverse semigroup with zero. Then the following conditions are equivalent: ( I ) S is a primitive inverse semigroup; (2) for each x E S , e , f E E ( S ) , x e # O(fz # 0 ) implies x e = x ( f x = 2 ) ; (3)forallx,yES,x
Lemma 1.2. Let S be an inverse semigroup with zero which i s categorical at zero. Then (1) /3 is the minimum 0-restricted primitive congruence o n S(i.e. Op = { O } and S/p is a primitive inverse semigroup). (2) p is idempotent pure i f and only if S is E*-unitary. By the Proposition 14 in page 91 of [9], we have the following fact:

Lemma 1.3 Let S be an E*-unitary categorical inverse semigroup. Then p n R = , B n L = {(s,s) E S x S : s E S } , where R and L: are Green’s relations o n S .

2. A characterization theorem

As we stated in the introduction, the aim of this section is to prove that every E*-unitary categorical inverse semigroup can be constructed from a primitive inverse semigroup acting on an E-unitary inverse category. Our method is a generalisation of an idea used by Kellendonk and Lawson [6]. In [6], they provided the categorical basis for the construction of inverse semigroups from any tiling in terms of a group acting partially and without fixed points on an inverse category associated with the tiling. We shall make use of the following notation concerning categories. If C is a category and u,v are objects of C , then let C(u,w) denote the

412

set of arrows of C which begins at u and ends at v. It is convenient to view C as the set of all arrows of C . If z E C then d ( z ) denotes the right identity of z, and r(z) denotes the left identity. Our convention is that d ( z ) = r(y). We write 3zy to mean "the product zy exists in the category". A category C is said to be inverse if for each z E C there exists a unique element z-' such that the product xx-'x and x-'xx-' exist and satisfy z = ~ z - ~ andz z-' = x-'zx-'. C is said to be E-unitary, if for every 2,y E C ,3zy, zyzy = zy(3yx, yzyz = yz) and zz = z imply yy = y. Let T be a primitive inverse semigroup, C a category and 0 : T* x C + C , ( t , z ) H 0 ( t , z ) a partial mapping. If O(t,z) is defined, then we shall write 0 ( t ,x) = t . 2. We write 3 t . x to mean that t . is defined. We say that T acts injectively on the category C if the following conditions hold: (Al) 3 s . ( t . z ) if and only if 3 ( s t ) . z and in which case ( s t ) . z = s . ( t . z ) ; (A2) 3 .z implies that 3-' . (t . z) and t-' . ( t . z) = z; (As) 3 t . x if and only if 3 t . d ( z ) ,in which case d ( t . z) = t . d ( z ) ;3 .z if and only if 3 t . r(z), in which case, r(t . z) = t . r(5); (A4) If the product zy is defined in C , then 3 t . (zy) if and only if 3 t . z and 3 .y; in which case t . (zy) = (t . z ) ( t. y); (A5) If 3 s . x and 3 t . z such that s . z = t .z, then s = t ; (AG)For each z E C, there exists t E T* such that t .x exists. Define a relation on C by x y if and only if there exists t E T* such that y = t . z . It is easy to check by using (Al)) (A2) and ( A 6 ) that is an equivalence relation on the set C . Denote the -equivalence class containing z by [z] and the set of equivalence classes by C / T . Let W = (C/T)O be the set C / T with the new symbol 0 adjoined. Let [z], [y] E C / T . If there exist a , b E T* such that product ( a . z ) ( b . y )exists in the category C then put [z]@[y] = [(u.x)(b.y)], otherwise put [z]@[y] = 0, and define O @ 0 = 0 and [z] @ O = 0 = 0 @ [z]. N

N

N

First, let us give some properties of W . Theorem 2.1 Let T be a primitive inverse semigroup acting injectively on a category C . Put W = (C/T)O. Then (1) W is a semigroup which is categorical at zero with respect to @. (2) If C is an inverse category, then W is an inverse semigroup. (3) If C is E-unitary, then W is E*-unitary.

413

Proof (1) By the same argument used in the proof of Theorem 4 [6], we have that (C/T)O is a semigroup. It remains to show that (C/T)O is categorical at zero. Let [z],[y], [z] E W such that [z] @ [y] # 0, [y] @ [z] # 0. Then there exist a, b, c,d E T* such that both ( a . z ) ( b . y) and (c y ) ( d . z ) exist. Thus d ( a . z) = r(b . y). By ( A ~ ) , u d (. z ) = b . r(y) and so b-la # 0 and (b-'a).d(z) = r(y). Since (c.y)(d.z)exists, r((c.y)(d.z)) = r(c.y) = c.r(y). It follows that (cb-la) . d(z) exists. By ( A s ) ,3(cb-la) . z and d((cb-'a) . z ) = (cb-la) . d ( s ) = c . r(y) = r ( ( c . y)(d. z ) ) . Thus 3((cb-'a) . z ) ( c . y ) ( d . z ) . Hence [z] @ ([y] @ [z]) # 0 and so W is categorical at zero. (2) If C is an inverse category, then for each z E C we have z = z z - ' ~ . Thus [z] = [z] @ [z-'] @ [z], and so W is regular. It remains to show that the idempotents commute. Suppose that [z] @ [z] = [z]. Then there exist a,b E T* such that [z] = [ ( a - z ) ( b - z )Thus ] . there exists c E T* such that z = c . ( ( a . z ) ( b - z ) ) . Hence z = ((ca) .z)((cb).z). But by ( A 6 ) ,there is t E T* such that t . d ( z ) exists. By ( A 2 ) , d(z) = ( t - I t ) . d ( z ) . Thus cb = t-'t by ( A s ) and so cb is an idempotent. By (A1) and ( A z ) ,

(cb) . z = ((cb)-l(cb)). z = (cb)-l . ((cb). z) = z. Similarly, (a) . z = z. Thus z = ((ca). z)((cb). z) = 2'. Clearly, z = x2 implies [z] = [z] @ [z]. Thus [z] is a non-zero idempotent of W if and only if x is an idempotent in C . By the argument of the end of the proof of Theorem 4 of [6], W is an inverse semigroup. (3) Let [z], [y] E W such that [z] # 0 and [z] = [XI @ [z]. If [z] @ [y] is a non-zero idempotent of W , then there exist a , b E T* such that ( a . z ) ( b .y) exists and [z] @ [y] = [ ( a .z ) ( b .y)]. By the proof of (2), we have that (a.z)(b-y) is an idempotent of C. By (A4), a - z is also an idempotent of C . Since C is E-unitary, b . y is an idempotent of C . Thus [b.y] = [b.y] 8 [b.y]. That is [y] = [y] @ [y]. Hence W is E*-unitary. Now we can state the main result, of this section. Theorem 2.2 Let S be an inverse semigroup. Then the following conditions are equivalent: (a) S is an E*-unitary categorical inverse semigroup. (ii) There exists a primitive inverse semigroup T and a 0-restricted, idempotent pure surjective homomorphism 4 from S onto T .

414

(iii) S is isomorphic to a semigroup ( C / T ) ' , where T is a primitive inverse semigroup which acts injective19 o n an E-unitary inverse category C. Proof ( i )+ (ii).See Lemma 1.2. (ii)+ (iii).Define a category C as follows: the objects of C are T * , and if u,v E T* then C(u,v) # 0 if and only if uRv, in which case C ( u , v )= { ( u , m , v ): m E S*,u4(m)= v } , (u# v)

C ( u , u )= { ( u , m , u ): m E S*,uqb(m)= u} u {(u, 1,~)) where 1 @ S and S U (1) is a monoid with identity 1. The product in C is defined by

(u, m,v ) ( v ,n,20)

= (u, mn, w)

and undefined in all other cases. It is routine to show that C is an E-unitary inverse category. Define a partial mapping 8 : T* x C -+ C by

8 ( t , (u, m, v)) = (tu,m,t v ) where tu # 0. It is straightforward to check that 8 satisfies conditions ( A l ) to ( A s ) . Thus we obtain an E*-unitary categorical inverse semigroup W = (C/T)'. The non-zero elements of W are of the forms [u, m,v],(u, m, v) E

C. Define a mapping $ : W --+ S by $ ( [ u , m , v ] )= m, and $(O) = 0. By the definition of 8 , m is uniquely determined by [u,m, v ] and so is well-defined. Clearly, $ is surjective. Next, we show that .J, is injective. It suffices to show that if (u, m,v),(w, m, z) E C , then [u,m,v] = [w,m,z]. Assume that ( u , m , v ) ,( w , m , z )E C. Then u-lu = $(mm-') = W - ~ W and so w = tu for some t E T * . But z = wqb(m) and so z = tuqb(m)= tv. Thus

+

( w , m , 2 ) = (tu,m,tv)= e(t,( U , m , v ) ) . Hence [ w , m , z ]= [ u , m , v ] . It therefore remains to show that $ is a homomorphism. Let [ u , m , v ] , [ w , n , z ]E W . It is easy to show that [ u , m , v ]€4 [ w , n , z ] # 0 if and only if there exist a , b E T* such that av = bw # 0 , in which case

[u,m,w] @ [w, n , z ]= [ ( u u , m , a v ) ( h , nbz)] , = [au,mn,bz]. Thus $"U, m,I . €4 [w, n, 4) = mn = $(b,m,.I)$([w, n,4).

415

If [u,m,u]I23 [w,n, x] = 0, then for all c, d E T * ,cu # dw. I t follows that u-lu # w - h . Since u = u$(m) and w-'w = 4(nn-l),w-lu = 4(mV1m)# +(nn-l). By Lemma 1.1, 4(m)$(n)= 0. Since 4 is 0restricted, mn = 0. It follows that

$(I.,

m,4 8 [w, n,.I)

= $(O) = 0 = mn = @ U U , m,.I)$([w n , 4 .

Thus S is isomorphic to W . ( i i i )+ ( i ) . See Theorem 2. 1.

3.

A structure theorem

In this section, we will give a 'P-theorem' for E*-unitary categorical inverse semigroups in terms of primitive inverse semigroups acting by partial order-automorphisms on a partially ordered set. First, let us define O-dualprehoinomorphisms. Let S and T be inverse semigroups with zero. A mapping $ : S -+ a 0-dual-prehomomorphism, if the following conditions are satisfied:

T is

(1) @(O) = 0. (2) For all ~ 1 , Es S~, $ ( s 1 ) $ ( ~ 25 ) $(slsz). (3) For all s E S,$(s-l) = $(s)-'. 0-dual-prehomomorphisms have the following important property.

Lemma 3.1 If $ is a 0-dual-prehomomorphism from S into T and e E E ( S ) , then $ ( e ) E E ( T ) .

Now we show how 0-dual-prehomomorphism can be used in the construction of E*-unitary categorical inverse semigroups. A similar approach was also adopt by Petrich and Reilly in [12]. Let X be a partially ordered set containing a least element 0. Zx the symmetric inverse semigroup on X . For any element a of ZX, we denote the domain (range) of a by A(a)(V(cx)). We denote by C ( X ) the inverse subsemigroup of ZX consisting of those elements that are isomorphisms

416

of ideals of X onto ideals of X . Clearly, for every a E C ( X ) , a ( O )= 0. Furthermore, the zero element of C ( X ) is the mapping a such that A ( a )=

V ( a )= (0). If $ is a mapping of a semigroup T into C ( X ) ,then we shall denote the image of any element t of T under $ by Qt. However, we shall frequently abbreviate $t(a),a E X , to ta. Theorem 3.2 Let Y = ( Y , A ) be a semilattice with zero 0, T be a primitive inverse semigroup and $ : t + Gt be a 0-dual-prehomomorphism of T into C ( Y ) satisfying the following condition:

V e E E(T*),V'z,y E A($,) n Y * , zA y

# 0.......(*).

Let Q

= Q ( Y , T ; $ )=

{ ( a , t )E Y * x T* : a E V($t)} U { ( O , O ) } .

Define an operation on Q as following:

and ( a ,S ) ( O , 0)

=

(0,

). = (O,O)(O,O)

= (070).

Then Q , with respect to this multiplication, is a n E*-unitary categorical inverse semigroup.

Proof Let ( a , s ) , ( y , t )E Q . Since $(s-') = $(s)-', we have that a E V ( $ ( s )= ) A($(s-')). Then s - l a A y 5 s-la E V ( $ ( S - ' ) )= A($(s)) and s-la A y 6 y E V ( $ t ) . Hence s(s-la A y) E V ( $ J s t If ) . st # 0, then s-ls = tt-l. Thus s-la A y E A ( $ J ~ - I ~ and ) so, by condition (*), s-la A y # 0. Consequently, s(s-la A y) # 0. Thus the above operation is well defined. It is easy to show that Q is a semigroup with respect to this operation. Clearly,(s-la,s-l) E Q . For all e E E ( T ) ,by Lemma 3.1, $e is an idempotent of C ( Y ) . Hence $e is the identity mapping on its domain. By this fact, we have (1) ( a ,s ) ( s - l a , & ) ( a , s ) = ( a ,s); (2) ( s - l a , s - q a , s)(s-la, s-1) = (s-la,s-1); (3) E ( Q ) = { ( a , e ) E Q : e E E ( T * ) }U {(O,O)} and it is commutative. Hence Q is an inverse semigroup. Since T is primitive,

417

( a ,s ) 5 (y,t )

either ( a ,s ) = (0,O)or a 5 y and s = t.

It follows that Q is E*-unitary. Let ( a ,s ) , (y, t ) ,(6,u ) E Q such that

( a ,S)(Y, tH6, u ) = (070). Then stu = 0. Since T is categorical, st = 0 or tu = 0. ( a , s ) ( y , t ) = (0,O)or ( y , t ) ( S , u )= (0,O).Thus Q is categorical.

Thus

Theorem 3.2 shows how 0-dual-prehomomorphisms can be used to construct E*-unitary categorical inverse semigroups. Now, we give another construction of E*-unitary categorical inverse semigroups. Let T be a primitive inverse semigroup, X a partially ordered set with a least element 0 and X*=X\{O}. Let 4 : T --+ C ( X ) ,t H 4t be a 0-restricted homomorphism and let y be a subset of X . The triple ( T ,X ,Y ) is called a Primitive triple if conditions ( P I )to (P3) are satisfied: ( P I )y is a subsemilattice and order ideal of X ; ( P . )for all e E E(T*)and all P, Q E A(&) n y * ,we have P A Q # 0; (P3) q5t(Y*)ny* # 0 for all t E T * . Now let

M = M ( T ,X , Y ) = { ( P , t ) E J’* x T* : &-i(P)E J’*} U (0) where ( T ,X , Y ) is a Primitive triple. We define a multiplication on M as following:

and

(P,u)O = 0(P,u ) = 00 = 0. Then, it is straightforward to show the following theorem.

Theorem 3.3 Let M = M ( T ,X , Y ) be defined as above. T h e n M is a n E*-unitary categorical inverse semigroup. Now, we may state the ‘P-theorem’ for E*-unitary categorical inverse semigroups.

Theorem 3.4 Let S be a semigroup . T h e n the following conditions are equivalent:

418

(1) S is a n E*-unitary categorical inverse semigroup; (2) S is isomorphic t o some semigroup Q = Q ( Y , T ; $ ) , where Y is a semilattice with zero, T a primitive inverse semigroup and $ : t 4 $t a 0dual-prehomomorphism of T into C ( Y ) satisfying the following conditions: (2) y = UtET A($t); ( i i ) V e ~E ( T * ) , V x , y E A ( I / I ~ ) , X A ~ # O . (3) S is isomorphic t o M ( T ,X , Y ) f o r some Primitive triple (T,X , Y ) .

Proof (3) + (1). See Theorem 3.3. (1) 3 (2). Let E be the set of idempotents of S, the minimum 0restricted primitive congruence on S and .1c, a map of SIP into C ( E ) defined by A ( $ J ~=~ {b-'b ) : b E up} U {0} and $,p(b-'b) = bb-', Gap(0) = 0, Va E S . We first observe that $ap is well defined since, if e = b-lb = c-lc for some b,c E up, then ( b , c ) E P n L. By Lemma 1.3, b = c. Therefore bb-' = cc-l and q a p is properly defined. If 0 # e 5 f,f E A(qap),where f = b-lb, b E UP, then b-'be = f e = e # 0 and so c = be # 0. Clearly, c E up. Thus e = c-lc E A(?,hap). Hence is an ideal of E. Also q a p ( e ) = cc-l = beb-' 5 bb-I = ?,hap(f). Hence is order preserving. Since x E up if and only if 2-l E a - l p . It is clear that V($ap) = A(qa-ip),which is an ideal of E from the preceding paragraph. It is then evident from their definition that $(a-~p)($ap) is the identity mapping on A($,@) and, in particular, that Gap is one-to-one. Hence $ap E C ( E ) . Clearly, (?+hap)-' = $(ap)-i. Thus we have shown that satisfies the third condition for a 0-dual-prehomomorphism. Let e E A ( $ a p $ b p ) . Then, for some x E a 0 , y E bP, we have e = y-'y, yy-' = $bp(e) = s-'x. Then $J

$Iap$bp(e) = xx-I = ss-lxx-l = xyy-1s-1 = X Y ( X Y ) - l . Since x y E aPbP and (XY)-'XY

= Y - ' x - ' x ~ = y-'yy-'y

= y-1y = e ,

c

it follows that e E A ( $ & @and ) therefore that A('$"ap$bp) A($apbp). firthcrmore, $=p$bp(e) = $apbp(e) and so $ap'$bp 5 $apbp. Thus $ satisfies the second condition for a 0-dual-prehomomorphism. Clearly, $00 is the zero element of C ( E ) . Therefore $ is a O-dualprehomomor-phism of S into C ( E ) . Next, we prove that .1c, satisfies the condions (i) and (ii). For each Y E E , Y P E T and y = y-'y E A(&p). Thus condition (i) holds.

419

Let e E E(T*),x,y E A($e) n Y * .Then there exist a , b E S* such that e = up = bp and x = K ' a , y = b-lb. I t follows that there is c E S such that ca = cb # 0. Thus cab-' = ebb-' # 0 and so u(u-lab-'b)b-l = ab-' # 0. Therefore x A y = a-lab-lb # 0 and so condition (ii) holds. Now let Q = Q(E,SIP, $) be defined in Theorem 3.2. We shall show that S is isomorphic to Q. Let s E S . Then ss-l E V($'sp)and ( s s - l , s p ) E Q. Hence 4 : s ---t (ss-',sp) is a mapping of S into Q. By Lemma 1.3, 4 is one-to-one. Let s , t E S. Then

4(.)4(t)

( s s - l, s P ) ( t t - l , t P ) ($2'( s s - l ) A tt-' ), sP) = (GSp(s - s~ A tt-'), sptp) =

=

=

($sp(S-lStt-l),

sptp)

NOWst = 0 or stt-' E sp. If s t = 0, then $ ~ ~ p ( s - ' s t t - ' ) = qbSp(O) = 0 and sPtP = ( s t ) p = 0. It follows that 4 ( s ) 4 ( t ) = (0,O) = ( s t ( s t ) - l ,( s t ) P ) = 4

(4

Assume stt-l E sp. Since (stt-')-l(stt-l) = tt-ls-lstt-l

$Jsp(s-'stt-1)= stt-l(stt-l)-l

= s-lstt-',

= stt-ls-l.

Thus

4 ( S ) ( b ( t ) = (stt-5-1, sptp) = +(st). and so 4 is a homomorphism. To see that 4 maps S onto Q, let ( e , a p ) E Q. Then e E V(@ap) and so e = ss-l for some s E ap. Hence 4 ( s ) = (ss-',s/3) = (e,a/3).Therefore 4 is an isomorphism. (2) + (3). Without loss of generality, let

S = Q(Y,T,$)= { ( ~ , Et )Y* x T* : 2 E V($t)} U {(O,O)} where Y is a semilattice with zero 0, T is a primitive inverse semigroup and $ is a 0-dual-prehomomorphism of T into C(Y)satisfying the conditions (i) and (ii). For convenience, we write 3 t . x to mean that 5 E A(&). In this case, we abbreviate &(x) to t . x . I t is easy to show the following conditions hold:

(C1) V t E T,3t . O and t . O = 0; (C2)3t . x implies It-' . (t . x) and t-l . (t . x) = 5; (C3) Vs, t E T ,2 E Y ,3s . (t . x) implies 3(st). x and s . (t . x) = ( s t ) . x; (C4) 30 . II: if and only if x = 0; (C5)Vx E Y * ,3t E T* such that 3t ' x.

420

( C S )Vt E T * ,3x E Y * such that 3 .x. (C7) Ve E E ( T * ) , V xy, E Y * ,3 e . x and 3 e . y imply

2

Ay

# 0.

First, we have the following fact: Lemma 3.5 For each x E Y * , there exists uniquely e E E ( T * ) such that e . x = x . We dtnote the element e b y ex Proof By condition (CS),there exists t E T* such that 3t . x . By conditions (Cz),(C3),(t-’t) . x = x. Clearly, e = t-lt € E ( T * )and e . 2 = x. If there is f E E ( T * )such that f . x = x. Then by (C3),3(ef) . x . By (C4), e f # 0 and so e = f , as required.

-

-

Define the relation on the set T* x Y by ( t ,x ) (t’,x’) if and only if either ( t ,x ) = (t’,x’) or there exists s E T such that t = t’s and s . x = x’. It is direct to show that is an equivalence relation on T* x Y.Denote the -equivalence class containing the element ( t ,x ) by [t,21. Let N

x = { [ t , X ] : ( t , z )E T* x Y }u {el Y

= { [ e x ,x ] : x E Y * }u (0)

By Lemma 2 of section 3 [7],we have the following results. Lemma 3.6 With above notation: (1) Define the relation I on X b y B 5 [ u , x ]and [ u , x ]5 [w,y] if and only if there exist [w’,y’] E [w,y] and x’ 5 y’ such that (w’,x’) E [u,x ] , Then 5 is a partial order on X . (2) The semilattice Y is order-isomorphic with the partially ordered set y with the order I induced from X .

d

Corollary 3.7 If [u,a] 5 [w,b ] , then for every (w, c) E [w,b] there exists 5 c such that (w, d ) E [u,a ] . Now, for each

IL

E T , let us define a partial transformation

& on X by

& ( B ) = B and & ( [ t , x ] )= [ut,x]if ut # 0. It is easy to show that & is injective. Moreover the following statement holds. Lemma 3.8 The mapping 0-restricted homomorphism.

4

of T into C ( X ) define b y

4(t) = &

is a

42 1

Proof (1) +(t)is order-preserving. Assume [u,u],[w,b]E A(4t) and [u,a] I [v,b].By Corollary 3.7, there exists d 5 b such that (v,d)E [u,a]. Thus 4t([u,al) = 4t([v,dl) = [tv,dI

I [%bI

= 4t([v,bl)

and SO 4(t)is order-preserving. (2) Both A(4t) and V(4t) are order ideals of X . Clearly, A(g5t) = { [ s , 4 E X : t s # 0) u {e),V(4t) = A(dJ-1). Suppose [u,y]I [s,z] E A(4t). Then by Corollary 3.7, there exists k E Y such that k 5 2 and ( s , k ) C [u,y]. It follows that there exists v E T* such that s = uv and y = v . k . Since t s # 0,tu # 0 and so [u,y] E A(&). Hence A(4t) is an order ideal of X. Similarly, A(&i) is an order ideal of X.Hence V(q5t) is also an order ideal of X . It follows that 4 is a map of T into C ( X ) . (3) 4 is a 0-restricted homomorphism. Clearly, 4 is 0-restricted. It remains to show that 4 is a homomorphism. Let s, t E T . It is easy to show that A(4s4t) = A(4st). If st = 0, then A(&) = A(4s4t)= (0) and h ( 0 ) = 4dt(e). Then If st # 0, then A(&) = A($.&) # (0). Let [u,z]E 4st([u,zI) = [stu,zl = 4s([tu,4) = 4s4t([u,z]). Thus, in both cases, we both have restricted homomorphism.

4st = 4sq5t. It follows that 4 is

a 0H

Lemma 3.9 (T,X ,Y ) is a Primitive triple.

Proof (1) We claim that y is an order ideal of X . For suppose that [t,z] E X and [t,z] 5 [e,, y] E y . Then by Corollary 3.7, there exists y’ 5 y such that (e,,y’) E [t,z].Thus [t,z] = [e,,y’]. Since y E A($(e,)) and A(+(e,)) is an order ideal of Y , y’ E A($(e,)) and so 3e, . y’. Since 2 and [t,z] = [e,, y’], y’ # 0. Hence e,! exists. Thus e,

. y’

= (eye,) . y’ = eyl . (e,

. y’)

= y’ = e,t

#0

. y’.

By Lemma 3.5, e, = e,’. Hence [t,z]= [e,,y’] = [e,/,y’] E y . Thus y is an order ideal of X . ( 2 ) F o r a l l e ~ E ( T * ) a n d [ s , z ] , [ tE , yA] ( & ) n y * , w e h a v e e s # O , e t # 0 and there exist [e,, a ] ,[eb,b] E y such that [s, z] = [e,, a],[t,y] = [eb, b]. Thus e = e, = eb. From 3e, . a and 3eb .b, we have that a, b E A($(e)). By condition (C7), a A b E A($(e))nY*. Thus 3 e . ( a A b ) and so [e,aAb] E y*.

422

Clearly, [e,a A b] 5 [e,,a] = [s,x] and [e,a A b] 1. [eb,b] = [t,y]. Thus

1% 4 A [t,Yl # 6.

(3) For each t E T * , by condition (Cs) there exists y E Y*such that I t . y. Thus t-'t = ey and so [t-'t,y] E y . Clearly, $q([t-'t,y]) = [ t , ~=] [tt-l,t . y]. Since 3-'. (t . y) and 3t . ( t - I . (t . y)), so tt-' = et.y. Thus [tt-', t . y] E y*. I t follows that 4t(y*)n y* # 8. Turning now to the proof of Theorem. Define a map 6 from S t o M = M ( T ,X ,Y )by 6(0,0) = 0 and S(z,t ) = ([tt-l,x], t ) if x # 0 and t # 0. We first observe that S is well-defined. Since (2,t ) E Q(Y,T ,$), 3t-' . x and so Itt-' .x.Hence tt-' = e,. Thus [tt-',x] E y . Clearly, [tt-',z] E A(&i). and &.I [tt-', x] = [t-l, x] = [t-lt,t-' . x] E y*. Next, let us prove 6 is injective. Assume 6(x,tl) = 6(y,tz) for some (z,tl),(y,tz) E S*. Then [tlt,',~] = [tzt,',y] and tl = t 2 . It follows that there exists u E T such that tit,' = t&lu and u . x = y. Thus u = tit,' E E ( T ) and so y = u . x = x. Hence S is injective. Next, we shall prove that S is surjective. Let ([ey,y],t) E M* such that [cy,y] E A(&-i). Then t-le, # 0 and so ey = tt-'. I t follows that ([ey,y],t)= S(y,t) and so 6 is surjective. Last, let us show that 6 is a homomorphism. It suffices to prove that S(ab) = S(a)S(b) for all a, b E S*. If (x,t)(y, u) = (0,0), then tu = 0 and so

S(z,t)S(y,u) = ([tt-',x],t)([uu-',y],u) = 0 = 6(0,0) = S((x,t)(y,u)). If ( ~ , t ) ( y , u#) ( O , O ) , then tu

# 0 and so

6(s7t)6(y,u)= ([tt-',Zc] A [ t ~ u - ~ , y ] , t u ) . It therefore remains t o show that

[ t t - ' , ~ ] A [ t ~ u - ' , ~=] [Cu(t~)-',t.(t-' . x A Y ) ] . That is

[tt-', x] A [t,y] = [tt-', t . (t-' . 2 A y)]. Clearly, t-(t-'.xAy) 5 t-(t-'.x) = x. Hence [tt-',t.(t-'.xAy)] Also, t . (t-' . x A y) 5 t . y. Hence

[tt-',t . (t-' . z A y)] 5 [tt-', t . y] = [ t , ~ ] . It follows that

[tt-',t. (t-l . x Ay)] 1. [tt-',XI

A [t,~].

5 [tt-',~].

423

Conversely, suppose that [u,z] 5 [ t t - ’ , ~ and ] [ u , z ] 5 [ t , y ] . Then by Corollary 3.7, there exist z’,z” E Y such that z‘ 5 x,z” 5 y and [tt-l,z’] = [ u , z ] = [t,z”]. I t follows that there exists w E T such that t = tt-lw and z’ = w . z”. Thus w = t and z‘ = t . z“. It follows that zll = t-’ .z’ and so zll 5 t-l.2. Thus zll 5 t-’ . x A y and consequently zI = t.z” 5 t . ( t - l . x A y ) . It follows that [u,z]= [tt-’,z’] 5 [ t t - l , t . ( t - l . z A y ) ] . Thus [tt-’, t . (t-l . x A y)] = [tt-l,x ] A [t,y ] , as required.

References 1. S. Bulman-Fleming, J. Fountain and V. Gould, Inverse semigroups with zero: covers and their structure, J. Austral. Math.Soc. (Series A ) , 67,15-30(1999). 2. G.M.S. Gomes and J.M. Howie, A P-theorem for inverse semigroups with zero, Port. Math. 53, 257-278(1996). 3. Zhonghao Jiang, Some notes on strongly E*-unitary inverse semigroups, Periodica Mathematica Hungarica, 44, 75-80(2002). 4. Zhonghao Jiang, The structure of 0-bisimple strongly E*-unitary inverse monoids, submited. 5. Zhonghao Jiang, Xg-semidirect products of 0-semilattices with primitive inverse semigroups, their inverse 0-subsemigroups and 0-restricted homomorphic images, Groups, Algebras and Geometries, accepted. 6. J. Kellendonk and M.V. Lawson, Tiling semigroups, Journal of Algebra, 224, 140-150(2000). 7. J. Kellendonk and M.V. Lawson, Partial actions of groups, their globalisations and E-unitary inverse semigroups, preprint . 8. M.V. Lawson, The structure of 0-Eunitary inverse semigroups I: The monoid case, Proc. of the Edinburgh Mathematical Society, 42, 497-520(1999). 9. M.V. Lawson, Inverse semigroups: the theory of partial symmetries, World Scientific, Singapore ,1998. 10. S.W. Margolis and J.E. Pin, Inverse semigroups and extensions of groups by semilattices, Journal of Algebra , 110, 277-297(1987). 11. M. Petrich, Inverse semigroups, John Wiley and Sons, 1984 . 12. M. Petrich and N. Reilly, A representation of E-unitary inverse semigroups, Quart. J. Math. Oxford, 30,339-350(1979).

New examples of noncommutative and noncocommutative bialgebras* Jinqi Li Miaosen Chen Department of Mathematics, Zhejiang Normal University Jinhua, Zhejiang 321004, People's Republic of China

AbstractLet M be a k-vector space and R E Endk(M @ M ) . The Hopf equation: R23R13R12 = R12R23 and the following FRT type theorem: if M is a finite dimensional vector space and R E Endk(M @ M ) is a solution of the Hopf equation, then there exists a bialgebra B(R) such that ( M ,. , p ) EB(R) M B @ ) . were introduced and mentioned in the monograph of G. Militaru. In this paper, we will use the above to construct new examples of noncommutative and noncocommutative bialgebras which are different from the previous arising from quantum group theory. 2000 Mathematics Subject Classification: 16W30 Keywords: Bialgebra; Hopf equation; FRT type theorem;

1

Preliminaries

Throughout this paper, k will be a field. All vector spaces, algebras, coalgebras and bialgebras that we considered are over k , @ and Horn will mean @k and Horn,,. For a coalgebra C , we will use Sweedler's C notation['], that is, Ac = Cc1 @ c2, ( I @ A)A(c) = Cc1 @ c2 @ c3. Definition 111 ' Let M be a vector space and R E Endk(M @ M ) . Then R is said to be a solution for the Hopf equation i f

holds,where the maps of Endk(M @ M ) R12, RI3, R23are given by

R12 = R @ I , R23= I

@

R, R13 = ( I @ T ) ( R@ I ) ( I @ T )

*Research is partially supported by a National Natural Science Foundation grant of China.

424

425

Lemma 1[2] Let f , g E E n d k ( M ) such that f = f , g2 = g and f g = g f . Then R = f 8 g is a solution of the Hopf equation Lemma 2i21 Let {mi}i E I be a basis of the vector space M . Then a n , ~ Ic~ endomorphism R of M 8 M is given by a family of scalars ( ~ t , ) i , j , ~ , of such that R(m, 8 mu)= xC,rni 8 mj for all u,v E I . Also R is a solution of the Hopf equation i f and only if ( ~ ~ , ) i , j , ~ ,is, , a~ ~solution of the nonlinear equations

c

x ~ x $ x= ~cx;,x;; ~

a,Ar

i

f o r all j , k , I , u, v, w E I . Lemma 3 [2] Let M be a finite dimensional vector space with a basis { m l , m2,. . m,} and R E E n d k ( M 8 M ) is a solution of the Hopf equation, we denote 0

,

R(m, 8 mu)= xcumi 8 mj (1) for all u, v = 1, 2, . . ., n. 1. Let(C, A, E ) = M n ( k ) be the comatrix coalgebra of order n, i.e. C is the coalgebra with the basis {qjli,j= 1 , 2 , . . ., n } such that A ( C i j ) = C C i k 8 ckj, E ( C i j ) = dij Then T ( C ) has a bialgebra structure on the tensor algebra T ( C ) which extends A and E . 2. Let n n

V i , j , k , 1) =

c

&%kGl

-

u,u=l

c

%Cia

a=l

for all i , j , k , 1 = 1, 2, . . ., n, I be the two-side ideal of T ( C ) generated b y all A ( i , j , k , I ) , i , j , k , 1 = 1 , 2 , . . ., n. Then I is a bi-ideal of T ( C ) and hence B ( R ) = T ( C ) / I is a bialgebra. I n fact, the bialgebra B ( R ) is a free algebra generated by with the following relations n

c

u,u=l

c n

X ~ , C U k C , ~=

xgcia

a=l

for all i, j , Ic, 1 = 1 , 2 , . . ., n. The comultiplication A and the counity E are given by

(2)

426

2

New examples of noncommutative and noncocommutative bialgebras Let M be a two dimensional vector space with basis {ml,mz}. Let

f, E E n d k ( M ) , with respect t o the given basis, is defined by

fi

where q is a scalar of k. Then = f,. Thus R1 = f, €3 f,, R2 = f, €3 idM are the solutions of the Hopf equation with respect to the basis { m l g ml, ml €3 m2, m2 @ ml, m2 €3 m2) so that

R1=

(

q2

q2

Q ( 1 - Q) q(1 - Q ) (1 - 412

Q(1- 4)

q2 d 1 - Q)

41 - Q )

4 1 - 4) (1 - qI2

(1 - d 2

Q(1- Q ) Q(1- qj (1 - 4 )

0

4

Q

0

1-q

0

1-q

0

0

1-q

0

1-q

Q

R2=(

Proposition 1 Let q be a scalar of the field k and R1 the solution of the Hopf equation. Let HI be the bialgebru B(R1). Then the bialgebru H I is the free algebra generated by A , B with the relation B3 = B2, the comultiplication A and the counity E given by

AA = A €3 A

& ( A= ) 1;

AB = ( B - B2)€3 A + B2 €3 B,

E ( B )= 1

Proof. Let M be a two dimensional vector space with { m l , m2}a basis of M with respect to the ordinate basis {mlgml, m1@m2, m2@m1,m2@m2}. Let R1 be given by:

Rl(ml@ml) = Rl(ml@m2) = Rl(m2 @ ml) = Rl(m2 @mz) = q2ml @ m l +q ( 1 - q)ml € 3 7 4

+ q(1-

q)m2 @m1+(1 - q )2 ml

Bml

Now by (l),we get, among the elements (&), the following equalities

xu, 11

= q2

x”,’,= xi;

= q(1- 4)

xu, 22 =

(1 - d 2

427

for all u, w = 1 , 2 . By (2), The sixteen relations X ( i , j , k,I) = 0 are

+

q2z1= q2c11 Q(1 - q ) c 1 2 $23 = q2c11 4(1 - q ) c 1 2

= q2c11 Q(1- q ) c 1 2 4 0 - Q)2 2 = Q(1 - q)c11 (1 - Q ) 2 c 1 2 4(1 - Q b 4 = 4(1 - Q)C11 (1 - 4 ) 2 c 1 2 4(1 - 4 1 x 2 = q 2 c 2 1 q(1 - q ) c z 2 Q(1 - q).4 = q 2 c 2 1 q(1 - q ) c 2 2 (1 - 4 I 2 z 2 = 4(1 - q ) c 2 1 + (1- q ) 2 c 2 2 (1= Q(1 - q ) c 2 1 (1- q ) 2 c 2 2 q2z4

dl - 4 b 1 = 41 - Q ) C 1 1 + (1- 4 I 2 c 1 2 dl - 4 x 3 = Q(1- d C 1 1 + (1 - 4 I 2 c 1 2

+

q(1 - 4 x 1 = Q2C21 q(1 - 4 C Z 2 q(1 - 4.1.3 = Q2C21 + q(1 - Q)C22 (1 - 4)2zl= q(1 - Q)C21 (1- q ) 2 c 2 2 (1 - 4I2z3 = q(1 - 4)CZl (1 - Q)2c22

+ +

where 51 = cllcll+cllczl+c21c11+c21c21, 2 3 = C12Cll

+ q(1 - q)c12 + + + + + +

q 2 z 2 = q2c11

+

+ c12c21 + C22Cll + c22c21,

z4

2 2 = ~11c12+c11c22+c2lc12+c21c22,

= c12c12

+ c12c22 + c22c12 + c22c22,

If we denote z = c11, y = c 1 2 , z = c21, t = c 2 2 , then we have the following cases (i) If q = 0, then y = 0, so there are only relations (z

+ z ) 2 = t,

(z

t(z+ z ) = t,

+ z)t = t,

t2 = t

In this case t is an element in the free algebra generated by z and z. If we substitute t in the other three relations, then only the following relation (z

+4

3 =

(z

+

z)2

remains, and the other two are linear dependent from this one. Now, if we denote A = z, B = z z, then B3 = B2. Thus, the formulas for A and E follow from equation (3),and hence we obtain the description for HI. (ii) If q = 1, then z = 0. Hence, there are only relations

+

z2 = z , z ( y + t ) = z , ( y + t ) z = z ,

(y+t)2=z

This shows that z is an element in the free algebra generated by y and t. If we substitute z in the other three relations, then only the following relation (Y

+ t)3 = (y + t ) 2

remains, and the other two are linear dependent from this one. Now, if we let A = t, B = y t , then we obtain a description of HI. (iii) If q # 0,1, we denote p = qP1(1 - q ) , then p + 1 = q-l, p-l 1= (1- q)-'. so there are only relations

+

+ + z z + z2 = z +py,

z2 5 z

+

xy

+ zt + zy + zt = 2 + p y

428

y z +y z

+ tz + t z = z + p y ,

y2

+ y t + t y + t2 = 2 + p y ,

x +py

=t

+p-lz

Hence t is an element in the free algebra generated by z, y and z. If we substitute t in the other four relations, we obtain the following four relations: (z

+ z)2 = x + py

p ( y z + y z ) = zz

+ z2

p(xy

+ z y ) = Icz + z2

p ( z y + Y z ) = z2 + P2Y2

So p y = (z + z ) 2- z is an element in the free algebra generated by z and z. If we substitute t in the other three relations, only (z + z ) 3 = (x + z ) remains. ~ Now, if we denote A = 5, B = x + z , then we also obtain a description of H1. Corollary 2 Let J be the two-side ideal of H1 generated by B2- B . Then J is a bi-ideal of H I , so that H = H l / J is a bialgebra which is a free algebra generated by = A+ J , y = B+ J with the relation y 2 = y , the comultiplication A and the counity E are by Ax = x 8 z, A y = y 8 y , ~ ( x=) ~ ( y=) 1. If H is a Hopf algebra, then y = 1. Hence, H is the usual Hopf algebra k [ z ,z-l], with A X = z @ z, E ( X ) = 1. Proposition 3 Let q be a scalar of the field k and R2 be the solution of the Hopf equation. Let H2 be the bialgebra B(R2). Then the bialgebra H2 is the free algebra generated by A, B with the relation A B = B 2 , B3 = B2. The comultiplication A and the counity E are given by

A A = A B A , & ( A )= 1; A B = ( B - B 2 ) @ A +B 2 @ B , E ( B )= 1

Proof. Let A4 be a two dimensional vector space with a basis { m l , m 2 } with respect to the ordinate basis {ml @ m l , ml 8 m2, m2 8 ml, m2 8 m2}. Let R2 be determined by:

Ri(m18 m2) = &(m2 8 m2) = q m l 8 m2

+ (1- q)m2 8 m2

Now by (l),we get, among the elements (xCu),the following equalities zll 11

= z;

= z21 21 = z22 22 = q

Xll 12

= x;; = z21 22 = 2 222 2 = (1 - q )

429

By (2), the sixteen relations X ( i , j , k,1) = 0 are the following:

+ c11cz1) = + (1 + =0 + =0 + = + (1 + = (1 (1 + =0 (1 + =0 + = + (1-

Q(CllCl1

QCll

Q(ClZC11

c12cz1)

Q(CZlCl1

c21c21)

Q(CZ2Cll

c22c21)

(1 - Q ) ( C l l C l l

QCll

c11c21)

c12c21)

- Q)(C2lCll

c21cz1)

c22c21)

q(cllcl2

CllC22

dC22

= QCll

+ +

+ (1-

Q)ClZ

0 Q(CZlCl2 c21c22) = 0 Q(C22C12 c22c22) = qcll (1- q ) c 1 2 (1- Q ) ( C l l C l 2 CllC22) = Q C Z l + (1- q ) c 2 2 (1 - q)(c12c12 C l Z ~ Z Z )= 0 (1 - Q)(CZlClZ c21c22) = 0 (1 - Q)(C22C12 c22c22) = QCZl (1 - q ) c 2 2

+

- q)c22

QCZl

+

Q(C12C12

- Q)ClZ

QC21+

- Q)(C12C11

(1 - Q)(C22Cll

- Q)ClZ

+

C1zCzz) =

+

+ +

+

+

If we denote = ql, y = c12, z = cgl, t = cZ2, then we have the following cases (i) If q = 0, then y = 0, so there are only six relations 2(z

+z) =t

zt

=t

z(x

+z) =0

zt

=0

t ( z + z ) = t t2 = t

In this case, t is an element in the free algebra generated by z and z . If we substitute t in the other five relations, only the following three relations:

x 2 ( x+ z ) = z(x + z ) , remain. Now, if we denote A = z,

+ z ) = 0 zz(. + z ) = 0 B = z + z , so there are relations z(5

A2B = AB B2 = AB A2B = BAB Because AB = B 2 , there are relations AB = B2, B3 = B2. The formulas for A and E follow from equation (3). We obtain a description of H2. (ii) If q = 1, then z = 0, so there are only the following relations, namely, 2

2 =z

z(y+t)

=2

yz = 0

y ( y + t ) = 0 tz = 2

t ( y + t ) = 2.

Then x is an element in the free algebra generated by y and t. If we substitute z in the other five relations, only the following three relations:

t2(y

+ t ) = t(y + t ) ,

y(y

+t)= 0

yt(y

+t )= 0

+

remain. Now, if we denote A = t , B = y t , then there are only the following relations AB = B2, B3 = B2. We obtain a description of H2. (iii) If q # 0,1, we denote p = q-'(l - q ) , then p + 1 = q-l, p-l 1= (1 - q)-l. so there are only relations

+

+

z2 xz = z + p y

sy

+zt

=z

+py

yx + yz = 0

430

y2+yt=O tx +tz = x +py

zx+z 2 = o ty + t 2= x +p.y

zy+zt=O x +py

=t

+p-'z

Then t = x+py-p-lz is an element in the free algebra generated by x and y and z. If we substitute t in the other eight relations, we obtain six relations:

x(x+z)=x+py py2 = yz

zx

pxy=xz

+ z2 = 0

yx+yz=o pzy

=z

2

So p y = z ( x + z ) - x is an element in the free algebra generated by z and z. If we substitute p y in the other five relations, then only x 2 ( x z ) = x ( x z ) , z ( x + z ) = 0 , z x ( x + z ) = 0 remain. Now, if we denote A = x , B = x + z , then there are relations A B = B2,B3 = B2. We hence obtain a description of H2.

+

Proposition 4

+

Let R be the matrix of M4(k) given b y

Then: 1. R is a solution of the Hopf equation. 2. If the q = 0, then the bialgebra B ( R ) is the free algebra generated by x , y , t with the relations:

x 2 = x , xy = y x = y 2 = 0 , t x = X I t y = y , t2 = t . The comultiplication A and the counity e are given by:

3. If the q # 0, then the bialgebra B ( R ) is the free algebra generated by x , y , t with the relations:

x2

= X I

xy = yx = 0, y 2 = q y , t x = 2 , ty = y , t2

= t.

The comultiplication A and the counity c are given by:

A x = x @ x , E ( x ) = ~ A, y = x @ Y + + @ t , E ( Y ) = ~ , A t = t @ t , E(t)=1.

431

1. By direct computation, we obtain the following:

Proof.

R 12 =

’1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

‘1 0 0 0 0 0 0

o o o o o o q o

,

o o o o o o o q 0 0 0 0 0 0 1 0 o 0 0 0 0 0 0 1

,o

0 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 0 0

0 0 0 0 q 0 0 0

0 0 0 0 0 q 1 1

‘1 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0

R 12 =

o o o o o q o o ,

0 0 0 0 0 1 0 0 o o o o o o o q o 0 0 0 0 0 0 1

Then

(1 0 o 0

0 1 o 0

0 0 o 0

0 0 q 1

0 0 o 0

0 0 o 0

0 0 o 0

0 0 o 0

o o o o o o o q o o o o o o o q

o o o o o o o q

( 0 0 0 0 0 0 0 1 This shows that R is a solution of the Hopf equation. 2. Let M be a two dimensional vector space with { m l , m2} a basis of M . Then we have:

R(mz €4 m i ) = 0 R(mz €4 m2) = qm2 8 ml

+ m2 €4 m2.

We obtain, among ( x c v ) the only nonzero elements, the following results:

x::

=

1,

12 222

= q,

22

x22 = 1.

432

Now by (2), the sixteen relations A ( i , j , k,1) = 0 lead to only eight relations: CllCll

= c11

CllC12 = 0

c21c11 = 0 c21c12 = 0

C12Cll

C22Cll

= 0 c12c12 = qc12

= c11 c22c12 = c12

qc21c21 = c21 qc21c22 = 0 qc22c21 = 0 qc22c22 = qc22 c21c21 = 0 c21c22

If we denote x

=0

c22c21 = c21 c22c22

= c11, y = c12, z = c21, t = c22,

= c22

then we have:

x 2 = x x y = o y x = o y2 = q y z x = o z y = o tx = x t y = y q z 2 = z qzt

=o

qtz=O z 2 = 0 z t = O t z = z t 2 = t (i) If q = 0, then z = 0, so there are seven relations x2 = x x y = o y x = o y 2 = 0 t x = x t y = y t 2 = t (ii) If q # 0, then z = 0, and so there are also seven relations, namely, x 2 = x x y = o y x = o y 2 = q y t x = x t y = y t2 = t Hence, we obtain the description of B ( R ) . Corollary 5 Let J be the two-side ideal of B(R ) generated by t - 1. Then J is a bi-ideal of B ( R ) . 1. If q = 0, then H = B ( R ) / J is a bialgebra which is a free algebra generated by X = x J, Y = y J with the relation

+

+

x2= x,

XY

=Y

The comultiplication A and the comity

AX

=X

E

X

=

Y 2= 0.

are given by:

@ X , E ( X )= 1, AY = X @ Y+ Y @ 1, E ( Y = ) 0,

Furthermore, H is three dimensional with { 1, x, y} a k- basis. 2. If q # 0, then H = B ( R ) / J is a bialgebra which is a free algebra generated by X = x J, Y = y J with the relation

+

+

x2= x, X Y = Y X = 0,

Y 2= 0.

433

The comultiplication A and the counity AX

=X

B X , E ( X )= 1, AY

E

are given by:

=X

B Y +Y B 1, E ( Y =) O ,

Furthermore, H is three dimensional with (1, x, y } a basis. Remark: In [3], two examples of three dimensional bialgebras over a field of characteristic two, are given. Both of them are commutation and cocommuattion. We now give two examples of three dimensional bialgebras. Both of them are noncommutation and noncocommuattion for an arbitrry field k.

References [l] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.

[2] G . Militaru, The Hopf modules category and the Hopf equation, Communications in Algebra, 26( 10),3071-3097.

[3] I. Kaplansky, Bialgebras, University of Chicago Press. Chicago. 1995.

Generalized con-cos groups A.S.Muktibodh* Department of Mathematics Mohat,a science college,Umred rd. Nagpur, INDIA

Abstract Con-Cos groups are the groups for which two methods of counting viz. Counting by cosets and counting by conjugacy classes closely coincide.Consequentlypm abelian groups can be classified as Con-Cos and non Con-Cos groups. We have defined generalized Con-Cos groups wherein we have replaced the particu1a.r conjugacy class by an order class of the group.We have proved some special properties of generalized Con- Cos groups. It is observed that Dihedral groups

4, (p,a prime) are the

generalized Con-Cos groups.Lastly we have conjectured trhatDihedral groups D, are the only generalized Con-Cos groups.

AMS subject classification number:20Bxx20B05 KEY WORDS: Con-Cos group,generalized Coil-Cos group:near abelian group,half subgroup,order class

1

Introduction

Definition 1.0.1 Let G be a group.Let a E G. N be a normal subgroup of G such that

N = cl(a) u ( e }

.

Let there exists a one to one correspon,dence betuieen the cosets ofiV i n

G and the co7riplexe.s in G where one of the complex is N and all other complexes are the nontrivial conjugacy classes in G other than cl(a).Then,G is called a Con-Cos group.

If n = e then G is called a trivial Con-Cos group. Every abelian group is a trivial Con-Cos group in which N = cl{e} U { e } = { e ) serves

as our normal subgroup of G. Obviously, order of all the conjugacy classes other than *Authorto whom all correspondence should be directed; email: aniukti20000yahoo.com

434

435 cl(n) will be same; where as order of cl(a) is one less than the order of other conjugacy

c:lasses.We call such c l ( n ) ~3 a deficient conjugacy class. In a Con-Cos group when we have to distinguish between deficient conjngacy class and other conjugacy classes, we call ot,her conjugacy classes as non deficient conjugacy classes.

A simple group is not a Con-Cos group.

2

Examples of non trivial Con-Cos groups

Ex 1. DL1is a non trivial Con-Cos group.

U, V } . Here, the conjugacy classes Consider the Dihedral group D4 = { I , Q , H , T,X!Y, are ; cl(a) = { I } , c l { H }= { H } ! c l ( X )= { X , Y } , c l ( V )= { U , V } , C l ( Q )= { Q , T } We choose N = { I ,H } = c l ( H ) U { I } .

Then, Dq/N = { N !{ X , Y } ,{ U ~ V } ~ { Q , T } } = { N ~ ~ ~ ( X ) ~ ~ ~ ( V ) , C ~ ( Q ) } Thus ,there exists a one to one correspondence between the cosets in Dd/iV and complexes in G where one of the complex is N itself and all the other complexes are nothing but non trivial conjugacy classes in G other than cl(H).Hence Dd is a Con-Cos group. Ex 2. Adis a Con-Cos group. Consider the alternating group, Aq = (1, c, C2! a , b, ah, ca, cb, cab, 2 u , c2b,C2ub}. Then, the conjugacy classes are, cl(1) = { l } , c l ( a )= {a;b,ab},cl(c)= {c,ca.,cb,cab}, d ( c z ) = {c', c2b,c2a,cza,b}. Consider

N

= cZ(a) U

(1). Then,

A 4 / N = { N ,{c, co.?cb, cab}, {c', c2a,:2ab}} = { N , cl(c),cl(cz}. Thus, there exists a one to one correspondence between 44/M and the complexes in G where one of the complex is N itself and all the other complexes are nothing but the lion trivial conjugacy classes in G other than cl(a). Thus, alternating group Ad is a Con-Cos group. It can be verified that there is no non trivial Con-Cos group of order less than eight.

Definition 2.0.2 A group G is called a near ahelaan group if all the non trivial commutators of G are in a single conjuqacy class.

436

Property A : A group whose conjugacy classes have same number of eleineiits except for one conjiigacy class which has number of elements one less than the other conjugacy

classes is said to satisfy propert,y A. Definition 2.0.3 Let (G;*) be a group and S be a subset of G. Let

(S, *) be a halfgro.uppoid

(partially closed subset) of G siich that ; 1.

(0,

* b ) * c = a * ( h * c); 'da,b?c E S

uiheiiever the corresponding products are in S .

2. P E S.e ,is the identity element of G . 3.

a-1

E S,Vn E

s.

then, ( S ,*) is called

0,

h,alf svbgroup of the yrorrp G .

Definition 2.0.4 Let G be a group. A subset 0, of G de,fined as; 0 , = { b E G / o ( b ) =

.(a), n

E G } pis called an order class of a

Proposition 2.0.5 If G i s

0.

n,on tri?,%dCon-Cos group then order of G is a. coir1,posite

number.Also, o(G)=[(Total nuiriber of conjugacy classes in G)-1][ order of a non deficient conjugacy class].

Proof. Let, G be a non t,rivial Con-Cos group.Let o(C)=r. Since, G is Con-Cos group there exists a normal subgroup ili in G siich that N = cl(ci) U {e} and

N, = cl(z)f o m # e.

Let

o(N)=m. Then m=order of a noii deficient coiijugacy clilss.

Thus,o(G)=r=in[No. of cosets of N in GI. = m[(Tot,al no.

of coxijugacy classes in G )-11

hence. the result . Proposition 2.0.6 If G is a Con-Cos group such that

N

=

cl(a) U { e } then G/IV is

abelian. Proof. G is a Con-Cos group

+ NrNY = cl(x)cl(y)~Vz,?jin G and at least. one of 1 and y

.

is not e where ATz and N , are the cosets of

classes of

Let

3;

2 E

and y respectively.

cZ(z)cl(y) = cl(z) (say)

N in G aiid c l ( . ~and ) cl(y) are the coiijugacy

437

Also; z E c l ( z ) + z-' E cl(z)

But, y;'~c;~ E cl(y)cl(z) Therefore, z-l E cZ(y)cl(s) =+ 3

E cl(y)cl(z)

Thns,cl(.r)cl(y)

C cl( y)cl(z)

Similarly, it can be proved that cl(y)cE(.z)C cl(.r)cl(y)

Therefore, cl(z)cl(?y)= c l ( y ) c l ( z ) Hence, GIN is abeliaii.

Proposition 2.0.7

If G

is a Con-Cos group then all the n o n trivial commutators of G

are in only one conjugaq class, and N i s th,e minimal normal subgroup which contains all the commutators. Proof,Let G he a Con-Cos group.Then G satisfies property A.

+ 3 a normal subgroup N = cl(a) u ( e } such that G I N is abelian. +Set of all commutators in G is contained in N

As the set of all comniutators in G is the union of conjugacy classes in G , cl(a) is the set of all lion trivial commutators and thus,iV = cZ(a) U { e } must be the minimal normal subgroup which contains all the commutators.

Corollary 2.0.8 Euery Con.-Cos group i s a near abelian, group.

438 Proof. Let G be a Con-Cos group. By above theorem all noii trivial commutators lie in a single rorijugacy class.Thus G is a near abelian group. Corollary 2.0.9 If G is a Con-Cos group then number commutators in G is equal to the

order of a non deficient conjugacy class.

Proof. Follows from the above theorem. Corollary 2.0.10 If G is a Con-Cos group then all the non trivial comm,utators i n G are of same order.

Proof. Let G be a Con-Cos group. Since all the noxi trivial commutators in G are in a single conjngacy class; they are of same order. The above cmrallary helps us in eliminating certain non abeliaii groups as non con-Cos groups. Theorem 2.0.11 If G is a Con-Cos group th,en there exists a h,alf subgroup H uiithout

identity i n G such that H U { e } is a derived svb,qroup of G.

Proof. G is a Coil-Cos group. =+ N

= d ( n ) U { e } , such

that,

N is a normal subgroup of G. cl(a) is a partially closed subset of G. Note that a6 E cl(a)foraZlb # a-'.If a E cl(a) =+ a-' E cZ(n).Clearly, H is a half subgroup without identity.

N is a, derived subgroup as all the commutitt.ors are in ilr. Hence the theorein Theorem 2.0.12 If G is a Con-Cos group then there exist two consecutive integers which

divide order of G Proof. Let G he a Con-Cos group.Then by definition there exists a deficient conjugacy class cl(a) such that N = cl(a) U { e } is a normal subgroup of G. Ohviously, o(cl(n))/o(G).

439 Also, o(iV) = o (cl (n ))

+ 1, and o ( N ) / o ( G ) .

Thus. there exist two coiisecutive integers which divide order of G.

Generalized Con-Cos groups Definition 2.0.13 Let G be a group. Let a E G. N be a norniml subgroup of G such that

N

= 0, u { e } = 0,.

Let there exists a, one to o'ne correspondence between the cosets of ilr

in,G and the complexes in G where one of the complex is 0 an.d all other com,plexes aTe the n.on trivial conjugncy classes in C: other than those in 0,.Then G is called a generalized Con-Cos group. N o k that

1) All abelian groups are generalized Con-Cos groups where a = e

:

2) If 0,1is a single conjngxy class then the group is a Con-Cos gronp.

Examples of generalized Con-Cos groups

3

Ex.1 Dihedral group

0 3

is a generalized Con-Cos group.

Consider the Dihedral group D3 = { C J I ~(72, ~

where 01,

(TZ, 03

w e t,he rotations and

T

3 ~~ 7u ,27 , ~ 3 )

denotes t,he refleckion.

Here,O,, = cl(u2) = { c J ~ , o ~ } Now,iV = { P I ,0 2 ,

u3)

is the required 1iornia.1subgroup whose cosets iii

0 . 7are

in one to

one correspondence with the conjuga.cy class cl(r).

Thns,Ds is a generalized Con-Cos group. Of rourse

0 3

is also ii Con-Cos group.

Ex.2 Dihedral group D7 is a generalized Con-Cos group. Consider the Dihedral group D7 = {e, a, nb, a!?,ab3, a,b4,ab5,a,b6, b, b2,b3, b4! b5, bG}

Then,cl(b) = {b?b6} , cl(b2)= { b 2 ,b5}),cl(b3)= {b",b4} Also, cZ(n) = { n , ab, ab2,&, a.b4;ab5jub6}

o,,= {b?b'! b 3 ) b", b", bfi} and N = { e , b, b2,h",b4, b5,be} is the reqiiired normal subgroup of

0 7

whose cosets in D7 are in one t o one correspondence with the conjugacy class cl(a,).

Thus, D7 is a generalized Con-Cos group.

440

Theorem 3.0.14 If G is a generalized Con-Cos group then there em&

an order class

with unity (a, h,alf .sul~group)in G which is a, cyclic group of prime order. Proof. Let G be a generalized Con-Cos group. Hence,there exists an order class O,in G such that N

=

0, U ( e l i s a normal subgroup of G . A n order class with .unity can be a normal

s~~bgroup on,ly when it ,is a, cyclic group of prime order. Thus,N is a, cyclic group of prime order.

Theorem 9.0.15 Dihedral groups D p (p3aprim,e ) is a generalized Con-Cos group. Proof. Consider a dihedral group D, where p is a prime. Then D, will have only two order classes.viz. order class of elements of order 2 and order class of elements of order p. Order class of elements of order p with e, will form the required nor-ma1 subgroup.Hence,D, will be th,e generalized Con-Cos group.

Corollary 3.0.16 A generalized Con-Cos group contains a sample group. Prmf.

EbllOlliS

from the theorem.

We have already shown, that all Dihedral groups D p (p: a prime) are the generalized Con-Cos groups. Now we make the following Conjecture. Conjecture: G is a generalized Con-Cos group @ G is a Dihedral group D p (p,a p r h e ) . Thus, Dihedral groups D,

hi,

a prime) can be characterized as Generalized Con-Cos

groups. References: 1: BAUMSLAG B. and CHANDLER B. Theory and problems of Group Theory SCHAUM'S

OUTLINE SERIES, McGRAW-HILL BOOK COkIPANY 1968 2: Bruck R . H. Sumiey o,f binary systems.Sprin,ger Verlag,Berlin.

9: Fraleigh John B. First coiirse in -4bstract Algebra. Narosa publishing house 1998.

441

4:

Gallian Joseph.

Coiiteinporary AbstractAlgebra.Narosa p,ubltshing house,4th edi-

tioii1999. 5: Mackdonald Ian. D. Theory of ,qroups. Oxford vniversaty Press, 19G8. 6: Muktibodh A.S. Con-Cos groups. Nagpur,2002

7: Zassen,haus Hans. The Theory of Groups

On Some Non-Associative Algebras Using Additive Groups * Ki-Bong Nam t January 8, 2003

Abstract

We define the Lie admissible non-associative algebra NA,,,,, in this paper. We show that NA,,,,, is simple. The corresponding Lie algebra of NA,,,,, is studied in the paper [3]. The matrix ring M,(F) is a subalgebra of N A , , , ~ , ~ . We find all automorphisms of the algebra NA,,,,,~. We define a Jacobi algebra by Theorem 4 in this paper. Key words: simple, non-associative algebra, left diagonal, annihilator, automorphism, Jacobi algebra

1 Preliminaries Let N be the set of all negative integers and Z be the set of all integers. Let F be a field of characteristic zero. Let F [ x l ,. . . ,z], be the polynomial ring with the variables zl, . . . ,2,. Let Al, . ' . ,A, be additive subgroups of F. Let us define the commutative, .. ,eAnzn,zfl, . . .,z,*1 ,z,+I,. . . ,z,] with the associative F-algebra FA,,^,, = F[eAIZ1,. standard basis

{ealsl , . .ehxnx: . . . x2lai E Ai,... ,a, E A,, zl,...,irn E Z,irn+l,...,is E N , l 5 n,m 5 s } and with the obvious addition and multiplication of it. Let us define the vector space N ( A , , m, s) over F which is spanned by

(ealZ1 . . .eanznzt

. . . xy&(al E AI,...,~, E An,il,...,im E Z, '

i r n + ~ , . ~ . , i a E N , ~ ~ ~ , ~ < ~ , 1 < t ~ s(1) } 'Mathematics Subject Classification: Primary 17B40, 17B56. tDepartment of Mathematics and Computer science, University of -Whitewater, Whitewater, WI 53190

442

443

where at, 1 5 t 5 s is the usual partial derivation with respect to xi.Thus we may define the multiplication * on N(A,, m, s) as follows:

. . e a n x n x ? . . .x-at * . . .e b n x n d . 1 . . .eav . . .eanxn+bnxnxii+31 .. .,:+L-& 1 + j t e a i z i + b i x i ...eanxn+b,xnZii+3i ,,.z:+j8z.lav 1 ealxl.

= bw,tbteaixi+bixi

(2)

where S,,,t is the Kronecker delta for w E { 1,. . . ,n } , 1 5 t ,v, 5 s, and eaiX1. . .eanzn ....?at, ehzl . . .e 6 n ~ n x f . . .xFav E FA,,",,. Thus we have the non-associative algebra NA,,",~with the multiplication * in (2) and with the set N(A,, m, s). The set of all right identities of NA,,",, is {Cllulsxu& Cllu5, ~ d , ) c , E F}. There is no left identity of N A ~ , O ,Let + . (A, +, .) be an associative F-algebra. Then (A[,],+, [ , I ) is a Lie algebra using the commutator [,] which is defined as follows; [a,b] = a . b - b . a, for any a, b E A. Let us call the above Lie algebra as A[,][2]. Let ( N , *) be a non-associative F-algebra. Let us define a semi-Lie algebra (",I, using the commutator [a,b] = a * b - b * a , for any a,b E N such that N and N[Jhave the same additive groups with respect to +. If (",I, [,I) holds the Jacobi identity, then (N[,l,+, [,I) is a Lie algebra over F. If ( N I J , [,I) is a Lie algebra, then we call the non-associative algebra N as a Lie admissible algebra. The algebra N A , , ~ ,is, Lie admissible. The algebra N A , , ~ , ,is Al x . . . x A,-graded as follows: '

+

+,

+, [,I)

+,

NA,,m,s =

@

+,

N[a1,-.,%)

(al,-,%)

where N(al,...,an) is the vector subspace of NA,,,",~with a basis {ealzl

...e a n x , Z y

. . . x:atlil,.

. . ,i, E Z,im+l,...,is E N,n5 s,m 5 s , l 5 t 5 s } .

An element in N(al,_..,an) is called ( a l , . . . ,a,)-homogenous and N(al,...,&) is called the (al,.. . ,a,)-homogeneous component. For basis elements of NA,,",~,we may define the lexicographic order > on Al x . . . x A, x Z" x N"-" x (1,. . . ,s}, where we used the natural order of { l , . . ., s } , 1 5 2 5 . . . 5 s - 1 5 s. For any standard basis element ealxl . . . eanxnx: . . . of NA,,",+, let us define the homogeneous degree of it as follows: 5 h 4 e a 1 x 1 . . . eanxnZy . . . .Cat) = i,.

C

u=l

For any element 1 E N A , , m , s , we may define deg(1) as the highest homogeneous degree , , ,may define hd(1) as the highest of the basis terms of 1. For any element 1 E N A ~ , ~we homogeneous degree of each monomial of 1. Let Aut(fV~,,,,,) be the set of all F-algebra automorphisms of NA,,,",~.Note that the set of all right annihilators of NA,,",~ is the subalgebra T, of NA,,",~ which is spanned by {a,ll 5 i _< s}, Let A be an F-algebra with a basis B. An element 1 of A is a left diagonal with respect to B , if ~ * ( C I ~ ~ ~ C(i, , , C0 )(b ~ i) = ,O C )I ~ ~ S ~ , C C(i, ( ~1 ) ,b iI ) ~ F

444

holds for any element Clsisn,C(i,O)EF C(i,O)bi E A The elements of all left diagonals with respect to the basis B of A is denoted by LB(A).The set of all left diagonals with respect to the standard basis of the non-associative algebra N & , O ,is~ {EH1cilcidil~,E F } . Similarly, we can define a right diagonal with respect to the basis B of an F-algebra A. The matrix algebra M,(F) is isomorphic t o the subalgebra of N A , , ~spanned ,~ by {z&uIl

2

5 u,'u I s}.

Main Results

T h e o r e m 1 The F-algebra N A , , ~is, ~simple.

Proof. Since N A , , ~ , is , A1 x . . . x A,-graded and any non-zero ideal I of N A , , ~ , ~ contains {&(l5 w <_ s}, this implies that I = N A , , ~ ,This ~ . completes the proof of the theorem. 0 Lemma 1 For any 0 E E ~ ~ ( N A , , , , ~i f)S, ( x F . ...:at) N A , , ~ ,where , ~ il,. . . ,is E N and 1 5 t 5 s.

= 0,

Proof. Let 0 be the endoniorphism in the lemma such that loss of generality, we may assume il # 0 and have that

O(dl * Z? . . . .:at)

=

then 0 is the zero m a p of

S(lc:2,1.

. ..$at)

o = ile(z";'-l.. ..:at).

= 0. Without

(3)

Since il # 0, 0(x?-'. . . x?&) = 0. By applying 6'1,. . . ,as appropriate times as (3), we have that 0(&) = 0. Without loss of generality, we may assume that t = 1. From S(a1) * Q(r18j)= 0(t$) = 0, where for any f E F[eazsz,. . . ,eanzn,5 2 , . . . ,xs].Thus we have that 0 ( f ) = 0 for any f E F[eazzz,.. . ,eanzn,xl,.. . ,zs],by induction on the degree of r1 o f f . Inductively, we may assume that e(ealzl

. . . eanxnx2;1. . .x2aw) = 0

(4)

<

for al E Al,. . . , a n E A,,i,+l, . . . , in E Z, and in+l,... ,is E N such that il p and i l 2 -p for some fixed positive integer p where 1 5 u s. Both 6'(al) * q e a m . . , eo,,z,xn+l x2 iz . . . x:&,) = Q(&*ealxl . . . eanznxn+lx?iz ...z:a,) = OandB(al)* 6(ealzl. . . e"nzn,x;nx$ . . . = e(& * eaIzl . . . eanxnx;nxy . . .~ $ 8=~0 )show that (4)holds for any al E Al, . . . , a, E A,, 21,. . . , in E Z, infl,. . . , is E N by induction. This implies that the map 0 is the zero map of N A , , ~ ,Therefore, ~. we have proven the lemma. 0

<

.>aw)

Lemma 2 For any monomorphism 6 of N A , , ~ ,o(C& ~ , xiai) = holds where ci E F , 1 5 i 5 s.

'

cbl xiai + ctl ciai

Proof. It, is straightforward by the fact that Ctlziai is a right identity of N A , , , ~ , ~ .

445

Lemma 3 For any 0 E A U ~ ( N A ~ ,8(Ts) ~ , += ) ,T, f o r any n,m E Z, s E N.

Proof. Let 6' be the automorphism in the lemma. Since Ts is the unique maximal right annihilator of N A , , ~ , +we , have that 8(Ts)= T,. Therefore, we have proven the lemma. 0

- -

Note that the subalgebra NA,,~,O of N A , , ~ ,has , the standard basis {x?

. . . x ; a w l i l , . . . ,is E z, 1 5 u I s}

Theorem 2 For any 6' E Aut(iV~,,,,,), 8(&) = cVa, for 1 5 u , u 5 s, where c, is a non-zero scalar.

Proof. Let 6' be an automorphism of N A ~ , ~Let , + .us assume that O(&), be written as follows:

O(&)

= C(a1,. . . , an, 21, . . . ,i,)ealzl . . . eanxnxF . . .

1 5 u 5 s, can

.:aar + . . . ,

(5)

where ealxl...eanxnz;'. . . x $ & is the maximal term of O(&) and ... is the sum of remaining terms of Q(&) such that a1 # 0, and 0 # C(a1,.. . ,a,, 2 1 , . . . ,is) E F. Since 0 is an automorphism of NA~,O,+, there is an element 1 E NA~,O,, such that O ( l ) = $81, p >> 1. This implies that 0 = 8 ( l * &) = $al * # 0. This contradiction shows that al and il = 0. Similarly, we have that a2 = .. . = a, = i2 = . . . = i, = 0 in (5). This implies that Q(&) = xi=,&, Q E F such that at least one of c l , . . . , ~ is non-zero by the maximality of ealzl . . . eamznz;l . . .z$& of all terms of O(&). Let us assume that there is a,, 1 5 u 5 s, such that O(&) # cudv, 0 # c, E F. Otherwise, there is nothing to prove. So there is I c (1,.. . ,s} = J such that

@(a,)

@(a,)=

C(U,0, .)av,

C(u,0, v) E F,

1 1 1L 2.

VEI

Let us assume that 1 is a subset of J = {l,...,s} such that 11)_> 2. From O(&) * 8(z,&) = Q(a,,), we have that CVEJ C(u,O,u)& * O(zu&) = CvEJ C(u,0, u)& where C(u,0, u ) E F. This implies that S(zu&) = CVEI z,d,+CvEI C(u,0, u)&, since 8(xu&) is an idempotent. There are three possible cases t o prove the theorem as follows:

Case I. Let us assume that I = J. Take an element u in J such that

20

# u.We have that

There is no element S(&) which holds the equality (6). This contradiction shows that I is a proper subset of J. Case 11.

446

Let us assume that I is a proper subset of J , u E I , and 1I1 2 2. Take an element W E I" such that w # u where I" is the complement of I in J. By 0 = e(x:,& * z,d,), we have that S(zJW) = CkEKCIC(x& + ck&), since e(zw&,)is an idempotent where ch E F, k E K , and K is a subset of I". This implies that there are hl,. . . ,h, E I" such

where 0 # qs E N, s E I", and q E F. Since III 2 2, there is r E I such that r have that

# u.We

Since

@(ar),

can't be zero for any non-zero element we have a contradiction. This implies that 11)= 1. Therefore, we have proven the theorem in this case. Case 111. Let us assume that I is a proper subset of J , u E I", and / I ]2 2. Take an element w E I". By 0 = S(zwaw * xuau) = S(zJ:,) * O(x,&), we have that t9(zWdw) = CtEKCIC(zt ct)dt, since S(zw&,) is an idempotent where K is a subset of I". We may find w l , . . . ,wg E I" such that

+

(zW1aw, + + zwgawg + zua,)

=

' ' '

c xsas+ c

z,a,

sEKCIc

UEI

+ ccvav. VEI

Take an element z in I ,

0 = e(az)* e(z:,,a:,, =q a z )

*(

c

%as

SEKCP

+ . . . + ~ : , ~ a+: zuau) ,~ +c cc a w ) . GL&+

UEI

(7) (8)

VEI

The far right hand side of (7) is never zero, since B(&) # 0. This contradiction shows that there is no such automorphism 0 of NA,,o,+such that e(&) = CvEIcvav, where 0 111 2 2. Therefore we have proven the theorem. By Lemma 3 and Theorem 2, we know that the abelian subalgebra {Cb, uiailu;E F } is auto-invariant, i.e. it is stable under any automorphism of N A ~ , ~[2]. ,o Proposition 1 For any injective endomorphism 0 of N A ~ , ~O(x&) ,o,

15 i , j 5 s.

= xjaj holds for

447

Proof. Let Q be the monomorphism in the proposition. By Theorem 2 , O(8,) = c,&, 1 5 u,w 5 s, 0 # c," E F. This implies that B(Cbl&) = EL1ci& where c l r . .. , c, are non-zero scalars. By Q(Cbl&*z,&) = = ha,, we have that I: c&*O(z,&) = = c,&. This implies that B(z&) = z,& c:&, since B(z,&) is an idempotent where

@(a,)

+

d, E F. ~ , B(C%,zi&*xEP,a,) = Inductively, we may assume that O(zt&) = C ~ - P ( Z , , + C ~ ) P ~by pO(zE&) for any p E N where 1 5 u,w 5 s. By B(ze& ti: z;'d) = -O(&), we have that (c;'x:a 2c;'c,z,d C,,(C:)~~,) O(z;'a) = -c,d,. This implies that O(z;'a) = c,z;ld and c: = 0. So we have the required forms in the proposition. Therefore, we have proven the proposition. 0

+

+

Proposition 2 For any B E Au~(NA,,,,,~), ZfO(z$i) c& where cj i s a non-zero scalar.

= xjaj, 1 5

i,j 5

s,

*

then O(&) =

Proof. Let Q be an automorphism of N A , , ~ , ,By . O(&) = O(&) * B(zp3,)= O(3) * z&. This implies that Q(&) = c& by Theorem 2 where cj is a non-zero scalar. Therefore 0 we have proven the proposition. Note 1. For any automorphism B of N A , , , , ~we , have that O(zi6'i) = zjb'j, 1 5 i , j 5 s, by Proposition 1. This implies that O corresponds t o the unique element u in the symmetric group S, 011 the set { a l , . . ', a,}. So we may define an autornorphism On,=of N A ~ , as , , ~follows: Qu,c(&)

= cU(,)&(,),

1 5 u 5 s, by Theorem 2 and

Qu,,(xu&) = zu(,)&(,) by Proposition 2

where c,,(~),1 5 u 5 s, are fixed non-zero scalars, u E S,, and c = ( ~ 1 , .. . ,c,) E R" such that all c l r .. . , c, are non-zeroes. By Proposition 2, we know that the symmetric 0 group S, is a proper subgroup of Au~(NA,,,,,).

Theorem 3 For any B , , E Aut(NA,,,p) and for a n y basis element $ ...x?at

of

NAo,s,O,

Qu,c(z- . . . % PsQ ) t holds wh,ere Qo,c(&,)

= c-P1 . . . 1-n o(1)

= cn(w)8u(w),1 5

-m+1

4 ( t ) Cu(t+l)

-p

2'

. . . %(,; u(1) . . .q , ) a o ( t )

(9)

w 5 s, and u E S,.

Proof. Let, Q,, be the automorphism and . . .zT&be the basis element in the t h e e rem. Let us prove the theorem by induction on the homogeneous degree hd($ . . ..pat) of the element zy . . . .pat. By Theorem 2 , we have proven the theorem when hd(l) = 0. For hd(l) = 1, it is enough t o prove that Bu,c(z,&)= ~ ~ ( ~ ) c ~ f , ~ z ~ (1. .5) au,w ~ (5 , , S, ), by Proposition 2. By

e,,(a, * zua,,)= Q

~ , ~ ( ~ J

(10)

448

a,

at,

we have that c,(.)~,(~) * Bu,c(z,a,)= c ~ ( & , ~ ( & ) .By replacing with 1 5 t I s, we have that Bo,c(z,&,) = cu(,)c~~)xu~&,(,).Thus we may assume that (9) holds for $ . . . @$ such that hd(xy . . . .pat) = n by induction for some fixed non-negative integer n. Let zy . . . be an element of NA~,,,O such that hd($ . . ..pat) = n 1. By

+

e,,(a1* ~y . . ..:at)

= pleu,c($-l$

. . .X p , )

(11)

we have that

B,,,(zp'

. . ..?at)

= cig) ' ' ' c

$ ~ c $ ~ '~' )' c i & e ( l ) ' ' ' $(s)au(t)

+

dTaT, 1_
where dl, . . . , d, E F. By comparing the coefficients of two sides of

~ , , , ( ~* ~41a .~. .:at) we have that d l = 0, . . . , d , proven the theorem. 0

=

= pleo,c($

. . . :at),

0. Thus (9) holds for any n E N. Therefore we have

Lemma 4 For a n y 1 E N A ~ , , ,and O for any 0 E End(NA,,,,o), i f B ( 1 ) = 0, then 6 is the zero m a p of NA,,,,,o.

Proof. Let B be the endomorphism and 1 be the element in the lemma. By taking an appropriate element 1 5 t 5 s, we have the element at*l such that the homogeneous degree of the element * 1 is less than the homogeneous degree of the element 1. Similarly, we have that @(au)= 0, 1 I u 5 s. This implies that for any basis element . . . . x)aU of N,+,,,o,, we have that B(x: = 0 by induction on the homogeneous degree of xy . . . . . . This completes the proof of the lemma. 0

.";.

a,, at

.:a,.

.....-.:a,)

Theorem 4 For any injective B E End(NAu,s,O), 6' is surjective.

449

Proof. Let 8 be an injective endomorphism of NA,,+,o.By Lemma 4, O(&) # 0, 1 5 t 5 s. Since NA~,+,O is simple, 0 is injective. By reviewing the proof of Theorem 3, we did not use the fact that 8 is surjective. Every basis element of N A ~ , +is, Oin the image of 8 by Theorem 3 . This implies that 0 is surjective. This completes the proof of the theorem. 0 By Theorem 4, we may define an F-algebra A is Jacobi, if any F-monomorphism from A to itself is surjective. We call such a monomorphism of an algebra as a Jacobi map. It is interesting problem to find that an algebra is Jacobi or not. The algebra N A ~ is , ~Jacobi , ~ by Theorem 4.

References [l]D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras, Consultants Bureau, New York, 1986, 2-10.

[2] I. N. Herstein, Topics in Ring Theory, The University of Chicago Press, 1965, 1-10.

[3] T. Ikeda, N. Kawamoto, and K. B. Nam, A Class of Simple Subalgebras of Generalized Witt algebras, International Conference on the Theory of Groups, Edited by Y. G. Baik, D. L. Johnson, and A. C. Kim, Walter de Gruyter GmbH & Co. KG, 2000, 189-202. [4] V. G. Kac, Description of Filtered Lie Algebra with which Graded Lie algebras of Cartan hype are Associated, Izv. Akad. Nauk SSSR, Ser. Mat. Tom, 38, 1974, 832-834. [5] N. Kawamoto, Generalizations of Witt algebras over a field of characteristic zero, Hiroshima Math. J., 16, 1986, 417-426. [6] E. N . Ihz’min and I. P. Shestakov, Algebra VZ, Encyclopaedia of Mathematical Sciences, 57, Springer Verlag, 1995, 197-207. [7] Mohaiiimad H. Ahmadi, Ki-Bong Nam, and Jonathan Pakinathan, Lie admissible non-associative algebras, Algebra Colloquium, Accepted, 2002, 1-9. [8] IGBong Nam, Generalized W and H Type Lie Algebras, Algebra Colloquium 6:3, (1999), 329-340.

[9] D. Passman, S h p l e Lie Algebras of Witt-Type, Jour. of Algebra, 206, 1998, 682-692.

[lo] R. D. Schafer, In.troduction to nonassociative algebras, Dover, 1995, 128-138.

ON THE HOMOLOGY BIFUNCTORS OVER SEMIMODULES Nguyen Xuan Tuyen Hue University, Vietnam

ABSTRACT By developing the results on homology and cohomology of monoids (S. Maclane1963; Tuyen-1976) on cohomology of semimodules (Tuyen and Son 2001, 2002), and on tensor product of semimodules (Takahashi - 1982; Katsov 1997), we present here the homology bifunctors over semimodules. The main result of this paper is the establishing of the isomorphisms of n-dimentional homology monoids for two regular projective resolutions X and X’ of the same right semimodule M with the coefficient left semimoduleA . The paper is organized into two main sections. The former studies chain S-complexes of semimodules and their homology, regular resolutions of semimodules and the comparison theorem for semimodules. The latter gives homology of semimodules with coefficient semimodules, and homology bifunctors over semimodules.

$1. CHAIN S-COMPLEXES AND REGULAR RESOLUTIONS 1.1. Definition [Golan, 1). A semiring is a nonempty set /1 on which operations of addition and multiplication have been defined such that the following conditions are satisfied: (1) (A, +) is a commutative monoid with identity element 0; (2 ) (A, .) is a monoid with identity element ]A,. (3) Multiplication distributes over addition from either side; (4) Or = O = rO for all r in A (5) I # O 1.2. Definition [Inassaridze, 21 Let A be a semiring. A left A-semimodule is a commutative monoid (M; +) with additive indentity 0, for which we have a function /I x M -+ M, denoted by (r, x) H rx and called scalar multiplication, which satisfies the following conditions for all elements r, s of A and all elements x, y of M (l)r(x+y)=rx+ry; ( 2 ) (r + s)x = rx + sx; (3) (rs)x = r(sx); ( 4 ) 1 s = x;

450

451 ( 5 ) row = ,0

= oAx

Right semimodules over a semiring are defined in an analogous manner

1.3. Definition [Takahashi, 61. A left A-semimodule P is calledprojective if given enpinorphism a: B+ C of left A-semimodules, each semimodule A-homomorphism iY :P+ C can be lifted to a semimodule A-homomorphism ,& P+ B such that

afl= iY 1.4. Definitions [Takahashi,4] a). For two A-semimodule A and B a A-homomorphism f.A -+ B is a map f A -+ B such that always f(x+y) =f(x) +fh) and f ( h ) = ,If(.) b). For a A-homomorphism fl A -+ B, if f(a) = f(a 7 implies a = a’, we say that f is a monomorphism, while if b E B implies b = f(a) for some a in A , we say that f is an epimorphism. Finally f is an isomorphism if and only if f is both a monomorphism and an epimorphism, we write f A E B c). For a A-homomorphism fl A + B a Kernel ofJ; an image ofJ; and a proper image o f j denoted by Kerfl, Imfl and f(A) respectively, are difined as follows: Kerfl = { a g A I f(a) = 0) Imfl = {b EBI b + f(a) =f(a 7 for some a, a ’ in A ) f(A) = { b & b =f(a) for some a in A} d). Let f A +B be a A-homomorphism f is called i-regular if f(A) = Imfl; f is called k-regular if f(a) = f(a 7 implies a + k = a ’ + k’ for some k, k’ in Kerfl; f is called regular if it is both i-regular and k-regular.

The following proposition gives us equivalent definitions of a projective A-semimodule. 1.5. Proposition. Let A be a semiring and let P, A and B be left A-semimodules. Every A-homomorphism f A+B induces a map: Homn (P, = fe :Hornn (P, A)+ HOmn (P, B)

a Hfa P is A-projective if and only if it satisfies the following properties: (i) r f f is surjective then f. is surjective (ii) I f f is k-regular then fe is k-regular 1.6. Definitions a). A sequence of left A-semimodules (C,& ...

+cn+,

&*I

>

c, -L cn-1+...

452

such that d,,d,+l= 0 for all n

E

Z is called a chain S-complex

b). If (C,8)and (C’,3)are chain S-complexes, a chain transformation f: C -+ C’ is a family of semimodule A-homomorphisms fn: C,, + C , one for each n, such that = fnd,,, for all n. c). If (C,a), (C ’,2)are chain transformations, a chain homotopy s : f g is a family of semimodule A-homomorphisms s, : C, + C’,+l , one for each demension n, d’n+/sn + S n - l d , + gn =fn I,,

-

1.7. Definitions [Takahashi, 41 a). Let A be left A-semimodule. An equivalent relation p on A is a congruence if it satisfies the following condition: x p y, 9E A implies (x+ &J p (y+ &J b). A subsemimodule R of the direct product A x A is called symmetric if (u, b) E R implies (b, a) E R c). If R is a symmetrie subsemimodule of A x A, then there is a congruence relation p(R) on A as follows: x p(R) y if and only if there exists (u, b) E R such that x + a = y + b p(R) is called a congruence associated with the symmetric subsemimodule R of A x A. A/p(R) denotes the set of equivalence classes of A with respect to p(R). By defining adition and operation of A on A/p(R) by [x] + [y] = [x+y], A[x] = [h], A/p(R) is a A-semimodule . In particular, if B is a subsemimodule of A, then R = B x B is a symmetric subsemimodule of AxA. In this case AIp(R) is denoted by AIB and called the quotient A-semimodule of A with respect to B.

1.8. Notes a). From the condition d,,d,+l = 0 in Definition 1.4.a we have: I@+, c K e r d , , n E Z Moreover, according to Definition 1.5.c. we have left A-semimodules H,S(C) = KerdJIrnd,,,, n E 2 The quotient A-semimodule H,S(C) is called an n-dimensional homology A-semimodule of the S-complex C. b). The condition 3 n f n +=l fnd,+l in Definition 1.4.b. yields fn(Imd,+l)c I m a , , . i , n E 2 then we have A-homomorphisms Hn(A = VJ.:H,,S(C) +H,,S(C’), n E Z 1.9. Proposition. Every chain homotopy s: f- g: C -+ C’ induces Hn(A = H,,(d: H,,S(C) -+ H,,S(C’), n E Z This result is a direct proof from the definitions.

453

1.10. Definition. A semimodule A-homomorphism f A +B is called strongly k-regular if f(al) = f ( a 3 there is k ~Ker(’ such that al + k = a2 or a2 + k = al. f is called strongly regualar if f is i-regular and strongly k-regular. We can prove immediately the following statement. 1.11. Proposition. Every short exact sequence 0 + C’ 4C 4C” of strongly regular transformations induces semimodule A-homomorphisms A,: H,S(C”) +H,./S(C’), Vn E Z called the connecting A-homomorphisms.

*0

By using the above proposition we obtain the following. 1.12. Corollary. For a commutative diagram: 0

__f

C’ 1 , c 4 C”

_+

0

pJ y-l 0 _ j D ’& D -& D ” -0 a-l

with the exact arrows, we have the following commutative diagram:

with the exact arrows, where J g, f ’, g ’ are the strongly regular transformations. 1.13. Definition [Takahashi, 41. A sequence of A-homomorphisms A 4B C is called exact if m I’( = Ker(d. A sequence of A-homomorphisms

-+

...

+An+/

f,,+l

> A, f ,A,., +...

is an exact sequence if the sequence A,+/ for each n.

fm+i

> A , f ,A,,./ is exact sequence

1.14. Definitions a). Let A4 be a left A-semimodule. A positive chain S-complex C (C,=0, V n < 0) together with a chain transformation E : C +M (M, = 0, V n f 0, Mo = A4 and L$,= 0 b’n) such that ~d= 0 : CI +M is called an S-complex over A4. b). An S-complex E : C + M over M is said to be strong& regular if d and E are strongly regular A-homomorphisms, andprojective if C, are projective for n E Z c). A regular S-complex over M is called a resolution of M if it is an exact sequence.

454

1.15. Comparison Theorem [Tuyen-Son, 91. r f y : A + A ' is an A-homomorphism of left A-semimodules, while E :X +A is a projective S-complex over A and E' :X' + A ' is a regular resolution of A', there is, then, a chain transformation f:X -+X' with E' f = YE and, any such two chain transformations are chain homotopic. 52. HOMOLOGY AND HOMOLOGY BIFUNCTORS 2.1. Definition [Takahashi, 41. A left A-semimodule A is cancellative if a+x = a'+x for a, a' in A implies a = a' 2.2. Definition [Takahashi, 51. Let X b e a right A-semimodule and Y a left A-semimodule and let T be a commutative monoid. Then a mapping f:X x Y -+T is called balanced A-map if for all x, x' E X ; y, y ' E Y and there are ( 9 f(x + x '2)=f(xJYl+ f(x 'IY) (ii) f(X,Y + Y ? =f(%Y)+f(x,Y? (iii) f(xAy) =f(x,Ay)

;1 E

A

2.3. Definition. Let X be a right A-semimodule and Y a left A-semimodule. A cancellative commutative monoid together with balanced A-map f:X x Y + T is called tensor product of X and Y over A, denoted by T = X@SAY if for any cancellative commutative monoid A together with balanced A-map g :X x Y + A then there exists a unique monoidal homomorphism h : T + A such that hf= g 2.4. Proposition [Takahashi, 51. Let A be a right and B a left A-semimodules and let C a commutative monoid I f f : AxB + C is a balanced map, then there exists a unique homomorphism of monoids w :A @ SAB +c(C) such that the following diagram is commutative:

AxB

f-l

c

+A@SAB -Iw

c \c (C)

where c(C) s C/p(AC) is the maximal cancellative submonoid of C. The proposition 2.4 yields the following: A @ SAB = N[A x BJ / p("M]) where N[H] is a free commutative monoid generated by the set H, M is the subset of the commutative monoid N[AxB]xN[AxB] which consists of all elements of the following forms: 1. ((a+a',b),(a,b)+(a',b)),((a,b)+(a',b),(a+a',b)) 2. ((a,b +b'),(a,b')+(a, b' )), ((a,b)+(a,b'),(a,b+b')) 3. ( ( a , w , (a,a)), ((a,W , ( a U J ) )

455

Let X = {Xn, &; n E be a S-complex of right A-semimodules and A be left A-semimodule. We have induced homomorphisms of cancellative commutative monoids: x:xnQsd + X n - / @ s d n EZ x Q a H &(x)@a _ _ Then, 3, = 0 3 Imd,,l c K e r z Thus, we have the following complex of cancellative commutative monoids:

a,,,

~

~

2.5. Definition. The quotient monoid K e r z / I m x

is called an n-dimensional homology monoid of the complex Xwith coefficients in A, denoted by H,,S(XA) for all n E Z

The following theorem can prove immediately. 2.6. Theorem. Every chain homotopy s : h g :X +X’of complexes induces the following chain homotopy of complexes of cancellative commutative monoids: S: i; : X Q S A +XQS,,A

-

-

The Theorem 1.15 and 2.6 yield 2.7. Corollary. If (X, E) and (XI, E? are two strongly regular projective resolutions of the same right A-semimodule M, we have the following isomorphisms of cancellative commutative monoids: H,,S(X,A) :H,,S(X’,A) for each n E Z , where A is a left A-semimodule. 2.8. Notes a). Corollary 2.4 yields that an n-dimensionnal homology monoid of strongly regular projective resolutions of the same right A-semimodule M for coefficient left A-semimodule A is unique up to an isomorphism. Hence, we denote H,,S(M, A) = H,,S(X,A), V n E N and call it an n-dimensional homology monoid of M with coefficients in A. b). We denote by A-SMod* (SMod*-A)the category whose objects are left (right) A-semimodules and whose morphisms are strongly regular A-homomorphisms and by CCMon the category of cancellative commutative monoids. Then the equalities

H,,SV’J g ’g)= H,,Sf‘,g 7.H,,SCf;g) forf :M

+M’ +M” and A

+A

’

+A

”

456

yield the following 2.9. Theorem H,,S(-, -) :SMod*-A x A-SMod"

is bifunctor from the category SMod*-A covariant in both variables.

CCMon x

A-SMod* to the category CCMon,

REFERENCES 1. Golan J.S. The theory of semirings with applications in mathematics and theoretical computer science. Longman Scientific & Technical, 1992, 3 18 pp

2. Inassaridze H. Quelques points d' algeibre et d' algeibre homotopique at leurs applications. Proc. A.Razmadze Math. Institute, 48 (1979, 1-141 3. Katsov Y. Tensor Product and Injective Envelopes of Semimodules over Additively-Regular Semimodules. Algebra Collog. 4 (1997), n.2, 121-131

4. Takahashi M. On the Bordism Categories 11: Elementary properties of semimodules. Math. Sem. Notes, 9 (1981), 495-530 5. Takahashi M. On the Bordism Categories III: Functors Horn and 8 for semimodules. Math. Sem. Notes, Vol 10 (1982), 2 11-236

6. Takahashi M. Extensions of semimodules II. Math. Sem. Notes. Vol 11 (1983), 83-118

7. Tuyen N.X. On the extensions of order n of semimodules by monoids. Bull.Sci.Ped. Coll. Hue Univ. 1 (32), 1999, 93-99

8. Tuyen N.X and Son T.T. Extact sequences of cohomology semimodules. Bull.Sci.Ped. Coll. Hue Univ. 1 (37), 2001, 143-150 9. Tuyen N.X. and Son T.T. On the Representation of n-dimensional cohomology monoids of semimodules. Proc. ICAA, March 2002, Bangkok (to appear).

ON SIMPLE SUBSEMIGROWS OF GROUPS J.S. Ponizovskii (St. Petersburg, Russia) ABSTRACT A problem “when a simple subsemigroup of a group is a subgroup” is considered

In what follows a simple semigroup means a semigroup S with no ideals different from S. 1. In [ 11 the author constructed a theory of irreducible matrix semigroups (over fields). A problem of a special importance in that theory (noted also by Ohinski [2, p.891) is the following one.

PROBLEM 1. Does there exist a simple subsemigroup S of a group G such that S is not a subgroup of G? At that time the only result in lines of PROBLEM 1 was a theorem published in [ 1, (9. l)] without proof. We formulate that theorem in a slightly modified (but

equivalent) form: THEOREM 2. Let S be a simple subsemigroup of a group G, and suppose that S is finitely generated. Then S is a subgroup of G. A bit afterwards Kelarev proved the following

THEOREM 3 [3]. Let Q be a field of all rational numbers. There exists a subsemigroup S of a full linear group GL(2,Q) such that: (i) S is simple, (ii) S is not a subgroup of GL(2,Q), (iii) S is not finitely generated. THEOREM 3 solved PROBLEM 1 in the affirmitive and showed that the condition “S is finitely generated ” cannot be omitted in THEOREM 2. 2. In [4] the author presented a proof of THEOREM 2, but that proof turned out to be incorrect being based on the following statement that in fact is wrong: if S is a subsemigroup of a group G containing identity of G and S2=S, then S is a subgroup of G. That is not true. Really, let G be an infinite monogenic group generated by gEG, and let S be a subsemigroup of G consisting from identity e of G and from all positive powers of g. Then clearly S2=S, S contains identity e of G, though S is not a subgroup of G . 3. The aim of the papaer is to give a full proff of THEOREM 2. To do that we

employ a special approach that may be of use in other cases. Let S be a semigroup such that S2=S, and let A={a; I i d } be a generating set of S. Denote by T the hull transformation semigroup on a set I. By a decomposition of A we mean a presentation of elements aiEA in the form

457

458 Since S2=S, such a presentation does exist. Of course, for a given ai,neither bi nor XETare unique in general. Let X be the set of all XETappearing in (1). 4. Two lemmas. The keypoint of our proff of THEOREM 2 is the folowing . LEMMA 4. X is a subsemigroup of T. Proof. Let x , y ~ XBy . definition (l), for any i d , there exists ci ES such that a=c. a I 1 YO) Partiqularly a x(i)=h(i) ay(x(i)) S) Then x y ~ by T definition of X. Proof of LEMMA 4 is complete. LEMMA 5. Suppose S is a simple subsemigroup of a group G, and assume that identity e of G is in S. Then S is a subgroup of G. Proof. For any XES,x-' denotes an inverse of x in G. Take arbitrary aeS. Since S is simple, we have S=SaS. By assumption eeS, therefore there exist c,dES such that e=cad. Then c-' =adeS, d-' =casS, hence a-' = c" d-' ES. Thus, given x , y ~ S , we obtain Y-IES,and thus x y-' ES. Then S is a subgroup of G. Proof of LEMMA 5 is complete. 5. Proof of THEOREM2. Let S a simple subsemigroup of a group G, and let A={ai Ii €1) be a finite generating set of S. Let T,X be as above. Then I is finite, so T is a finite semigroup. By LEMMA 4, X is a subsemigroup of a finite semigroup T. Thus X is finite. Therefore there exists XEXsuch that x2=x. So there exists i d with x(i)=i. Taking this i in (1) we obtain ai=b, a, (b, ES).But then bi=e, so eES. Applying LEMMA 5 we get: S is a subgroup of G. Proof of THEOREM 2 is complete. Bibliography 1 Ponizovskii, J.S., On irreducible matrix semigroups. Semigoup Forum 24(1982), 117-148 2. Ohinski, J., On semigroup algebras satisfymg polynomial identities. Semigroup Forum 33(1986), 87-102 3. Kelarev, A.V., A simple matrix semigroup which is not completely simple. Semigroup Forum 37(1988), 123-125 4. Ponizovskii, J.S., On simple subsemigroupsof a group (Russian). SovremennayaAlgebra 4(24), 1999, 82

459

Interpolation theorems for the independence and domination numbers Narong Punniml Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand email: [email protected] .th

AMS Subject Classification (2000): 05C07 Abstract. For each graph parameter f = f ( G ) of a graph G, it is said that f interpolates over the set of all graphs with a fixed degree sequence d, denoted by R(d), if the following statement holds: If m and M are the minimum and maximum values (respectively) of f ( G ) over all G E R(d), then for any k , m I k 5 M , there is a graph G E R(d) such that f ( G ) = k . In this paper we show that the vertex independence number, the edge independence number (or matching number), and the vertex domination number are interpolation graph parameters.

Keywords: degree sequence, graph parameter, interpolation graph parameter.

1. Introduction. Let G be the class of all simple graphs, a function f : G -+ Z is called a graph parameter if f ( G ) = f ( H ) , whenever G H . I f f is a graph parameter and J’ C G, f is called an interpolation graph parameter with respect to J’ if there exist integers a and b such that { f ( G ): G E J } = { k E Z : a I k I b } . We say that f interpolates over the set J’ provided that f is an interpolation graph parameter with respect to

J, A number of graph parameters were shown to interpolate as follows. In [5] - [lo], Harary et. al., published some results of interpolation theorems on various kinds of graph parameters with respect to the set of all spanning trees and some classes of spanning subgraphs of a given graph. An interpolation theorem for the chromatic number, x , with respect to the class of connected regular graphs of order n was proved by Caccetta and Pullman [l]. In 1956, Nordhaus and Gaddum [12] proved the following theorem Theorem (Nordhaus and Gaddum) Let G be a graph on n vertices. Then (2)

(ii)

I x(G)+ x(G)I R. + 1, 5 x(G). x(G)5 (;(n

2 6

+

+

+

0

Furthermore, by using the following definition : ( x X ) ( G):= x(G) x(G)and ( x . X)(G) := x(G) . x(G) where G is the complement of G, Fink [a] established the existence of a graph with prescribe chromatic number. The author proved that ‘Work supported by The Thailand Research Fund, under the grant number BRG/09/2545

460

+

+

every pair of positive integers p and q with p q 5 n 1 and p q 2 n there exists a graph G of order n such that x(G) = p and x ( E ) = q. This result shows that x + X is an interpolation graph parameter with respect to the set of all graphs of order n. Moreover, it is easily seen that the graph parameter x .XI in general, does not have interpolating property with respect to the set of all graphs of order n. We proved interpolation theorems for some graph parameters with respect to R(d) where R(d) is the set of all graphs with degree sequence d: the chromatic number in [15] and [16], the maximum order of induced forest in [17], and the clique number in [18]. In this paper we extend our work on interpolation theorems for the independence numbers and the domination number. In addition, we obtain the interpolating property of the vertex covering number and the edge covering number. In section 2 we introduce the graph of realizations. The proofs of the interpolation theorems are shown in section 3.

2. The graph of realizations. In this section, we shall define basic graph-theoretic terms that will be used. We also obtain some facts concerning relationships between the graph of realizations and the range of graph parameters with a given graphic degree sequence. Only finite and simple graphs are considered in this paper. Most notation and terminology can be found in the textbook of Parthasarathy [14]. Let G = (VIE) denote a graph with vertex set V = V(G) and edge set E = E(G). Since we deal only with finite graphs, we will use the following notation and terminology for a typical graph G. Let V(G) = { q , v2,. . . ,vn} and E ( G ) = { e l , e2,. . . , e m } . The cardinality of a set S is denoted by (SI. and therefore n = (VI is defined as the order of G and the size of G is m = /El. To simplify writing, we write e = uv for the edge e that connects the vertices u and v. The degree of a vertex v of a graph G is defined as d(v) = I{e E Ele = uv for some u E V}l. The maximum degree of a graph G is usually denoted by A(G). A graph G is said to be r-regular if all of its vertices have the same degree r. Let G be a graph of order n with V(G) = {WI,w2,.. . , vn}. The sequence ( d ( q ) , d ( v 2 ,... ) ,d(v,)) is called a degree sequence of G. A graph H of order n is said to have the same degree sequence as G if there is a bijection f from V(G) to V(H) such that d(vi) = d(f(vi)) for all i = 1 , 2 , . . . ,n. A sequence d = ( d l , d2, . . . ,d,) of non-negative integers is called a graphic degree sequence if it is a degree sequence of some graph G and the graph G is called a realization of d. There is a useful result, due to Have1 [ll] and Hakimi [4],yields a recursive algorithm to test whether a given sequence d is graphic. We state their result in the following theorem.

46 1

Theorem For n > 1, let d = ( d l , d 2 , . . . ,d,) non-negative integers and denote the sequence (d2 - l,d3 - 1,.. . ,

-

1,

be a non-increasing sequence of

. . ,d,)

= d'.

0 T h e n d i s graphic i f and only i f d' is graphic. For a given graphic degree sequence d = (dl ,d2, . . . ,d,) in non-increasing order, it is necessary that dl 5 n - 1 and Cy=ldi is even. These two conditions become sufficient when d is a constant sequence. In other words, it can be shown that (e.g., see [19]) for any positive integers r and n , there exists an r-regular graph of order n if and only if nr = O(mod 2) and n 2 r 1. A switching on a graph G is a replacement of any two independent edges ab and cd of G by the edges ac and bd, where a c and bd are not edges in G, written a(u,6;c, d ) . It is easy to see that the graph obtained from G by a switching will have the same degree sequence as G. Thus, we may say that switching is a degreepreserving operation. It would be convenient to denote by G"(a~b;c~d) for the graph obtained from G by the switching a ( a , b; c, d).The next theorem was shown by Have1 [ll]and Hakimi [4].

+

Theorem Let d = ( d l , d2,. . . ,d,) be a graphic degree sequence. If G1 and G2 are any two realizations of d, then one can be obtained f r o m the other by a finite sequence of switchings. 0 As a consequence of the above theorem , we can define the graph R(d) of realizations of d as a graph whose vertices are those graphs with degree sequence d. Two vertices in the graph R(d) are adjacent if one can be obtained from the other by a switching. Thus we obtain the following corollary. 0 Corollary T h e graph R(d) is connected.

3. Interpolation theorems. In this section, we prove interpolation theorems for three graph parameters: the vertex independence number, the edge independence number, and the vertex domination number. Some relevant graph parameters are also discussed. We begin with definitions of some graph parameters and state some related known results as follows.

A subset U of the vertex set V of a graph G = (V, E ) is said to be an independent set of G if no two vertices of U are adjacent in G. An independent set of G with maximum number of vertices is called a m a x i m u m independent set of G . The number o f vertices in a maximum independent set of G, written ao(G),is called the independence number of G. A subset M of the edge set E of a graph G = (V, E ) is an independent edge set or matching in G if no two distinct edges in M have common vertex. A matching M

462 is maximum in G if there is no matching M' of G with IM'I > [MI.The cardinality of a maximum matching of G, written cq(G), is called the matching number of G. A vertex of a graph G = (V,E ) is said to cover the edges incident with it. A vertex cover of a graph G is a set of vertices covering all the edges of G. The minimum cardinality of a vertex cover of a graph G, written ,Bo(G), is called its vertex covering number of G. An edge of a graph G is said to cover the two vertices incident with it. An edge cover of a graph G is a set of edges covering all the vertices of G. The minimum cardinality of an edge cover of G, written Pl(G), is called its edge covering number of G. A dominating set (or domset, for short) of a graph G = (V,E ) is a subset D of V such that each vertex of V \ D is adjacent to at least one vertex of D. The domination number y(G) is the cardinality of a minimal dominating set with least number of elements. 0 Gallai ([3]) proposed a result concerning the relationship between follows.

010,

and Po as

+

0 Theorem For any graph G of order n , ao(G) ,&(G) = n. In the next theorem, Norman and Rabin [13] also showed the relationship between ~ 1 and , &.

Theorem For any graph G of order n and b 2 1, a l ( G )

+ /31(G) = n.

We now prove the results on interpolation theorems on graph parameters and y as follows:

0 QO,

a1

Theorem 3.1 If G is a graph and n(a, b;c,d) is a switching on G, then )> - ~ o ( G) 1. ao(Gu(%b;Cd) Proof. Let G = (V,E ) be a graph and let U be an independent set of vertices of V with IUI = ao(G). Let o(a,b;c,d) = 0 be a switching on G. Since U is an independent set of vertices, the induced subgraph of U" in G' contains at least 0 ao(G) - 1 independent vertices. Therefore ao(G") 1 ao(G) - 1.

Corollary 3.2 If o is a switching on G, then Jcyo(G)- a,-,(G")J5 1. Proof. Since a switching is symmetry, we may assume that ao(G) 2 ao(G"). By Theorem 3.1, ao(G") is either ao(G) - 1 or ao(G). In both cases we have lao(G)ao(G")I 5 1 . 0

Theorem 3.3 If G is a graph with a1(G) = a1 and u is a switching on G, then a1(GU)2 - 1. Proof. Let M be an independent set of edges of E with [All = al(G). Let a(a,b;c,d) = 0 be a switching on G. If {ab,cd} n M = 0, then \MI = lM"1. If {ab,cd} C M , then [MI = IM"I. Finally, if M contains exactly one edge from the 0 set {ab,cd}, then IM"I = JMI- 1. Therefore al(G") 1 a1 - 1.

Corollary 3.4 If o is a switching on G, then Ial(G) - al(G')I 5 1.

463 Proof. Since a switching is symmetry, we may assume that q ( G ) 2 a1(GU).By Theorem 3.3, cq(G") is either cul(G) - 1 or al(G). In both cases we have Ia1(G) (Yi(G")J 5 1 . 0 Theorem 3.5 If G i s a graph with y(G) = y and u i s a switching on G , then

r(G") i Y + 1.

Proof. Let D be a domset o f vertices o f V with ID1 = y(G), Let u ( a , b; c, d ) = o be a switching on G. If { a , b , c , d } n D = 8, then ID1 = ID"/. I f { a , b , c , d } C D , then ID1 = 1D"I. I f a E D and b,c,d E V\ D , then D u { b } is adomset o f G". I f a , b E D and c, d E V \ D , then D is a domset of G". If a , c E D , then D U {b} is a domset of G". Finally, if a , b, c E D and d E V \ D , then D U { d } is a domset of G". Thus 0 r(G") 5 r(G) 1. Corollary 3.6 I f u i s a switching on G, then ly(G) - y(G")I 5 1.

+

Proof. Since a switching is symmetry, we may assume that y(G) 5 y(G"). By Theorem 3.5, y(G") is either y(G)+l or y(G). In both cases we have ly(G)-y(G")I 5 1. Combining the results in this section and with the fact that the graph of realizations is connected we can conclude the following theorem.

Theorem 3.7 Let d = ( d l ,dp, . . . ,d n ) , d l 2 dp 2 . . . 2 d, 2 1 be a graphic degree sequence. T h e n ( Y O , ( ~ 1 PO, , PI, y are interpolation graph parameters with respect t o R ( d ) . 0 Corollary 3.8 Let f E

d, there exist integers a

{ c Y o , ( Y ~ , P o T, ~h e~n ,f ~o r}any . graphic

degree sequence

:= a ( f ) and b := b ( f ) such that d has a realization G with

f ( G )= c i f and only i f c is a n integer satisfying a 5 c 5 b.

0

References [l] L. Caccetta and N.J. Pullman, Regular graphs with prescribed chromatic number, J. Graph Theory, 14 (1990), 65-71. [2] H. J. Fink, Uber die chromatischen Zahlen eines Graphen und Seines Komplements, Wiss. Z. T. H. Ilmenau. 12 (1966) 243-251. [3] T. Gallai, Uber extreme Punkt-und Kantenmenger, Ann. Univ. Sci, Budapest, Eotvos Sect. Math. 2 (1959), 133-138. [4] S. Hakimi, On the realizability of a set of integers as the degree of the vertices of a graph SIAM J. Appl. Math. 10 (1962), 496-506. [5] F. Harary, and P. KOVBCS, Regular graphs with given girth pair J. Graph Theory 7 (1983), 209-218. [6] F. Harary and M. Plantholt, Classification of interpolation theorems for spanning trees and other families of spanning subgraphs, J. Graph Theory 13 (1989), no.6, 703-712.

464

[7] F. Harary and S. Schuster, Interpolation theorems f o r invariants of spanning trees of a given graph: covering numbers. 250th Anniversary Conference on Graph Theory (Fort Wayne, IN, 1986). Congr. Numer. 64(1988), 215 - 219. [8] F. Harary and S. Schuster, Interpolation theorems f o r the independence and domination numbers of spanning trees. Graph theory in memory of G.A. Dirac (Sandbjerg, 1985), 221 - 227, Ann.Discrete Math.,41, North-Holland, Amsterdam-New York, 1989. [9] F. Harary and S. Schuster, Interpolation theorems f o r the invariants of spanning trees of a given graph: edge-covering. Eighteenth Southeastern International conference on Combinatorics, Graph Theory, and Computing (Boca Raton, Fla, 1987).Congr. Numer. 59 (1987), 107-114. [lo] F. Harary, J. Robert and M. Plantholt, Interpolation theorems f o r diameters of spanning trees, IEEE Trans. Circuits and Systems 30 (1983), No. 7, 429-432. [ll] M. Navel, A remark on the existence of finite graphs (in Hungarian), Casopis Pest. Mat., 80 (1955), 477-480. [12] E. A. Nordhaus and J. W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177. [13] R. Norman and M. 0. Rabin, An algorithm f o r a minimum cover of a graph, Proc.Amer. Math. SOC.10 (1959), 315-319. [14] K. R. Parthasarathy, “Basic Graph Theory”, Tata McGraw-Hill Publishing Com. Ltd. New Delhi: 1994. [15] N. Punnim, Coloring regular graphs, Proceedings of General Algebra and Discrete Mathematics, Universitat Potsdam, (1998), 161 - 174. [16] N. Punnim, Degree sequence and chromatic number of graphs: J. of Graphs and Combinatorics, 18 (2002), 597 - 603. [17] N. Punnim, On m a x i m u m induced forest in graphs: to appear in SEAMS Bull. Math. [18] N. Punnim, T h e clique numbers of regular graphs, Journal of Graphs and Combinatorics, 18 (2002), 781 - 785. I191 I. Tomescu, “Problems in Combinatorics and Graph Theory” (translated from Romanian by R. A. Melter), A Wiley Interscience Publication, New York, 1986.

A Note on Serial Multiplication Modules J. Sanwong and S. Srisook Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.

Abstract. Let R be a commutative ring with identity, M a unitary R-module, and S the ring of R-endomorphisms of M . In this paper we give some characterizations of serial multiplication modules and we look at some ring theoretic and module theoretic properties of such modules and try to see how they relate among R, S , MR and M s in case M is a multiplication module. Also the characterization of the Jacobson Radical of the ring S is given. 1. Introduction. Wisbauer [8] called an R-module M a uniserial module if its submodules are linearly ordered by inclusion, and called M a serial module if it is a direct sum of uniserial modules. The ring R is called right serial if RR is a serial module. In 1981, Barnard [2] called an R-module M a multiplication module if each submodule N of M has the form M I for some ideal I of R. Later in 1995, Naoum and Al-Aubaidy [7] studied multiplication modules and their rings of endomorphisms S and found that if M is a faithful multiplication R-module, then the following statements are equivalent: (1) M is artinian as an R-module; (2) R is an artinian ring; (3) S is an artinian ring; (4) M is artinian as an S-module. In this paper we give some characterizations of serial multiplication modules and we look at some ring theoretic and module theoretic properties of such modules and try to see how they relate among R , S , M R and M s in case M is a multiplication module and S is the ring of R-endomorphisms of M . In the last part of this paper the Jacobson Radical of the ring S is characterized. 2. Results.

Throughout the paper R is a commutative ring with identity and all modules are unitary right R-modules. Let S = E n d ( M ) be the ring of R-endomorphisms of M . It is known that if A4 is a multiplication module, then S is a commutative ring [6], and if M is a finitely generated multiplication module, then S is isomorphic to R / r R ( M ) [5]. Also, every cyclic module is a multiplication module. Recall that for any module M , a submodule N of A4 is small in M , if for every submodule L of M , N L = M implies L = M , and the radical of M , denoted by R a d ( M ) ,is defined to be

+

465

466 the intersection of all maximal submodules of M , in case M has a maximal submodule, otherwise, R a d ( M ) = M . In general, submodules and factor modules of serial modules need not be serial (see [S]). However, we can state:

2.1 Proposition. Let M be a multiplication R-module and {Ma / a E A } a non-empty collection of R-modules. Then (1) If M is a serial module then every submodule and factor module of M are again serial. (2) An R-module M = @ M, is a serial module if and only if each M , is a ,€A

serial module. ( 3 ) If the short exact sequence 0 --t K M 4N serial module if and only if K and N are serial modules.

4

+ 0 is split, then M

is a

Proof. (1) Suppose M is a serial module and M = @ M , with M, is uniserial for a€A

all a E A. Let N be a submodule of M . Since M is a multiplication module, there exists an ideal I of R such that N = M I . Then N = ( @ M a ) I = @ ( M J ) and aEA

M / N = @ M a / @ ( M J ) 2 @ (M,/M,I). aEA

aEA

aEA

As a submodule of M a , each MaI is

,€A

uniserial and as a factor module of M a , each Ma/M,I is uniserial. Therefore N and M / N are serial. (2) (+) Assume M = @ Ma is a serial module. Let

B

=

{ ( z , ) , ~ A E @ M, / z,

aEA

=

0 for all a # p }. Since Mp S B is a submodule of

,€A

@ M,, B is a serial module by (l),and hence Mp is a serial module for all ,Ll E A aEA

(+) Since M , is a serial module, Ma = @ Mp; Mp is uniserial. PEA,

So we have

M = @ Ma = @ ( @ Mp); Mp is uniserial. aEA

a E A PEA,

Therefore A 4 is a serial module. ( 3 ) (+) is a direct consequence of (1). 2 lmf 5 M and N. Thus I m f and M l l m f are serial. Since the sequence is split, I m f is a direct summand of M . So M = I m f @ K ' for some submodule K' of M and then I m f n K' = 0. Hence we have

(e) Suppose K and N are serial modules. We have K

MlImf

M / I m f = ( I m f @ K ' ) / l m f E K ' / ( l m f nK ' ) = K ' . Since M / I m f is serial, K' is serial. Thus I m f and K' are serial. By (2), M = I m f @ K ' is a serial module.

0

467

Next proposition gives us some properties of the module which is multiplication over a serial ring. In general, if R is a commutative ring with identity and M is a multiplication R-module then M may not be a serial R-module. For example, the module Z is a multiplication Z-module but Z is not serial Z-module.

2.2 Proposition. Let R be a serial ring. Then we have: (1) A right R-module is multiplication if and only if it is cyclic. (2) Every multiplication R-module is a serial module. (3) If M is a multiplication R-module, then M is a finite direct sum of cyclic uniserial modules.

Proof. (1) Since R is a semilocal ring, (1) is a direct consequence of Proposition 4 in

PI. (2) Let M be a multiplication module. From (l),M is cyclic. Assume M = mR where m E M . Thus M R/TR(m) = R/TR(M). Since R is a serial ring, R/TR(M) is serial. Therefore M is a serial module. (3) By (2), M is a serial module. Assume M = @ Ma where Ma is uniserial for LYEA

all a! E A. By (l),M is finitely generated. Then the direct decomposition of M is finite, n

that is M = @Ma, with

c A for some n

(a1, ...,an}

E N . Since M is multiplication,

i=l

each Ma, is a multiplication R-module by Theorem 2.2 in [3]. Thus each Ma, is cyclic. m

Hence M = @Ma; where Ma, is cyclic uniserial. i=l

2.3 Proposition. Let M be a faithful finitely generated multiplication R-module and N a submodule of M . Then N is uniserial if and only if there is a uniserial ideal I of R such that N = M I .

Proof. (+) Since M is multiplication and N 5 M , there is an ideal I of R such that N = MI. We will show that IRis uniserial. Let A and B be ideals of I . Then M A and M B are also submodules of M I since R is a commutative ring. Because N is uniserial, so M A C M B or M B c M A . Then A c B or B c A [3, p.7681. Hence I is uniserial.

(+) Suppose I is uniserial and N = MI. Let K and L be submodules of N . Then there exists ideals A, B of R such that A, B C I and K = MA, L = M B by [3, p.7591. Since I is uniserial and both A and B are also ideals of I, A c B or B c A 0 which implies MA c M B or M B c M A . Hence N is uniserial. 2.4 Proposition. Let M be a faithful multiplication R-module and let ( R a ) a E ~be a non-empty collection of ideals of R. If M = @ (MR,), then ( R a ) a E ~is direct and

@ (MRa) = M ( @ Ra). ,€A

LYEA

aEA

468

C (MR,) = M ( C R,). We show that R, n(C Rp) = 0 for all ,€A B#a E A . Since MR, n(C MRp) = 0, Theorem 1.6 in [3] shows that M(R, n Rp) = Of, Of, MR, n ( M c Rp) = MR, n(c M R p ) = 0. AS M is faithful, R, n(C Rp) = 0. Proof. It is clear that

,€A

(Y

Of,

Pf,

Of,

0

Hence (R,),EA is direct and @ (MR,) = M ( @ R,). ,€A

,€A

In the previous proposition, we see that every multiplication module over a serial ring is a serial module. On the other hand, if M is a finitely generated multiplication serial R-module, then R may not be a serial ring. For example, the module Z, as Zmodule is a finitely generated multiplication serial Z-module, but Zis not a serial ring. We will prove that if M is a faithful finitely generated multiplication serial R-module, then R is a serial ring. For a module M without faithfulness, we have R/TR(M) is a serial ring. 2.5 Proposition. Let M be a faithful finitely generated multiplication R-module. Then M is a serial module if and only if R is a serial ring. n

Proof.

(3) Assume

M = @Mi with Mi uniserial for all i E ( 1 , . . . ,n}.Since Mi 5 M i=l

and by Proposition 2.3, there is a uniserial ideal Ri of R such that Mi = MRi. By n

n

n

TI

Proposition 2.4, MR = M = @Mi = @MRi = M(@Ri). Then R = @Ri with Ri i=l

i=l

i= 1

i=l

is uniserial. Therefore R is a serial ring.

(e) Since R is serial and M is a multiplication R-module, Proposition 2.2 shows that M is a serial module. 0 2.6 Lemma. Let M be an R-module. Then M is uniserial as an R-module if and only if M is uniserial as an R/rR(M)-module.

Proof. Since the lattices submodules S ( M R ) = S ( M R / ~ ~ ( [l, M )p.351, ) we get MR is uniserial if and only if MR/,,~(M) is uniserial. 0 2.7 Proposition. Let M be finitely generated and multiplication. If M is serial, then M is cyclic. n

Proof. Since M is a finitely generated serial module, M = $Mi where Mi is uniserial i=l

for all i. So M/Rad(M)

&(Mi/Rud(Mi)). From M is multiplication, we get Mi i=l

is multiplication for all i. Because every proper submodule of Mi is contained in a maximal submodule of Mi, thus Mi has unique maximal submodule. Then we have Mi/Rad(Mi) is simple for all i, so each Mi/Rad(Mi) is cyclic. Hence M/Rad(M) is a finite direct sum of cyclic submodules. Since M/Rad(M) is multiplication, it is cyclic

469 by Corollary 2.4 in [3]. Let % := M/Rad(M) and assume % = 3 R where 3 E %. Then it follows that M = XR Rad(M). Because Rad(M) is a small submodule of M , so M = xR. Hence M is cyclic. 0

+

2.8 Proposition. Let M be a finitely generated multiplication R-module. Then M is a serial module if and only if R/rR(M) is a serial ring.

Proof. (+) By above proposition M is cyclic. Assume M = mR where m E M . Then M R/rR(M). Since M is a serial R-module, R/rR(M) is a serial R-module. Let R := R/rR(hf). w e prove that R is a serial ring. As RR is a serial module, R = 11@ . . .@ f k where is a uniserial R-module. By Lemma 2.6, is a uniserial R-module. Thus R is a serial R-module. Hence R is a serial ring.

(+) Assume R := R/rR(M) is a serial ring. We show that ME is a multiplication module. Let N be a submodule of MR. It is clear that N is a submodule of MR. Then k

there exists an ideal I of R such that N = M I . And M I = { C m i ( r i i=l

+ rR(M)) / m i E

k

M , ri E I , k E Z+} = { C m i r , / m i E M , ri E I , k E Z+}

=

M I = N Thus M R is a

i=l

multiplication module. By Proposition 2.2 Mfi is a serial module, and hence M is serial 0 as an R-module.

2.9 Theorem. Let M be a finitely generated multiplication R-module and S = E n d ( M ) . Then the following statements are equivalent: (1) M is serial as an R-module; (2) R/rR(M) is a serial ring; (3) S is a serial ring; (4) M is serial as an S-module.

Proof. (1)H (2) By Proposition 2.8. (2) H (3) Since &f is finitely generated multiplication, R/rR(kf) Therefore R/rR(M) is a serial ring if and only if is a serial ring.

s as ring.

s

(4) We have M s is faithful finitely generated multiplication [6, p.2241. So (3) 0 by Proposition 2.5 Ms is a serial module if and only if S is a serial ring.

2.10 Corollary. Let M be a faithful finitely generated multiplication R-module and S = E n d ( M ) . Then the following statements are equivalent: (1) M is serial as an R-module;

(2) R is a serial ring; (3) S is a serial ring; (4) M is serial as an S-module.

470

Proof. Since M is faithful, T R ( M )= 0. Thus the corollary is a direct consequence of the above theorem. 2.11 Corollary. Let M be a multiplication module and S = E n d ( M ) . Then M is serial and finitely generated if and only if S is a serial ring. Proof. (+) follows from Theorem 2.9.

(e) Since S is semilocal [4],M is cyclic by Corollary 2.12 in [6]. Again by Theorem 2.9 M is serial.

0

3. The endomorphism ring of serial multiplication modules. Recall that for any module M , the radical of M , denoted by Rad(M), is defined to be the intersection of all maximal submodules of M , in case M has a maximal submodule, otherwise, Rad(M) = M . It is known that Rad(M) is the sum of all small submodules of M . Moreover, if M is a multiplication module then Rad(M) is the unique largest small submodule of M . Now, let S = E n d ( M ) and T ( M )= { f E S / f ( M ) is a small submodule of M } . Then we have: 3.1 Theorem. Let M be a finitely generated serial multiplication R-module and S = E n d ( M ) . Then (1)For each f E J ( S ) , f ( M ) is a small submodule of M ; (2) J ( S ) = T ( W ; (3) J ( S ) = H o m ( M , Rad(M)).

Proof. (1) We have M is cyclic. Assume M = xR where z E M . For each f E S there exists a E R such that f ( m ) = mu for all m E M [5, p.2521. Let fr : M + M be defined by fr(m) = mr for all m E M . Then S Z R / r ~ ( z )via fr H r T R ( Z ) . So we have fr E J ( S ) @ r ~R(zE ) J(R/TR(~)); (1 - yr) r ~ ( zhas ) an inverse for all y E R; for each y E R there exists zy E R, (1- yr)zy - 1 E T R ( Z ) . Hence for each fr E J ( S ) , and each y E R there exists zy E R such that (l-yr)zy -1 E

+

+

+

TR(X).

Let fa E J ( S ) . We prove that fa(M) is a small submodule of M . Let N be any submodule of M such that N + f , ( M ) = M . Hence z = n+fa(m) = n+fa(zt) = n+zta where n E N , m E M , t E R. Thus z(1 - t a ) = n. Since fa E J ( S ) ,there exists zt E R such that (1 - ta)zt - 1 E T R ( Z ) . Then x(1 - ta)zt - x = x((1 - ta)zt - 1) = 0 which implies z = z(1 - ta)zt = nzt E N , so z E N . Hence M = N .

47 1

(2) Let f E J ( S ) . By (l),f ( M ) is a small submodule of M . Hence f E T ( M ) . Conversely, let f E T ( M ) ,then f ( M ) ( I - f ) M = M , where I is the identity endomorphism on M . Since f E T ( M ) ,f ( M )is a small submodule. Hence ( I - f ) M = M , that is I - f is surjective and I - f is an isomorphism [5, p.2801, then f is a quasi-regular element. Thus f E J ( S ) . (3) If f E J ( S ) , then by (1) f ( M ) is a small submodule and thus f ( M ) c Rad(M). Conversely, iff E Hom(M,R a d ( M ) )then f ( M )C Rad(M).Because Rad(M) is small in M , thus f ( M ) is also small in M . Therefore f E T ( M ) = J ( S ) . 0

+

3.2 Corollary. Let M be a finitely generated serial multiplication R-module and S = E n d ( M ) . Then J ( S ) = 0 if and only if Rad(M) = 0.

Proof. (+) Let y E Rad(M). Since M is cyclic, let M = X Rfor some z E M . Define f : M + Rad(M) by f ( n ) = Y T for all T E R. Then f E H o m ( M ,R a d ( M ) )= J ( S ) = 0, and hence y E y R = Imf = 0. (+) For each f E J ( S ) , f ( M ) is small in M . Because Rad(M) is the unique 0 largest small submodule of M . Therefore f ( M ) c Rad(M) = 0 and that f = 0.

Reference [l]F.W. Anderson and K.R. Fuller, Ring and Categories of Modules, Spinger-Verlag, New York, 1992.

[2] A.D. Banard, multiplication modules, J. Algebra, 71(1981), 174-178. [3] Z.A. El-Bast and P.E. Smith, Multiplication modules, Comm. in Algebra, 16(1988), 755-779. [4] A. Facchini, Module Theory, Birkhauser Verlag Badel, Boston, Berlin, 1996. [5] A.G. Naoum, On the ring of endomorphisms of a finitely generated multiplication module, Periodica Mathematica Hungarica, 21( 1990), 249-255. [6] A.G. Naoum, On the ring of endomorphisms of multiplication modules, Periodica Mathematica Hungarica, 29(1994), 277-284. [7] A.G. Naoum and W.K.H. Al-Aubaidy, A note on multiplication modules and their ring of endomorphisms, Kyungpook Math J. 35(1995), 223-228. [8]R. Wisbauer, Foundations of Modules and Ring Theory, Gordon and Breach, London, Tokyo, e.a. 1991.

An Inverse Heat Conduction Problem A. Shidfar’ and K. Tavakol? Department of Mathematics

I m n University of Science and Technology, Narmak, Tehmn- 16, Iran Abstract

In this payer, wi inverse heat conduction problem will be considered. By reducing this inverse problem to an integral eyuation iuid using an overspecified condition: it is shown that the solution to this problem exists, and this solution is unique. Key words: inverse problem, Volterra integral equation, ill-posed problem, parabolic type.

1. Introduction For any constant number T consider the following problem ‘Ut(5:

1 ) = u,,(5: 1 )

= { ( x : t ) l z> 0:0

> 0 and U

+ .(t)u(.:

1)

20 = h ( l ) , h ( t ) # 0;

,u(z:O) = f(z);x *U(O,1)

t)E 20

(2:

x

u

0 5 1 <_ 7’;

< t < 7‘):we (1) (2)

(3)

where j’ and h are give11 sufficiently smooth functions 011 their domains. For any function ~ ( 1 ) the : problem (1)-(3) has a unique solution for u ( z , 1 ) [l].

If u ( t ) is a n unknown function: then the above problem is a n iuverse problem with two unknown function u ( x :1 ) and a ( t ) . ‘Yo solve the problem (1)-(3): we meed an extra condition and we use the following heal flux condition

-u,(O: 1 ) = &):

‘Email: [email protected] ‘Email: mathktavako1yQyahoo.com

472

0

5 1 51’

(4)

473

where y(z) is a given function: the condition (4) is called an overspecified condiIiou, The Probleiri (1)-(4):has many applications in heat problems (12]-[S]). 1Jsing the transformation v(z: 1 ) = u(c:t)e--luCa(+

(5)

the problerIi (1)-(3) becomes

with the overspecified condition -0,(O:

t ) = q(t):

where

and

The problem (6): (7): and (9) has the solution

where

where t

e(t) =

J,

(I(T)dT.

(9)

474

If we use the overspecified data(4): theri we have 4 0 , l ) = 2 f k ( 0 , l - .)exp(O(l)

- 8(7))g('j-)d.

+ laN ( O , g , l)exp(O(l))j(y)dy

= h(t);

arid

Consequeutly, we have the following second Volterra integral equation of second kind

(14) where

2. Existence and Uniqueness in this section: by reviewirig the followirlg theorerri we prove terice aud uniqueues results.

S O I exis~

Theorem 1. for any ~iecewise-coritiIiuosfuiiction +( 1 ) the integral equation @ ( l )= * ( 1 ) + J t0H ( l ; r : Q ( l ) ) d T

possesses a uuique; I)iecewise-coritiIluous solution Q ( 1 ) for all 0 H(L:r,@(1))is conlinuous for all Q ( 1 ) and for T < 1 < II': if

(16)

< t < 1':if

IH(L:T:QI)- H([,7-;@~)1 5 L(L,.T)I@1- @217

where

for some rrioriotone-irlcreasirlg furiction cr such that lirrL,,,u+ a(71) = 0 aud if

475

for some Iion-negative function

with l i ~ r ~ , , ~ ~=+0.~ ( q )

The proof of this by a slightly differeut is given in [l]. Now: for the integral equation (16) we note that

where

and

Since h is a continuous function, then for any number N , iV 5 h ( t 2 ) we have

Therefore

cy

is an increasing continuous function aud we obtain

Conseque~itly,H and $1 satisfy in the condition theorem 1. Therefore the integral equation(l4) possesses a unique coIitirluous solution @; for all 0 < 1 < 1'.Now: from G ( t ) = e-u(t) we obtain d ( t ) = - h ( @ ( t ) ) . By difkrentiation from d ( l ) = 1; U ( T ) ~ Twith respet to 1: we fiid w ( l ) = -.

dl

Substitution (17) in (1);this problem becomes a direct problem with a unique solution in the for111 (13) for .u(z:t ) . Therefore the iriverse problem (1)-(4) has a unique solution.

References [l] Cannon. J. Co. 1984.

K, The one-dirnentional heat equation, Addison-Wesley

476

[2] Tikhoriov, A., and ArseIiin: V., Solution of ill-posed probleiiis. Winsloii aud Sons, WashiIigtori UC, 1977.

[3] Can1i011, J. K.: Determination of an unknown coefficient in a parabolic differential ey. Math. J.: 30: 313-323: 1963. [4] Beck, J., Blackwell, B., and Clair, C . K. St., Inverse heat conduction ill-posed problems, Wiley-Iiilerscieiice, New k'ork: 1985.

[5] hlclaughlin, X., lriverse problems, SIXM-AMS Proc. 14, AMS, Providence, 1984. [6] Shidfar, A.. and Azary, H.. An inverse problem for a noIiliIiear difiusiori equation. Noriliriear Analysis, Theory, Methods arid Applications, "01. 28, No. 4, 589-593, 1997.

On a Class of Azumaya Galois Extensions George Szeto and Lianyong Xue Department of Mathematics, Bradley University Peoria, Illinois 61625, USA Email: [email protected] [email protected]

Abstract. Let B be a ring with 1, G a finite automorphism group of B , C the center of B , K = ( 9 E G I g ( c ) = c for all c E C}, and Jg = { b E B I bx = g ( z ) b for all x E B } for each g E G. Then, B is an Azumaya Galois extension with Galois group G and Jg = (0) for each g # K if and only if B is a commutator Galois extension of BK with Galois group K and B K is a DeMeyer-Kanzaki Galois extension of BG with Galois group G I K . More equivalent conditions are also given in terms of Azumaya skew group rings. 2000 Mathematics Subject Classification: 16835, 16W20

1 Introduction

Let B be a Galois algebra with Galois group G, C the center of B , and K = ( 9 E G I g ( c ) = c for all c E C}. In [3], it was shown that if C is indecomposable (that is, C contains no idempotents but 0 and l), then B is a central Galois algebra with Galois group K and

C is a commutative Galois algebra over CG with Galois group G I K . In [5], the result was generalized t o a Galois algebra B with nontrivial idempotents: B is a central Galois algebra with Galois group K and C is a commutative Galois algebra with Galois group

G / K if and only if Jg = (0) for each g

#K

where Jg = { b E B J b x = g ( x ) b for all

x E B } for an g E G. The purpose of the present paper is t o give a further generalization to the class of Azumaya Galois extensions for which

Jg = (0)

for each g @ K where an

Azumaya Galois extension B with Galois group G is a Galois extension of BG which is an Azumaya Cc-algebra (for more about an Azumaya Galois extension, see [2] and [S]). We

477

478 recall that a Galois extension B with Galois group G i s a commutator Galois extension if the commutator subring of BG in B , VB(BG),is a Galois extension with Galois group induced by and isomorphic with G ([S]), and a DeMeyer-Kanzaki Galois extension if B is an Azumaya algebra over C which is a Galois algebra over CG with Galois group induced by and isomorphic with G (13],[5]). We shall show the following equivalent conditions: (1) B is an Azumaya Galois extension of BG with Galois group G and Jg = {0} for each

g@K. (2) B is a commutator Galois extension of B K with Galois group K and B K is a DeMeyer-

Kanzaki Galois extension of BG with Galois group G / K .

(3) B 2 BGC @C V B ( B ~such ) that BGC is a DeMeysr-Kanzaki Galois extension of BG with Galois group G/K and V B ( B ~is )a central Galois algebra over C with Galois group K .

(4) The skew group rings B * G and ( B G C )* ( G / K ) are Azumaya CG-algebras. Similar results are also obtained for commutator Galois extensions.

2 Basic Definitions and Notations Throughout, we assume that B is a ring with 1, C the center of B , G a finite automorphism group of B , K = { g E G I g(c) = c for all c E C}, and J, = { b E B I bx = g(x)bfor all x E

B } for each g E G. We denote BG the set { a E B 1 g ( a ) = a for all g E G}. For a subring

A of B , we denote the commutator subring of A in B by V B ( A )and { a E A I ax

= g(x)a

for all x E A } by JjA’ for each g E G. As given in [7], B is called a Galois extension of BG with Galois group G if there exist elements { a i , bi in B , i = 1 , 2 , ..., m for some integer m } such that

EL,aig(bi) = 61,,,

and such a set { a i , b i } is called a G-Galois system for B. A Galois extension B with Galois group G is called an Azumaya Galois extension if BG is a n Azumaya algebra over

CG ([2],[6]), and a DeMeyer-Kanzaki Galois extension if B is an Azumaya algebra over C

479 which is a Galois algebra over CG with Galois group induced by and isomorphic with G

([3],[5]).A Galois extension B with Galois group G is called a commutator Galois extension of BG if the commutator subring of BG in B , V B ( B ~is) ,a Galois extension with Galois group induced by and isomorphic with G ([8]). We note that the class of DeMeyer-Kanzaki Galois extensions is contained in the class of Azumaya Galois extensions which is a subclass of commutator Galois extensions. Throughout this paper, for an G-invariant subring A

of B , that A is a Galois extension of AG with Galois group G means that A is a Galois extension of AG with Galois group induced by and isomorphic with G.

3. Characterizations In this section, keeping all notations of section 2 we shall give several characterizations of the class of Azumaya Galois extensions for which Jg = (0) for each g $ K . We begin with an expression of an Azumaya Galois extension, a DeMeyer-Kanzaki Galois extension, and a commutator Galois extension, respectively.

Lemma 3.1

If B is a DeMeyer-Kanzaki Galois extension with Galois group G, then B = BGC 2

BG &G

C where BG is an Azumaya CG-algebra and C is a commutative Galois algebra

with Galois group G.

Proof. It is given in Lemma 2 in [3].

Lemma 3.2 If B is a commutator Galois extension of BG with Galois group G, then

B = BG . V B ( B ~such ) that V B ( B ~is )a Galois algebra over GG with Galois group G. Proof. Since B is a commutator Galois extension of BG with Galois group G, V ~ ( l 3 ~ ) is a Galois algebra over CG with Galois group G by Lemma 3.1 in [8]. But then B and

480

BG. V B ( B ~are ) Galois extensions of BG with Galois group G such that BG. V B ( B ~c)B ; and so B = B G .VB(BG).

Lemma 3.3 If B is an Azumaya Galois extension with Galois group G, then B = BG . V , ( B G ? ) i

BG @ ' c ~VB(BG)where BG is an Azumaya CG-algebra and V B ( B ~is)a Galois algebra over CG with Galois group G.

Proof. Since B is an Azumaya Galois extension with Galois group G, BG is an Azumaya CG-algebra and V B ( B Gis ) a Galois algebra over CG with Galois group G ([l],Theorem 2). Hence B is a commutator Galois extension of BG with Galois group G. Thus by Lemma 3.2, B = B G .V B ( B ~ Z )BG @CG V B ( B ~ ([l], ) Theorem 2 ) .

We need one more lemma on patching two different Galois systems t o one.

Lemma 3.4

If B is a Galois extension with Galois group K and C is a Galois algebra with Galois group G I K , then B is a Galois extension with Galois group G. Proof. Let { a i , bi in B , i = 1 , 2 , ..., m for some integer m} be a K-Galois system for B

and {cj, dj in C, j = 1,2, ..., k for some integer k } be a GIK-Galois system for C. Then

CiI1 m aibi = 1, C z , aig(bi) = 0 for g # 1 in K, x:=l cjdj

=

1,and

C,"=,c j g ( d j ) = 0 for

g @ K . Hence m

r n k

k

m

m

k

i=l

j=1

k

Thus {aicj,bidj in B , i = 1 , 2 , ..., m and j = 1 , 2 , ...,k } is a G-Galois system for B. Therefore B is a Galois extension with Galois group G.

481

Theorem 3.5 The following statements are equivalent:

(1) B is an Azumaya Galois extension of BG with Galois group G and Jg = (0) for each g

# K.

( 2 ) B is a commutator Galois extension of BK with Galois group K and BK is a DeMeyer-Kanzaki Galois extension of BG with Galois group G / K .

( 3 ) B 2 BGC@cV s ( B G )such that BGC is a DeMeyer-Kanzaki Galois extension of

BG with Galois group G / K and V B ( B ~ is )a central Galois algebra over C with Galois group K .

(4) The skew group rings B * G and (BGC)* ( G / K )are Azumaya CG-algebras.

Proof. (1) + (2) Since B is an Azumaya Galois extension with Galois group G, B =

BG.VB(BG) 2 B G @ cV~s ( B G such ) that V s ( B G is ) a Galois algebraover CG with Galois ), group G by Lemma 3.3. From the expression B = B G .V B ( B ~ )BG @CC V B ( B ~we can see that the center of V B ( B ~is) C and Jg = JjVB(BG))for each g E G, so that

Jg = (0) for each g

#K

implies that JjVB(BG))= (0) for each g @ K . Hence V B ( B ~ )

is a central Galois algebra over C with Galois group K and C is a commutative Galois algebra with Galois group G / K ([5],Proposition 3 ) . This implies that both BGC and B" are Galois extensions of BG with Galois group G / K (for C

c B K ) .Therefore, noting that

BGC c B", we have that BK = BGC which is a DeMeyer-Kanzaki Galois extension with ) was already shown Galois group G / K , and then V B ( B ~=)VB(BGC)= V B ( B ~which to be a central Galois algebra with Galois group K , that is, B is a commutator Galois extension of B" with Galois group K . Thus ( 2 ) holds.

(2)

* (1) Since B is a commutator Galois extension of B"

with Galois group K ,

B =BK.V~(BK by)Lemma 3.2 such that V B ( B ~ is a ) Galois algebra over C K with Galois group K and has center C. By hypothesis, B K is a DeMeyer-Kanzaki Galois extension of

BG with Galois group G / K , so BK = BGZ

BG @ Z G Z where Z is the center of BK

482 by Lemma 3.1, BG (= ( B K ) G / Kis) an Azumaya ZG-algebra, and Z is a commutative and ) Z Galois algebra over ZG with Galois group G / K . Noting that B = BK . V B ( B ~ is the center of B K , we have that Z is contained in the center of B , that is, Z C C. But

C c B K ,so C c Z . Thus Z = C ; and so BG is an Azumaya CG-algebra. Thus to show that B is an Azumaya Galois extension of BG with Galois group G, it suffices t o show that

B is a Galois extension of BG with Galois group G. In fact, since B is a Galois extension of BK with Galois group K and C (= 2)is a Galois algebra with Galois group G / K , B is a Galois extension of BG with Galois group G by Lemma 3.4. Next we claim that Jg = (0) for each g @ K . Since V B ( B ~is) a Galois algebra over C K with Galois group K , V B ( B ~=)@CgEK J j V B ( B K )([5], ) Theorem 1). But B is a Jg ([5],Proposition 1). Galois extension of BG with Galois group G, so VB(BG)= @I CgEC K

Noting that BK = BGC and JjvB(B

))

= Jg for each g E K , we have that

v , ( B G )= VB(BGC)= VB(BK)= @ICgcKJ 9( V B ( B K )= ) each g

@I

CgEG Jg =

exgEK Jg. Thus Jg = (0)

for

6K .

(1) ==+ ( 3 ) By Lemma 3.3, B = BG . V B ( B Gg ) BG @ C G V B ( B ~"=)BG @CG C @C V B ( B G2 ) BGC@cVg(BG). By the proof of (1)==+ (2), BGC = B K which is a DeMeyerKanzaki Galois extension of BG with Galois group GIK and V B ( B ~is )a central Galois algebra over C with Galois group K , SO statement (3) holds.

(3) ==+ (1) Since B

g

B G C @ cV B ( B ~C) ,is the center of BGC. Hence C is a Galois

algebra with Galois group GIK for BGC is a DeMeyer-Kanzaki Galois extension of BG with Galois group GIK. By hypothesis, VB(BG)is a central Galois algebra over C with Galois group K , so B is a Galois extension with Galois group K . Hence by Lemma 3.4,

B is a Galois extension with Galois group G. Moreover, since BGC is a DeMeyer-Kanzaki Galois extension of BG with Galois group G I K , BG is an Azumaya algebra over CG by Lemma 3.1. Thus B is an Azumaya Galois extension of BG with Galois group G. Next we show that Jg = (0) for each g @ K . Since B is a Galois extension of BG with

483 Galois group G, V B ( B ~=)@ CgEG Jg ([5], Proposition 1). By hypothesis, V B ( B ~is )a central Galois algebra over C with Galois group K , so V B ( B ~=)@ CgEK JJvB(BG)) (151, Theorem 1). Since BGC and V s ( B G are ) Azumaya C-algebras, that

B

BGC@cV~(BG)

implies that B = B G C . V B ( B ~=)B G .VB(BG)by the multiplication map ([4], Theorem

4.4, page 58). This implies that JjvB'B(BG)) = VB(BG) =

Jg

for each g E G. Hence €3 CgGG Jg =

@ C Jg(gV ~ B ( B~G )=) @CgEK Jg. Thus Jg = (0)

for each g @ K .

(1) ==+ (4) Since B is an Azumaya Galois extension of BG with Galois group G, B*G is an Azumaya CG-algebra ([2], Theorem 1). Moreover, by (1) ==+ (3), BGC is a DeMeyerKanzaki Galois extension of BG with Galois group G / K , so BGC is an Azumaya Galois extension of BG with Galois group G / K . Thus, by Theorem 1 in [2] again, ( B G C )* ( G / K ) is an Azumaya CG-algebra.

(4) ==+ (2) Since B*G is an Azumaya CG-algebra, B is an Azumaya Galois extension of BG with Galois group G ([2], Theorem 1). Hence, by Lemma 3.3, B = BG . V B ( B ~Z)

BG @CG VB(BG)Z BG @CG C @c VB(B~) Z BGC @c V B ( B ~ Thus ). both BGC and V B ( B ~have ) center C. Therefore V&C(BG)= V p C ( B G C )= C. Moreover, by hypothesis, (BGC)* ( G / K ) is an Azumaya CG-algebra, so BGC is an Azumaya Galois extension

of BG with Galois group G / K by Theorem 1 in [2] again. Hence, by Lemma 3.3, C

(= VB,,(BG)) is a Galois algebra with Galois group G / K . This implies that BGC = B K . Thus B K is a DeMeyer-Kanzaki Galois extension of BG with Galois group G / K . Next we claim that

B is a commutator Galois extension of BK with Galois group K ,

that is, VB(B") is a Galois extension with Galois group K . In fact, since B is an Azumaya Galois extension of BG with Galois group G, by Lemma 3.3, Ve(BG)is a Galois algebra over CG with Galois group G. Hence Ve(BG)is a Galois extension with Galois group K . But vB(BG)= vB(BGC)= V B ( B K )so , V B ( B Kis) a Galois extension with Galois group

K . This completes the proof. By using the similar argument as given in Theorem 3.5, we can show two equivalent

484 conditions for a commutator Galois extension for which

Jg = (0) for each g $ K . Recall

that B is called a center Galois extension of BG with Galois group G if C is a Galois algebra over CG with Galois group G ([7]).

Theorem 3.6 The following statements are equivalent: (1) B is a commutator Galois extension with Galois group G for which

Jg = (0) for

each g @ K . (2) B = B G C . V ~ ( B G such ) that BGC is a center Galois extension of BG with Galois is )a central Galois algebra over C with Galois group K . group G / K and V B ( B ~ (3) B is a commutator Galois extension of B K with Galois group K and BK is a center Galois extension of BG with Galois group G / K .

Remark. The equivalent conditions of (l),(2), and (3) of the Theorem 3.5 generalize the Kanzaki’s result ([5], Proposition 3) to Azumaya Galois extensions, and Theorem 3.6 is a further generalization of Theorem 3.5 to commutator Galois extensions.

Acknowledgement. This paper was written under the support of a Caterpillar Fellowship at Bradley University. The authors would like to thank the Caterpillar Inc. for the support.

References [l] R. Alfaro and G. Szeto, Skew Group Rings which are Azumaya, Comm. in Algebra, 23(6)(1995), 2255-2261. [2] R. Alfaro and G. Szeto, On Galois Extensions of an Azumaya Algebra, Comm. in

Algebra, 25(6) (1997), 1873-1882.

485

[3] F.R. DeMeyer, Some Notes on the General Galois Theory of Rings, Osaka J. Math., 2(1965), 117-127. [4] F. R. DeMeyer and E. Ingraham, Separable algebras ower commutative rings, Volume 181. Springer Verlag, Berlin, Heidelberg, New York, 1971.

[5] T. Kanzaki, On Galois algebra over a commutative ring, Osaka J. Math., 2(1965), 309-317. [6] M. Ouyang, Azumaya Extensions and Galois Correspondence, Algebra Colloquium,

7:1 (2000), 43-57.

[7] G. Szeto and L. Xue, On Characterizations of a Center Galois Extension, International Journal of Muthematics and Mathematical Sciences, Vol. 23, No. 11(2000), 753-758.

[8] G. Szeto and L. Xue, On Central Commutator Galois Extensions of Rings, International Journal of Mathematics and Mathematical Sciences, Vol. 24, No. 5(2000),

289-294.

STRONGLY EVENTUALLY INVERSE SEMIGROUPS WHOSE LATTICE OF FULL EVENTUALLY INVERSE SUBSEMIGROUPS FORM A CHAIN*

Z. J. TIAN AND K . M. YAN School of Sciences Gansu University of Technology Lanzhou, Gansu 730050, China E-mail: [email protected]

A semigroup S is called eventually inverse if for every element a of S there exists m E Z+(the set of positive integers) such that am is regular and every regular element of S possesses a unique inverse. We call an eventually inverse semigroup S strongly eventually inverse if the set E s of all idempotents of S is a semilattice. In the paper, necessary and sufficient conditions are found t o be a chain results. Some reviews obtained by Tian in 1998 and 2000 are extended and enriched. 2000 Mathematics Subject Classification: 20M10 Keywords: Strongly eventually inverse semigroups; Lattice of full eventually inverse subsemigroups

1. Preliminaries Let S be a semigroup. Denote the set of all idempotents of S by E s , the set of all group elements of S by GrS, the set of all regular elements of S by Regs, the subsemigroup of S generated by A by (A). A semigroup S is called eventually regular (or n-regular) if for every element a of S there exists m E Z+(the set of positive integers) such that am is regular. Let .(a) be the least positive integer m such that am is regular of S and call it the regular index of an element a. If every regular element of an eventually regular semigroup S possesses a unique inverse (the unique inverse of a regular element a is denoted by a - l ) , then S is called eventually inverse (or 7r-inverse). We call an eventually inverse semigroup S strongly eventually inverse if E s is a semilattice. A semigroup 'Supported by the Development Program for Outstanding Young Teachers in Gansu University of Technology, China.

486

487

S is strongly eventually inverse if and only if it is eventually regular and Regs is an inverse subsemigroup of S (see, [l]).Let A be a subsemigroup of an eventually inverse semigroup S. We say that A is an eventually inverse subsemigroup of S if for any a E A, a-l exists, and a-1 E A. A full eventually inverse subsemigroup of an eventually inverse semigroup S is the subsemigroup containing all the idempotents of S. Obviously, A is an eventually inverse subsemigroup of S if and only if A n Regs = RegA. For an eventually inverse semigroup S , the set SubnS of all eventually inverse subsemigroups (including the empty set) of S forms a lattice with respect to set intersection denoted as usual and union denoted, denoted it by n(,), where T ( A ,B ) is the eventually inverse subsemigroup generated by the union of subsets A, B of S. The set SubfnS of all full eventually inverse subsemigroups of S forms a complete sublattice of SubnS, with n(Es) as its zero element. In this paper, we will characterize a strongly eventually inverse semigroup S for which SubfnS is a chain. In [7,8,9,10], Tian has already characterized an eventually inverse semigroup S for which SubnS is modular or complemented or 0-distributive or 0-modular or A-semidistributive, respectively. Inverse semigroups for which full inverse subsemigroups lattice is distributive or a chain were determined by Jones in [4]and [5]. For notation and terminology not mentioned in this paper, the reader is referred to [2,3,6]. Lemma 1.1. The nilsemigroup is not 0-simple. Proof. Suppose that the nilsemigroup S is 0-simple. Then for all a , b E S\{O} with a # b, there exist x,y, s, t E S1 such that a = xby, b = sat and at least one of x,y, s, t not the identity 1. Thus a = xby = z(sat)y = ( x ~ ) ~ a (= t y.). .~= ( x ~ ) ~ ( ~ ~ ) ~ ( ~ y ) a ( t y )=~ 0, ( "a~ contradiction. )~(~Y) 0 Let J be a 3-class of an eventually inverse semigroup S . We define the principal factor PF(J ) as the set J with a zero 0 adjoined, such that the product of two elements of J being their product in S if it lies in J , and all other products are zero. Thus, if a, b E J , ab = 0 in PF(J ) if and only if Jab < J in s. Obviously, for any 3-class J of an eventually inverse semigroup S , the principal factor PF(J ) is a 0-simple eventually inverse semigroup or is null, and we denote the lattice of full eventually inverse semigroups of PF(J ) by L J .

Lemma 1.2. Every principal factor P F ( J ) of a n eventually inverse semi-

488

group is 0-simple i f and only i f PF( J ) contains nun-zero idempotent. Proof. We only need to prove the necessity part because the sufficiency part is trivial. Suppose that the principal factor P F ( J ) is 0-simple, but PF( J ) does not contain any non-zero idempotent. Then for any a E J , we $ J , there exists a E J such that aT(,) E J . In this case, have either we have 0 # a'(a)(aT(a))-l E J c P F ( J ) , i.e. P F ( J ) contains a non-zero idempotent so that PF( J ) is nilsemigroup, which contradicts to Lemma 1.b

Lemma 1.3. Every principal factor P F ( J ) of a semigroup is null i f and only if 2 1 x 2 . . . x , $ J f o r any x l , x 2 , . . . , x, E J and n 5 2. Proof. We only need to prove the necessity part as the sufficiency part is obvious. Let the principal factor PF( J ) be null. Now suppose that there exist a l , a 2 , .. . , a n E J such that a = ala2 .. . a , E J . Then J = J , 5 0 Ja,a2 5 J,, = J,, SO J , = J,,,,, that is, 0102 E J , a contradiction.

Let S be an eventually regular semigroup and J a 3Lemma 1.4.a class o f S that contains an idempotent. T h e n every element of J is regular, or equivalently, any 0-simple eventually regular semigroup is regular. T h e principal factors of an eventually regular semigroup are either null (corresponding to irregular 3-classes) or regular.

Lemma 1.5. ([5], Proposition 1 and Corollary 1) Let S be a n eventually inverse semigroup. T h e n the following statements hold: (1) ( E s ) is a periodic semigroup. ( 2 ) ( E s ) is an eventually inverse subsemigroup of S , i.e. ~ r ( E s=) ( E s ) . (3) If a E Rs n ( E s ) ,then a E Es. Lemma 1.6. Let S be an eventually inverse semigroup. A , B E SubfIrS, T ( A ,B ) = ( A ,B ) .

T h e n f o r all

Proof. Let A, B E SubfIrS. We need to show that any element of ( A ,B )n Regs is regular in ( A ,B ) . In fact, let a E ( A ,B ) nRegs. Then, there exist a l , a 2 , . . . ,a k E A LJB such that a = a l a z a . . . arc, k E Z+. Put

ei

= aiai+l. . . a k a - l a l a g . . . ai-1

aThe Lemma is offered by P. Jones, where in the simple (rather than 0-simple) case this was first proved by Bogdanovich et al.

489

and xi = ( a i a i + l . . . a ~ a - 1 a l a 2 .. . a i - l ) a i = eiai, where i = 1 , 2 , . . . ,k . Clearly, a

= 21x2.. . Xk

and

. . .a k a - l a l a z . . . ai-1)2 = (UjUi+l . . . a k a - l a l a 2 . . . ai-l)(aiai+l . . . aka-lala2 . . . ai-1) = aiai+l.. . a k ~ - ~ u l .a.a+l ~ . = ei,

es = (aiai+l

i.e. ei E Es and so xi E A U B for any i = 1,2,. . . ,k (since E s Now, since we have

. .ai-l)xi . . . a k a - l a l a 2 . . . a ~ - l a i ) ( a i + l a i + z... a k a - l a l a z . . (aiai+l . . . a k a - l u l a 2 . . . Uz-lai) = aiai+1 . , . a k a - l a l a z . . . ai-laz) = xi,

A n B).

2Z(Ui+lUi+2.. a k a - l a l a 2 . .

=

(UZUi+1

we see that Putting

xi

. a2-1)o

is regular and hence x i 1 E A U B for any i = 1 , 2 , . . .,k . y = x ~ l e k x ; ~ l e l c - .l . .x1 -1 e l .

Then we see that y E (A,B ) and we can verify that y is the inverse a-l of a. This shows that the element a of (A,B ) n Re g s is regular in (A,B ) . Thus n(A,B ) = (A,B ) . 0

Lemma 1.7. For each 3-class J of an eventually inverse semigroup S and any A, B E Subf7rS if J 5 RegS, then ( A ,B ) n J = ( An J , B n J ) n J. Proof. Let J be a 3-class of S with J C RegS and A, B E SubfrS. Now, for any x 6 (A,B ) n J , x can be expressed as a product x = 21x2.. . x,, n E Z+, with each xi E A u B , i = 1 ,2 ,. . . ,n. Then x = (xx-121)x2 . . .x,. Putting y1 = Z Z - ~ X ~and , x = ylxz .. .x,. Because A and B are full eventually inverse semigroup and J = J , 5 JY1 5 J,, we see that y1 E ( Au B ) n J , and thereby, x = y 1 ( y 1 1 y 1 x 2 ) . . .x,. If n = 1, the result holds obviously. If n 2 2, we can suppose that x = y 1 . . . yjxj+1 . . .x, (1 5 j < n ) ,with 91,. . . ,y j E ( A u B ) n J . Then, we have x = y 1 . . . yjyj+lxj+z . . . x,, where yj+1 = y j y ; ' x j + ~ E ( A U B ) n J . By induction, we obtain that x = y l y 2 . . . yn for some 91,. . . , y, E ( AU B ) n J , that is, 2

Thus (A,B ) n J

E

( ( A u B )n J ) n J = ( A n J, B n J ) n J.

C (AnJ,B n J ) n J , and hence the result follows immediatelp

490

In any J-class J of an eventually inverse semigroup S , we let N ( J ) = S / J : K < J } and I ( J ) = ( E s ) u { KE S / J : K 5 J } . Then, it is clear that N ( J ) and I ( J ) are full eventually inverse subsemigroups of S . In this case, [ N ( J ) I, ( J ) ]is an interval of SubfrS. For A E SubfrS, we let AJ = (An J ) u N ( J ) . Then AJ = ( A n I ( J ) )U N ( J ) E [ N ( J ) , I ( J ) ] . Clearly AJ f~ J = A n J . Thus, for all A , B ESubfrS, we have AJ = BJ if and only if A n J = B n J , furthermore, if A E [ N ( J )I, ( J ) ] ,then we have AJ = A.

( E s )u { K

Lemma 1.8. Let J be a regular J-class of an eventually inverse semigroup S . Then, [ N ( J ) , I ( J )E ] LJ.

-

Proof. For any regular J-class J E S / J and A E [ N ( J )I, ( J ) ] ,we define a mapping ‘pJ : A ( A n J ) U (0). Then ‘pJ is a mapping which maps the form interval [ N ( J ) I, ( J ) ]to L J . Let A, B E [ N ( J ) I, ( J ) ] .If A ‘ ~ = J B ~ Jthen , A n J = B n J . This leads to AJ = B J . However, since A, B E [ N ( J )I,( J ) ] ,we have A = A J = BJ = B. Let C E LJ and D = (C\{O}) U N ( J ) . Then D E [ N ( J ) , I ( J )and ] C = D n J u { 0 } , i.e. D ’ ~ = J C. This shows that ‘pJ is a bijective mapping. Furthermore, if A B , then A n J 5 A n J . Hence A ‘ ~ 5 J B ~ Jand , so ’ p order ~ preserving. Therefore, the mapping ‘pJ is a lattice isomorphism Lemma 1.9. ( [ 5 ] , Theorem 1.5) If the lattice SubfiS of full inverse subsemigroup of a completely O-simple inverse semigroup S is a chain, then S is either a cyclic p-group or a quasicyclic p-group with zero for some prime number p , or is the Brandt semigroup B2 = ( a ,blaba = a , bab = b, a2 = b2 = 0 ) , which is a combinatorial semigroup with five element. For an eventually inverse semigroup S , we let

J * = { J E S / J : IJI 2 2, or IJI = 1 but ES n J

= 8).

Then, J * is clearly a partial order subset of S / J . 2. Main results

Theorem 2.1. For a strongly eventually inverse semigroup S , the lattice SubjirS is a chain i f and only i f 5’ satisfies the following conditions: (1) J * is a chain; (2) Every non-trivial J-class J of S is either a cyclic p-group or a quasicyclic p-group f o r some prime number p , or has some principal factor

49 1

P F ( J ) which is the Brandt semigroup B2, or has a null principal factor PF( J ); ( 3 ) For all x,y E S\Es,J, < J y implies that x E ~ ( E sy)., Proof. +) Let the lattice SubfnS be a chain of a strongly eventually inverse semigroup S. Then for all z, y E S\Es, we have ~ ( E sx), 5 n ( E s ,y) so that z E " ( E s ,y) and J , 5 J Y , or ~ ( E sy), 5 n ( E s , z ) so that and JY 5 J,. This shows that condition (1)holds. If the non-trivial 3-class J of S does not contain any non-zero idempotent, then its principal factor P F ( J ) is null. If 3-class J of S contains some non-zero idempotents, then its principal factor PF( J ) is a 0-simple inverse semigroup and L J is a chain. Therefore, by Lemma 1.5, J is either a cyclic p-group or quasicyclic p-group, or J has a principal factor PF( J ) which is a Brandt semigroup B2. This proves that condition [2) holds. so that Again, let x,y E S\Es with J , < JY. Then, y $ n(Es,x), n(Es,x) 5 n(Es, y), that is, x E n(Es,y). This shows that condition [3) holds. -+) Suppose that S is an strongly eventually inverse semigroup satisfying conditions (l),(2) and ( 3 ) , and let A, B ESubfnS, which A g B. 1) If A\B has regular elements, we let a E A\B be such a regular element. Then a $ Es and its principal factor PF(J a ) is a Brandt semigroup B2 or Ja is a cyclic p-group or a quasicyclic p-group and, obviously, L J , is a chain. Hence J , CRegS. Now, by Lemma 1.4 and Lemma 1.7, we have (A V B ) n J a

=

((An J a )

V

( Bn J a ) ) n J a .

Hence, for any C, D ESubfnS, we can define a mapping p : C J,) u (0) satisfying the folloeing conditions:

-

(C fl

(C n D)p = ((Cn 0) n J a ) u (0) = ( ( Cn J a ) U (0)) n ( ( Bn J a ) U (0) = Cp n Dp, (C v D)p = ((C v D ) n J a ) u ( 0 ) = ( ( Cn J a ) V ( B n J a ) ) u (0) = ( ( Cn J a ) U (0)) v ( ( Dn J a ) U (0)) = Cp V Dp. Hence, the mapping p is a lattice morphism of SubfnS onto L J , . Since A g ( B n J , ) u ( O ) . Thisshowsthat ( B n J , ) u ( O ) G B , wehave(AnJ,)u(O} ( A n J a ) U {0), that is, B n Ja C A n J,. Let b E B\Es, but b $! J,. Then we have Jb < Ja or Jb > Ja (by (1)). If Jb < J a , then b E n(Es,a) G A,

492

s

and so B 2 A. If Jb > J,, then we have a E r ( E s ,b) B, and so a E B , a contradiction. Thus, if A\B has regular element, then we have B C A . 2 ) If A\B does not has any regular element, then we let a E A\B. In this case, PF(J a ) is clearly null, by Lemma 1.4. Now, for any C,D ESubfrS, we can easily see that

( ( Cn J a ) v ( D n J a ) )n Ja c ( C v D ) n J,. B y L e m m a 1 . 6 , f o r e a c h x ~(CVD)nJa,thereexistx1,x2,..., x, E C U D such that x = xlxz .. . x,, and so J , 5 J,;, for n E Z+ and i = 1 , 2 , . . . ,n. Then we claim that there exists k with 1 _< k _< n such that J , < J x k . For if otherwise, we will have J , = Jxl = Jx2 = . . . = Jxn = Ja, and SO x = 21x2.. . x, E J a , which is a contradiction. Thus our claim holds and so for 2 E r ( E s ,xk), we have x E (C U D ) n .la.This proves that

-

(C V D ) n Ja = ( ( Cn Ja) V ( D n J a ) )n Ja. Thus for every C, D ESubfrS, the mapping p : C the following conditions:

(CnJ a ) u ( 0 ) satisfies

(C n D ) p = ( ( Cn D ) n J a ) U ( 0 ) = ((C n J a ) U (0)) n ( ( Bn J a ) U (0) = Cp n D p , ( C v D ) p = ( ( Cv D ) n J a ) u (0) = ((C n J a ) V ( B n J a ) ) U (0) = ( ( C n J a ) U (0)) v ( ( Dn J a ) = Cp V D p .

u (0))

This shows that p is indeed a lattice morphism of SubfrS onto L J , . Since ( B n J a ) U ( 0 ) . Hence, we have A $Z B , we see that ( A n J a ) U ( 0 ) ( B n Ja) U ( 0 ) 5 ( A n J a ) U ( 0 ) , that is, B n Ja G A n J,. Let b E B\Es, but b $! J,. Then we have Jb < Ja or Jb > Ja (by (1)).If Jb < J,, then b E r ( E s , a ) & A and so B C A. If Jb > J,, then a E r ( E s ,b) B and so a E B , a contradiction. Thus, if A\B does not contain any regular element, then we have B C A. This proves that Subf7rS is a chain. 0

References 1. S. Bogdanovic, Semigroups with a System of Subsemigroups, Novi Sad, Yugoslavia (1985). 2. A. H. Clifford and G. B. Preston, The Algebraic Theory Semigroups, Vol.1 and Vol. 11, AMS publicayion (1961 and 1967).

493 3. J. M. Howie, A n Introduction to Semigroup Theory, Academic Press (1976). 4. P. R. Jones, Distributive inverse semigroups, J . London Math. SOC.17(1978), 457-466. 5. P. R. Jones, Inverse semigroup whose full inverse subsemigroups form a chain, Glasgow Math. J. 22 (1981), 159-165. 6. L. N. Shevrin and A. J. Ovsyannikov, Semigroups and their Subsemigroups Lattice, Mathematics and its application 379, Kluwer Academic Publishers, Dordrecht (1996). 7. Z. J Tian, 7r-Inverse semigroups whose lattice of eventually inverse subsemigroups is modular (Chinese), J . System Sci. Math Sci. 17 (1997), 226-231. 8. Z. J Tian, 7r-Inverse semigroups with complemented 7r-inverse subsemigroups lattices, Northeast. Math. J. 10 (1994), 330-336. 9. Z. 3. Tian, 7r-Inverse Semigroups whose lattice of eventually inverse subsemigroups is 0-modular or 0-distributive, Semigroup Forum 56 (1998), 334-338. 10. Z. J. Tian, Eventually inverse semigroups whose lattice of eventually inverse subsemigroups is A-semidistributive, Semigroup Forum 61 (2000), 333-340.

RADICAL THEORY: MAIN ISSUES AND RECENT DEVELOPMENTS

RICHARD WIEGANDT The main issues of radical theory of rings in my personal opinion - are t o prove structure theorems, t o study and compare properties of rings via closure operations, t o discover hidden but useful properties of rings, and to construct rings which distinguish properties of rings. In this talk these issues will be discussed with emphasis on recent developments. At the genesis of radical theory in 1930 Kothe posed the famous problem: Does the maximal nil ideal of a ring contain all nil left ideals?. Recently approximation of Kothe's Problem by radical classes led to interesting results between rings, polynomial rings and radicals. We shall focus our attention to these investigations. 411 rings considered are associative and need not possess a unity element. As usual, I o A denotes that I is an ideal of the ring 4 . ~

1. RADICALS OF

RINGS

The primary aim of introducing radicals is to determine the structure of rings with 0 radical. Therefore, various radicals, for instance, the nil, Jacobson, Baer, Brown-McCoy radicals, were introduced in the thirties and forties of the last century. In the early fifties Kurosh and Amitsur independently recognized that all these radicals share certain common properties, and based on these properties, they introduced the notion of general radicals as follows. A class y of rings is called a radical class (or briefly only a radical) in the sense of Kurosh and Amitsur, if i) y is homomorphically closed: '4 E y and f + f ( A ) imply f(,4) E 7 , ii) yf.4) x(10.4 I I E y) E y for all rings A , for all rings .4. iii) y(A/y(A)) = 0 The ideal ?(A) is called the y-radical of the ring A . It is obvious that

sf

=

{ A I ?(A) = A ) .

Every radical y determines its semisimple class

Sy = { A I y(A) = 0). Beside the semisimple operator on classes of rings by

S we define also the upper radical operator U acting

Ue = { *-l I '4 has no nonzero homomorphic image in

Q} .

The support of the Chinese University of Hong Kong and of the Hungarian research grant OTK.4 # TO34530 is gratefully acknowledged.

494

495

If the class Q is hereditary (that is, 10.4 E e implies I E e ) , then Ue is a radical class. Moreover, if y is a radical class then U S y = y,and if u = S y is a semisimple class then SUu = u . The above mentioned concrete radicals and further ones will be defined in the subsequent sections. All these concrete radicals are examples of Kurosh-Amitsur radicals. Developing the theory of Kurosh-Amitsur radicals, a major obstacle is that, the relation I a A is not transitive. Anderson, Divinsky and Suliriski [a] proved a theorem of fundamental importance which works for the missing transitivity of the relation I a A.

ADS-THEOREM 1.1. For ever). radical y and every ring 4 the relation I a A implies ?(I) a 4 . An immediate consequence of the ADS-Theorem is

COROLLARY 1.2. Every semisimple class S y is hereditary. Anderson, Divinsky and Sulinski [2] proved the validlty of Theorem 1.1 also for alternative rings. In the variety of all not necessarily associative rings the ADSTheorem is false. Gardner [19] proved

THEOREM 1.3. In the variety of all not necessarily associatire rings a radical 7 has a hereditary semisimple class if and only if y satisfies the following condition:

A E y and A+

B+ imply B E y,

where A and B are rings with additive groups A+ and B+ (that is, y depends only on the additive group of the rings). Also the radical theory of involution rings features interesting results. An associative ring A is said t o be an involution ring, if there is defined a unary operation * : z ++z*, called involution, sa,tisfying the identities z** = z,

(z

+ y)* = 2* + y * ,

(zy)* = y * z *

vz, y E '4.

Examples a.re commutative rings with identica.1involution z H 2 , real a,nd complex matrices with transposition a.nd a,djoint ma,trix, respectively, and polynomial rings A[z] over an involution ring subject to z* = z. Working in the category of involution rings with homomorphisms preserving also involution, we have less mappings which makes the situation more difficult. So, the ADS-Theorem 1.1 is not true in general, but we have the following

THEOREM 1.4 ([as]). For a. radical y of involution rings the followings are equivalent : (i) I a -4 implies y ( I ) a =1 for all ideals in every involution ring *4; (ii) if *4* E y and A2 = 0 then -4" E y for every involution o on -4.

496

2 . STRUCTURE O F RINGS

A central problem of ring theory is t o determine the structure of rings in terms of linear transformations. Wedderburn (1908) suggested and Kothe (1930) implemented an ingenious technique. They discarded or ignored a. LLbad“ideal R of a ring (or algebra) ‘4 such that the factor ring AIR ha,s a ‘‘nice” structure (that is, representable by rings of linear transformations on vector spaces). Kothe considered the unique hrgest nil ideal

N ( A )=

X ( I A 1 I is a nil ring) o

a,s a “bad” ideal and determined the structure of -4/,V‘(.4) which has no nonzero nil ideals. To each element a E ?V(A) there exists an exponent n > 1 with u ’ ~ = 0, whence the elements of N ( A )can be viewed as the n-th root of 0. Root in Latin is radix, so Kothe called the “pathological” ideal N ( A ) the (nil) radical of 4 . A ring A is said t o be nil semisimple, if N ( A )= 0. The classical Wedderburn-Artin Structure Theorem describes the structure of a nil semisimple ring under the assumption that A is artinian (tha.t is, A satisfies the descending chain condition (dcc) on left ideals).

WEDDERBURN-ARTIN STRUCTURE THEOREM 2.1. A ring 4 is artinian and nil semisimple if and only if A is a direct sum of finitely many simple rings A 1 , . . . , ,4, and each Ai is isomorphic to a matrix ring hlki(Di) over a division ring Di. This theorem explains also the origin of the attribute “semisimple”. The most general but satisfactorily efficient structure theorem describes the structure of Jacobson semisimple rings as subdirect sums of dense rings of linear transformations. The nil radical N ( A ) is in some extent not big enough to determine the nil semisimple rings in general. Jacobson [22] introduced a s0mewha.t bigger radical via quasi-regularity. A ring A is quasi-regular, if for every element z E A there exists an element y E A such tha,t z + y - zy = 0. The Jacobson radical J ( A ) of a ring .4 is the unique largest quasi-regular ideal of A. A ring A is said to be (left) primitive, if A contains a maximal left idea.1 L such that zA C L implies z = 0, or equivalent,ly, A has a. faithful irreducible A-module (for instance A I L ) . The Ja.cobson radical J ( A ) can be represented as the intersection J(A) = n ( r o 4 I A / I is a primitive ring).

THEOREM 2.2. -4 ring A is Jacobson semisimple (that is, J ( A ) = 0) if and only if A is a subdirect sum of primitive rings. A ring A is primitive if and only if A is a dense subring of a ring Hom(V, V ) of linear transformations on a vector space I/, that is, to any finitely many linearly independent elements X I , .. . , x,, E 1‘ and arbitrary elements y1,. . . , yn E V there exists an element t E A such that tz1 = y1,. . . ,tzn = yn. We mention two more theorems which describe the structure of simple rings. A subring B of a ring A is called a. biideal, if B.4B B . We say that a ring .4 is strongly locally ma.trix ring over a division ring D , if every finite subset C of ,4ca.n be embedded into a biideal B of 4 such tha.t B ? M , , ( D ) for some nat,ural number n depending on the size of C.

497

LITOFF-ANH THEOREM 2.3. A ring A is simple, A’ # 0 and 4 has a minimal left ideal if and only if A is a strongly locally matrix ring over a division ring. In the original Litoff Theorem [22] only embeddability i n subrings was demanded, and therefore it was not an “if and only if” theorem. The present version is due to Anh [6].

Theorem 2.4 (Beidar [S]). A ring 4 is a matrix ring over a division ring if and only if A is a prime ring such that to each nonzero element a E 4 the subset ail contains a nonzero idempotent and the degrees of nilpotency of the elements in A4 are hounded. Every ring A has a unique largest ideal r ( A ) whose additive group is torsion. T ( A )is called the torsion rudicok of A.

Theorem 2.5 (F. Szasz [45]). Every artinian ring decomposes into a direct sum

A

= r ( A )@ F

where F is a uniquely determined ideal of ’4 whose additive group is torsionfree. Theorem 2.5 was extended to alternative rings with dcc on principal left ideals in [14].

3. CLASSES OF

RINGS CLOSED UNDER CERTAIN PROPERTIES

In radical theory two types of classes are of fundamental importance; these are the radical and semisimple classes. A natural question to ask is the characterization of radical and semisimple classes (maybe with some additional property) in terms of closure properties. We compile here a list representing such results.

THEOREM 3.1. A class y of rings is a radical class if and only if (i) y is homomorphically closed; (ii) y has the coinductive property: if I1 2 . . . C I , C . . . is an ascending chain of ideals in a ring A such that Ix E y for all X and .J = U I x then A E y; (iii) y is closed under extensions: I E y and A / I E y imply A E y. Semisimple classes can be characterized by dual conditions (cf. [27]and [35]).

THEOREM 3.2. A class 5 of rings is a semisimple class [of the radical class y = Uu) if and only if (i) u is hereditary; (ii) n has the coinductive property: if I1 2 . ’ . 2 1, 2 . . . is a descending chain of ideals of a ring A such that A l I x E CT for all X and = 0 then .-1 E 5 ; (iii) 5 is closed under extensions. Among radical cla,ssestjhe hereditary ones are important a,nd common. To cha.racterize the semisimple classes of hereditary ra.dical classes, we need the following notion. An ideal I of a ring A is essential in -4, if I n K # 0 for every nonzero ideal K of ’4.

498

THEOREM 3.3 ([7], [32]). A radical class y is hereditary if and only if the corresponding semisimple class u = Sy is closed under essential extensions: if I is an essential ideal in A and I E u then A E u . The “good” radical classes are those which are hereditary and contain all nilpotent rings. Such ra.dicals are said to be supernilpotent.

THEOREM 3.4 ([3]). Let u be the semisimple class u = Sy of a radical class y # 0. The radical 7 is supernilpotent if a.nd onlji if the semisimple class u is weaklJhomomorphically closed: A E u , I a A and I 2 = 0 imply A / I E u. The best description of semisimple rings is achieved by special ra.dica’ls. A cla’ss of prime rings is said to be a. special class, if e is hereditary a.nd closed under essential extensions. The radica,l class y = UQ is called a special radical. It is known that every special radical y is supernilpotent, and if y(A) = 0 then A is a subdirect sum of prime rings taken from the class u (see [4]).However, not every supernilpotent radical is special.

e

THEOREM 3.5 ([21]). A class y of rings is a special radical if and only if y is homomorphica.lly closed, hereditary and satisfies the following condition (A) if every nonzero homomorphic image of a ring 4 has a nonzero ideal in y then

A E y. THEOREM 3.6 ([33]). Let u be the semisimple class of a hereditary radical y. The class y is a special radical if and only if u satisfies the following condition (a)if A E u then A is a subdirect sum of prime rings in u .

A subclass e of rings is called a variety, if e is closed under taking homomorphic images, subrings and direct products. As is well known, the varieties are just the subclasses of rings which can be defined by identities (e.g. zy = yz or zn = z for a. fixed n ) . THEOREM 3.7 ( [ 7 ][20], , [28],[49]). For a class e ofrings the following conditions are equivalent: (i) e is a variety which is closed under extensions; (ii) e is a. radical class closed under subdirect sums; (iii) e is a radical and a semisimple class; (iv) e is a homomorphically closed semisimple class; (v) e is a radical class closed under essential extensions; (vi) e is a variety closed under essential extensions. These classes have been determined explicitly by Stewa,rt [43].

THEOREM 3.8. A cla,ss e satisfving any of the conditions of Theorem 3.5 is either e = ( 0 ) or Q = {all rings} or e is the semisimple class of a special radical y = UF where F = { F I ,. . . ,F,} is the special class of finite fields, F1, . . . , F,, such that if G is a subfield of some Fi E F then also G E F. For any chss y of rings we define its essential cover ly a.s

ly = {all rings

1 .4

ha.s a.n essential idea.1 I E y}.

In view of Theorem 3.7 (iii) and (r) a. radical chss y is a semisimple class if and only if &? = y.

499

THEOREM 3.9 ([16]). The essential cover &y of a radical class y is a semisimple class if and only if y is hereditary and has a complement in the lattice of all hereditary radicals. The classes of Theorem 3.9 are generalizations of radical and semisimple classes and have been explicitly determined [9] and [lo) in terms of matrix rings over finite fields.

4 . USEFULLEMMAS

OF RING THEORY

One of the benefits of radical theory is the discovery of hidden properties of rings, mainly on the behavior of the ideal structure of rings, which can be successively used in answering various questions of ring theory. We shall list a sa.mple of them. These results are of technical character, and therefore - not diminishing their importance we shall call them lemmas.

~

ANDRUNAKIEVICH LEMMA 4.1 ([4]). lf K a I a A a n d z denotes the ideal of A generated by K , then CK.

z3

KREMPA’S LEMMA 4.2 ( [ 26] ) . Let K a I a A and I = R, the ideal of A generated by K . Then every nonzero homomorphic image of the ring I contains a nonzero homomorphic image of K . SANDS’ LEMMA 4.3 ([36]). For a ring R the following assertions are equivalent: (i) RaR # 0 for every 0 # a E R, (ii) R I R # 0 for every 0 # I Q R, (iii) for every ring A, if K a I a A and I I K E R then K a A.

A subring B of a ring A is called an n-accessible subring of A , if there exists a, sequence C1,. . . , C,, of subrings of A such that B = CI aC2a . . . aCn = A. In this terminology the ideals are the 2-accessible subrings.

STEWART’S LEMMA 4.4 ([44]). Let S be a homomorphically closed class of nilpotent rings. If K E 6 is a nonzero n-accessible subring of A, then A has also a nonzero 3-accessible subring J E 6. We define the class if S

E .4

is an n-a,ccessible subring of B - 1 accessible subring of B

, , , = { A 1 then S is an n

1

for some n 2 3. For a ring A , we denote by A+ the additive group of A and by .Ao the ring with zero multiplication built on A’.

ANDRUSZKIEWICZ LEMMA 4.5 ([5]). For n 2 4 a ring .A is in the class K,, if and only if (,A/A2)+ is a divisible torsion group. Thus ~4 = K S = . . . . GARDNER’S LEMMA 4.6 ([18]). A nilpotent ring A belongs to a radical class y if and only if A0 belongs to y.

500

5 . RINGCONSTRUCTIONS Constructing rings with peculiar properties exhibits how delicate or nasty (UP to the reader’s taste) the behaviour of rings may be. This activity contributes substantially t o the better understanding of the structure of rings and gives impetus for further researches. To decide that two radica.1 classes (or semisimple classes, or other properties of rings) are different, one has to construct a ring which belongs to one class but not to the other. Two famous examples: i) Bergman [15] constructed a left primitive ring A which i s n o t right primitive. The Jacobson radical class is the class of all quasi-regular rings, so it is a. left- a.nd right-symmetric notion, and the same is true for its semisimple cla,ss. Hence in the constructed left primitive ring .4, being Jacobson semisimple, the intersection of all right primitive ideals is zero. ii) S F i a d a I371 (see also Svsiada and Cohn [38]) constructed a simple p r i m e ring which is also a Jacobson radical ring. In the sequel we shall list some recent constructions. Kasch [23] defined the total Tot(A) of a ring A as the set Tot(A) = { a E A I aA has no nonzero idempotents}. Tot(A) is not closed under addition] so it ca,nnot be an ideal in .4. The K a s c h radical class K has been defined in [14] as

K

I

= {all rings A no nonzero homomorphic image of A has 0 total}.

Observe that Beidar’s Theorem 2.4 describes the structure of certain simple rings with 0 total. The question as whether the Kasch radical K is a special one is equivalent t o the claim that K coincides with the radical class

K , = {all rings A I no nonzero prime homomorphic image of A has 0 total}. Beidar [14] gave a quite involved genuine construction for a ring G such that Tot(G) = 0 but Tot(G/P) # 0 for every prime ideal P of G. The existence of such a ring G proves that K # lc,, whence t h e Kasch radical K as n o t a special radical, though K possesses many nice properties (for instance] if L is a left ideal of a. ring A E K then also L E K , and if L = K ( L ) is a left ideal of a ring A then L W)). A long standing open problem of Levitzki asked: Does there exist a simple nil ring? Recently Agata. Smoktunowicz [41] answered Levitzki’s problem affirmatively. The existence of a simple nil ring readily shows that the nil radical class N i s n o t contained in the antisimple radical class p, which is the upper radical .,V is known of subdirectly irreducible prime rings. The opposite relation p, for long; SBsiada and Suliriski [39] gave a primitive ring which cannot be mapped homomorphically onto a subdirectly irreducible prime ring. Smoktunowicz a.nd Puczylowski [42] constructed a polynomial ring wh,ich is .lacobson radical and n o t nil. Considering any partition ( q ,C) of the chss of all simple rings, it is well known that. the upper radical class UC contains properly the lower radical class C I (t1ia.t ~

c

501

is, the smallest radica,l class containing T I ) . One may impose some constraints on the lower and upper radicals and prove that the containment rema,ins proper. For instance, the lower special radical determined by q is properly contained i n the upper radical of all subdirectly irredu,cible rings with heart i n C, as shown by Tumurbat and Wiegandt [48]. To distinguish these two classes, in [48] a.n infinite nmtrix ring M ( A ) was used where 4 was a. *-ring (that is, A is a. prime ring which has no proper prime homomorphic images) without minimal ideals. Halina, France-Ja.ckson [17] has taken a, non-simple idempotent *-ring for A, and answered a problem of G. Tziutzis (1987) by using properties of M(,4): the upper radical class of all rings without accessible subrings which are not ideals, contains properly the cla,ss of all idempotent Brown-McCoy radical rings R whose prime hornomorphic images have non-trivial center. In connection with pa,rtitions of simple rings, hl. Ferrero (2000) posed the following problem: Let y and S be special radicals such tha,t i) y(S) = 6 ( S )for all simple rings S , ii) y(A[x]) = S(A[z]) for all polynomial rings A[$]. Does then y coincide with 6 ? A negative answer was given by Tumurbat [46]. By making use of a commutative *-ring, he chose the Jacobson ra.dica1 J’ for y and constructed a. special radical S fulfilling the above conditions, but b c J’. For a. survey of rings and ring constructions which distinguish various radica.ls, the rea.der is referred to [50]. 6. KOTHE’SPROBLEM At the genesis of radical theory in 1930 Kothe [24] posed a problem which ha,s been one of the hardest problems in ring theory.

KOTHE’S PROBLEM. Does the nil radical N ( A )of any ring 4 contain every nil left ideal of 4 1 An affirmative answer has been verified for many different classes of rings. A radical y is said to be left strong, if L at A and L E y imply L C_ y(.4) for every ring A. Thus Kothe’s Problem asks: Is the nil radical N left strong? It is well known tha,t the Jacobson ra.dica1 J’ and the Ba.er radical p (the Baer ra.dical b(.4) of a ring .4 is the intersection of all prime ideals of A) a.re left strong. As mentioned in Section 5, also the Kasch radical K is left strong. We say that a. class 6 of rings is matric-extensible, if for all na.tural number n, .4 E 6 e M,L(A4) E 6 where M n ( A ) stands for the n x n ma,trix ring over .4.

PROBLEM 6.1. Is the nil radical hfmatric-extensible? Krempa. [25] proved that Problem 6.1 is equivalent to Kothe’s Problem. He gave also another surprising equivalent formulation of Kothe’s problem which exhibits a subtle connection between the nil radical Ar and Jacobson ra.dica.1J’.

PROBLEM 6.2. Is the polynomial ring -4[x] over any nil ring .4 a Jacobson radical ring? -4niitsur [I] proved that for the Baer radical 13 t,he implica.tion

502

holds. He asked as whether the analogous implica,tion is true for the nil ra.dical

N . (At that time Krempa’s equivalent formulation 6.2 of Kothe’s Problem wa’snot known.) This problem was solved recently by Agat)a Smoktunowicz [40].

THEOREM 6.3. There exists a nil ring .J such that the polynomial ring 4 x 1 is not nil. Since nil rings a,re always Jxobson radical rings, Theorem 6.3 ca.n be rega’rded as an approximation of Kothe’s problem from below. Kothe’s Problem has been approximated also from above. Notice that the Jacobson radical is contained in the Brown-McCoy ra,dical, the upper radica,l G = UA4 of the cla.ss M of simple prime rings with unity element.

THEOREM 6.4 (Puczylowski and Smoktunowicz [31]). If A is a. nil ring, then the polynomial ring A[x] is a Brown-McCoy radical ring.

A ring A is sa.id to be uniformly strongly prime, if there exists a finite subset F of A, called a unzforrn insulator, such tha,t aFb # 0 whenever 0 # a , b E A. The uniformly strongly prime radical is the upper radical u determined by the class of all uniformly strongly prime rings (cf. Olson [30]). We note that N c u but u is not comparable with the Jacobson radical. THEOREM 6.5 (Beidar, Puczylowski, Wiegandt [12]). A ring 4 is uniformly strongly prime if and only if the polynomial ring A [ x ]is uniformly strongly prime. Hence A E u , if and only if A[z] E u.In particular, .4E N implies 4x1 E u. In view of Problem 6.2 and Theorem 6.5 we have the following equivalent formulation of Kothe’s Problem.

PROBLEM 6.6. Does A E N imply

41 .

E ,j’n u?

Recently the approximation given in Theorem 6.3 ha.s been substantially improved by Beidar, Fong and Puczylowski [ll]. Let 13 be the upper radical of the class of all rings possessing a nonzero idempotent. B is called the Behrens radical. The position of B is known: 3 c 13 c G.

THEOREM 6.7. I f A E N then .4[z] E B Thus, by Theorems 6.5 and 6.7 the so far known best approximation of Kothe’s Problem is this: A E N ==+ 421 E 2~ n B. Further results on a.pproximatingKothe’s Problem by radicals have been achieved by Tumurbat [47] In mathematics approximation means to get closer a.nd closer to the goal but never reach it. Therefore, we should not expect to solve Kothe’s Problem by approximating it in terms of polynomial rings a.nd radicals. Nevertheless, such investigations ha,ve led to int,eresting and deep results on rings, polynomial rings a.nd radicals. In the Abstracts of the Algebra, Conference, Venice, June 2 - 9, 2002, 1. I. Sakhajev reported the negative solution of Kothe’s Problem.. He mnounced in [34, Theorem] tha.t ”there is an algebra. with a. right ideal 7 such tha.t either the ideal

a

503 -

B generated by 7 is not nil or is nil but the ideal 8,of the matrix ring n/12(2) is not nil”. (Sakhajev was regretfully not present a.t the Conference.) The position of radical classes occurring in Sections 5 and 6 is given in the following diagram, where 7 stands for the upper radical of division rings and L for the class of locally nilpotent rings (i.e. rings in which every finitely generated is nilpotent).

U

REFERENCES S. A. Amitsur, Radicals of polynomial rings, Canad. J . Math., 8 (1956), 355-361. (1) T. Anderson, N. Divinsky and A. Suliriski, Hereditary radicals in associative and alter-

(2)

native rings, Canad. J . Math., 17 (1965), 594-603.

(3) T. Anderson and R. Wiegandt, Weakly homomorphically closed semisimple classes, Acta Math. Acad. Sci. Hungar., 34 (1979), 329-336.

(4)

\;. A . Andrunakievich, Radicals of associative rings I (Russian), Mat. Sb., 44 (1958),

179-212; I1 (Russian), Mat. Sb., 55 (1961), 329-346.

(5) R. R. Andruszkiewicz, On accessible subrings of associative rings, PTUC.Edinburgh Math. Soc., 3.5 (1992), 101-107.

(6) P. N. Anh, On Litoff’s theorem, Studia Sci. Math. Hungar., 18 (1983), 153-157. (7) E. P. Armendariz, Closure properties in radical theory, Pacific J . Math., 26 (1968), 1-7. (8) K. I. Beidar, On rings with zero total, Beitriige Alg. und Georn., 38 (1997), 233-239. (9)

K . I . Beidar, Y. Fong and \V. F. Ke, On complemented radicals, J . Algebra, 201 (1998), 328-356. (10) K. I. Beidar, Y. Fong, W. F. Ke and K . P. Shum, On radicals with semisimple essential covers, Preprint, 1995. (11) K. I. Beidar, Y. Fong a.nd E. R. Puczylowski, A polynomial ring over a nil ring cannot be homomorphically mapped onto a ring with nonzero idempotent, J . Algebra. (12) K. I. Beidar, E. R. Puczytowski and R. Wiegandt, Radicals and polynomial rings, J. Austral. Math. Soc.. 47 (2002), 1--7. (13) K . I. Beidar and R. \\-iegantlt.; Split.ting theorems for nona.ssociative rings, Publ. Math. Debrecen 38 (1991). 121-143. (14_ K . I. Beidar a.nd R. Wiegandt, Radicals induced by the total of rings, Beitrage Al.9. und

504 Georn., 38 (1997), 149-159. (15) G. M. Bergman, A ring primitive on the right hut not on the left, Proc. Am.er. Math. SOC.,15 (1964), 473-475. (16) G. F. Birkenmeier and R. Wiegandt, I-kential covers and complements of radicals, Bull. Austral. Math. Soc., 53 (1996), 261-2fi6. (17) Halina France-Jackson, On a non-simole idempotent *-ring with zero centre, Acts Math. Hungar., to appear. (18) B. 3 . Gardner, Sub-prime radical classes determined by zerorings, Bull. Austral. Math. SOC.,12 (1975), 95-97. (19) B. J. Gardner, Some degeneracy and pathology in non-associative radical theory, Annales Univ. Sci. Budapest., 22-23 (1979/19XO), 65-74. (20) B. J . Gardner and P. N . Stewart, Ori semi-simple radica.1 classes, Bull. Austral. Math.

Soc., 13 (1975), 349-353. (21) B. J . Gardner and R. Wiegandt, Chat.acterizing and constructing special radicals, Acta Math. Acad. Sci. Hungar., 40 (1982). 73--83. (22) N. Jacobson, Structure of rings, Amet.. Math. SOC.Coll. Publ. 37, Providence, 1968. (23) F. Kasch, Partiell invertierbare Homomorphismen und das Total, Algebra Berichte 60, Verlag Reinhard Fischer, Miinchen, 1 ! K . (24) G. Kothe, Die Struktur der Ringe, dei-en Restklassenring nach dem Radikal vollstandig reduzihel ist, Math. Zeitschr., 32 (1930), 161-186. (25) J. Krempa, Logical connections among some open problems in non-commutative rings, Fund. Math., 76 (1972), 121-130. (26) J . Krempa, Lower radical properties for alternative rings, Bull. Acad. Polon. Sci., 23 (1975), 139-142. (27) L. C. A . van Leeuwen, C. Roos and R. Wiegandt, Characterizations of semisimple classes, J . Austral Math. Soc., 23 (1977), 1 7 2 ~182. (28) N . V. Loi, Essentially closed radical classes, J . Austral. Math. Soc., Ser A , 35 (1983), 132-142. (29) N. V. Loi and R. Wiegandt, Involution algebras and the Anderson-Divinsky-Sulinski property, Acta Sci. Math. Sreged, 50 (1986), 5-14. (30) D. M. Olson, A uniformly strongly prime radical, J . Austral. Math. Soc., 43 (1987),

95-102. (31) E . R. Puczyiowski and Agata Smoktuitowicz, On maximal ideals and the Brown-McCoy radical of polynomial rings, Comrn. in. Algebra, 26 (1998), 2473-2482. (32) J u . M. Rjabuhin, Radicals in categories (Russian), Mat. Issled. Kishinev, 2 (1967), No. 3, 107- 165. (33) Ju. M. Rja.buhin and R . Wiegandt, On special radicals, supernilpotent radicals and weakly homomorphically closed classes, J . Austral. Math. Soc., 31 (1981), 152-.162. (34) I. I. Sakhajev, Negative solution hypot,hesis of Koethe and some problems connected with it, Abstract of talks, Algebra Cotiference L'enezia 2002, pp 37-38. (35) A . D. Sands, Strong upper radicals, Quart. J . Math. Ozjord, 27 (1976), 21-24, (36) A. D. Sands, On ideals in over-rings, Pub/. Math. Debrecen, 35 (1988), 245 and 274G279. (37) E. Sgsiada, Solution of the problem of existence of a simple radical ring, Bull. Acad. Polon. Scz., 9 (1961), 257. (38) E. Sgsiada and P. h l . Cohn, An example of a simple radical ring, J . Algebra, 5 (1967), 373-377. (39) E. Sgsiada and A . Sulinski, A note on the Jacobson radical, Bull. Acad. Polon. Sci., 10 (1962), 421--423. (40) Agata Smoktunowicz, Polynomial rings over nil rings need not he nil, J . Algebra, 239 (2000), 127--436. (41) Agata Smoktunowicz, A simple nil ring esist,s; C0m.m. i n Algebra, 30 ( Z O O Z ) , 27--59. (42) Agata Snioktunoakz and E. R. Puczylowski; i\ polynomial ring that is Jacohson radical and not nil. Israel .I. Math,.: 124 (2001). 117-325. (43) P. N . Stewart,, Semi-simple radical ~ I R S S E S ~ Pacific J . Math,., 32 (1970), 249-254. (44) P. N. Stewart, On l.hc lower radical coiist.rucl.ion, Acta Math. Acad. Sci. Hungar., 25 (1974), 31-32.

505 (45) F. S z b z , Uber artinsche Ringe, Bull. Acad. Polon. Sci., 11 (1963), 351-354. (46) S. Tumurbat, On special radicals coincidig on simple rings and on polynomial rings, Preprint, 2001. (47) S. Tumurbat, T h e radicalness of po1ynomia.l rings over nil rings, Math. Pannonica, to appear. (48) S. Tumurbat and R. Wiegandt, A note on special ra.dicals and partitions of simple rings, Comm. in Algebra, 30 (2002); 1769-1777. (49) R. Wiegandt, Homomorphically closed semisimple classes, S t u d i ~Univ. Babes-Bolyai, Cluj, Ser. Math.-Mech., 17 (1972), No 2, 17-20. (50) R. Wiegandt, Rings distinctive in radical theory, Quaest. Math., 22 (1999), 303-328.

Author’s address: Richard Wiegandt A. RBnyi Instutute of Mathematics PO Box 127 1364 Budapest Hungary e-mail: wiegandt Qrenyi.hu

Some problems and conjectures in modular represent ations Jiping Zhang* School of Mathematical Sciences, Peking University, Beijing, China

1 Introduction Modular representation theory of finite groups was established by R. Brauer. His development of the theory was motivated by arithmetic properties of the classical Frobenius characters and by the problem of classifying the finite simple groups. Brauer’s theory used at a critical early stage of that classification, and continue to be a critical tool to understand the result. With the completion of the classification the problem turns around. The finite simple groups are there in nature, the task is to understand them better, particularly the subgroup structure and their representation theory. Also as is well-known, the existing proof of the classification is not perfect. Brauer’s theory and ideas are still very fundamental in revising the proof. This talk is intended to give a survey on some consequences of the classification theorem on modular representations. More results can be found in the expository papers by Broue and Michler.

M. Broue, Equivalences of blocks of group algebras. Finite-dimensional algebras and related topics, 1-26, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 424, Kluwer Acad. Publ., Dordrecht, 1994. *

*Supported by Cheung Kong Scholar’s Programme, National 973 project and RFDP.

506

507

G. Michler, Contributions to modular representation theory of finite groups. Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), 99-140, Progr. Math., 95, Birkhauser, Basel, 1991

2

Puig’ s conjecture and related problems

Let G be a finite group and B a p-block of G with defect group D. For any irreducible B-module S , let V ( S ) be the vertex of V. As is prove by Knoerr, we know that by properly choosing the defect group D , C,(V(S)) 5 V(S) 5 D. Erdmann proves that if V ( S )is cyclic then V ( S )= D . In 1970’s Puig proved in an unpublished paper that for any finite p-solvable group G the order of D is bounded from above in terms of the order of V(S). Later on, Bessenrodt and Willems provided first concrete bound, namely, ID\2 \V(S)\ps(s-1)/2for p-solvable C where s is the normal sectional p-rank of V(S) which is the maximal integer s such that ps = I M / N ( where M and N are normal subgroups of V(S) with M / N elementary abelian. Then I improved the bound so that we have ID1 5 IV(S)lpS-’ for any finitep-solvable G. However, for G = PSL(2 ,q ) with q = 1 (mod 4) the principal 2-block B of G contains an irreducible module with Klein four group as its vertex. Since the Sylow 2-subgroup of G can be as large as possible by a proper choice of q , the result is not true for prime 2.

Conjecture A ( Puig ) For odd prime p and any finite group G, the order of defect group D of a given p-block B is bounded from above in terms of the order of the vertex of irreducible B-module S , namely (Dl 5 f ( ( V ( S ) ( ) where f is a function from integer to integer. By generalizing Fong’s reduction theorem and the results on the p-rank I prove the following

Theorem 1 (Zhang): Puig’s conjecture is true provided it is true for all finite quasi-simple groups. It has been verified Puig’s conjecture is true for alternating groups. The conjecture is still open for groups PSL(n,q ) .

508

Let B be a p-block of a finite group G with defect group D = D ( B ) .Let be the maximal Cartan invariant of B.

CB

Conjecture B ( Hiss) There is a function g from integer to integer such that CB 5 g ( ( D ( B ) (for ) all p-block B. -

Olaf Duevel has reduced Conjecture A to quasi-simple groups.

Now let k(B) be the number of the ordinary irreducible characters in B and let [ ( B )be the number of irreducible Brauer characters in B .

Brauer’s Problem 21 Is there afunction h such that ID(B)I 5 h ( k ( B ) ) for all pblocks B ? Kulshammer provides a positive solution to Brauer’s problem in case G is p-solvable. By apply E. I. Zelmanov’s solution of the restricted Burnside problem, Kulshammer and Robinson prove that Alperin-McKay conjecture implies a positive answer to Brauer’s problem 21. Zeng and I prove the following Theorem 2( Zeng and Zhang). Let B be a pblock of a finite group G with defect group D. Suppose that D is a normal abelian subgroup of G then

ID1 5 l ~ ( B ) ~+/1, 4 if B has a cyclic inertial factor group T(b)/CG(D) and

ID\ 5 n k ( B ) ,if B has a non-cyclic inertial factor group, where n is the inertial index of B . If Broue’s conjecture on p-blocks with abelian defect group is true then the normality of D can be removed. Brauer’s celebrated k ( B ) - conjecture is one of the major problems in modular representation theory. It is stated that k ( B ) can be bounded from above by the order of the defect group D of B.

U. Riese will speak on this problem. Here I would like to talk about a generalization of a result by Michler and a new connection with the conjecture. Note that the significant progress made on the problem by Thompson and Robinson depends on the classification of finite simple groups.

509

Let B be a block of the finite group G with defect group D . Let C ( B )= (cij) be the Cartan matix of B . The multiplicity m ( B )of B is defined to be m ( B )= maxi(cii - 1). Michler studied the blocks with multiplicity one. Theorem 3 (Michler ). If a block B of a finite group has multiplicity one then B contains (DI ordinary characters which are all of height zero, and ID1 - 1 modular characters. If in addition G is p-solvable then D is elementary abelian.

Theorem 4 (Wang). Let B be a block of G with defect group D , then K ( B ) - l ( B )= 1 if and only if all nontrivial elements of D are conjugate in G and dimeBZ(FS) = 1 where eB is the block idempotent of B and S is the psection containing a nontrivial element of D . Theorem 5 (Wang). Notations as above. If K ( B ) - [ ( B )= 1 then all characters in B are p-rational and there exists an positive integer r such that r J x ( u )for any x E B and u E D \ l with -$C,,,(X(U))~ = ID[. In particular, k ( B ) I PI. Now let p(B) be the Perron-Frobenus eigenvalue ( i.e. the largest one) of C ( B ) .Then we have cij 5 p(B)and the equality holds if and only if [ ( B )= 1. For finite p-solvable group G, Kiyota and Wada prove that p ( B ) I (D(B)I.

Conjecture C ( Wada). For finite p-solvable group G, and a p-block B of G, we have k ( B ) I p ( B ) . Thus Conjecture C is stronger than Brauer’s k ( B ) - conjecture for finite psolvable group. Note that if D ( B ) is normal in G then p ( B ) = I D ( B ) Jso , the two conjectures coincide.

I would like to mention that one of the most fundamental problems for eigenvalues of the Cartan matix C ( B )is to study the case where eigenvalues and the elementary divisors of C ( B ) coincide. Since the largest elementary divisor of C ( B )is ID(B)I, the coincidence implies that p(B) = ID(B)I. Recently, the rationality of eigenvalues of C ( B ) has been known to be closely related the semisimplicity of the cellular algebras by the work of C. Xi.

510

3

TI subgroups and TI Blocks

Alperin’s conjecture and related problems have been verified by Blau and Michler for finite groups with a T I Sylow p-subgroup. Now we consider the analogues of T I blocks which have been investigated by Alperin, Broue and Olsson. For a finite group G, denote by BZk(G) the set of p-blocks of G. For a p-subgroup R of G and a pblock bR E BZk(RCG(R)),the pair ( R ,bR) is called a subpair of G. For a pblock B , if (1,B ) 5 ( R ,bR) we call ( R ,bR) a B-subpair. If R is a defect group of B then a B-subpair ( R ,bR) is said to be a Sylow B-subpair. A p-block B is called a T I block if each nontrivial B-subpair is contained in a unique Sylow B-subpair.

Theorem 6 (An and Eaton ) If B is a T I block then B has a T I defect group. The inverse of the theorem is not true, however it is true for principal blocks. Let u = Qo < Q1
Theorem 7 (An and Eaton ). A p-block B of G has a T I defect group if and only if either plr(B) = 0 or plr(B) = 2 and O,(G) = 1. It is well-known that defect groups are Sylow p-intersection. Now let D be a maximal Sylow pintersection of G then NG(D)/D has a T I Sylow p-subgroup. It was an open problem for a long time when D is a defect group. The following result depends on the classification of finite simple groups.

Theorem 8 (Zhang). Let D be a p-subgroup of a finite groupG such that N G ( D ) / Dhas a strongly pembedded subgroup then D is a defect group for

51 1

some p-block of G if and only if there exists a p-regular element z in G such that D is a Sylow p-subgroup of CG(Z).

Corollary 1 . If G is a finite group with an abelian Sylow p-subgroup then every maximal Sylow p-intersection of G is a defect group. Now we consider finite groups with all maximal Sylow p-intersections being TI sets. We call such groups pTI-groups. Finite groups with T I Sylow p-subgroup are pTI-groups, so are groups with a Sylow p-subgroup of order P2.

Question 1. Find all pTI-groups. Question 2 . Verify the conjectures (Alperin, Brauer . . .) for pTI-groups. Theorem 9 (Zhang). If G is a pTI-group then every nontrivial defect group of G is either a Sylow psubgroup or a maximal Sylow p-intersection. The following interesting result is proved by Zenkov and Mazurov : For finite simple group G, there exist two Sylow p-subgroups PI and P2 such that PI n P2 = 1 , and for any finite group G there are a t most three Sylow p-subgroups Pisuch that PI f l P2 n P3 = O,(G).

4

Group theoretical properties from algebras and modules

Let F be an algebraically closed field and G a finite group.

Question 3. Which properties of the finite group G are reflected from FG ? The problem has been intensively investigated by many. If the characteristic of F is zero then the nilpotency of G or Sylow 2-subgroup abelian are determined by FG. It is not known whether or not FG determines the solvability of G. Let Irr(G) be the set of ordinary irreducible characters of G and cd(G) = ( ~ ( 1:)x E Irr(G)}. For non-solvable groups we have the following

512

Conjecture D (Huppert). Let H be a nonabelian simple group and G a finite group. If c d ( H ) = cd(G) then G 2 H x A with A abelian. There are plenty of results on the character degree set cd(G), many by German algebraists. Here I report some recent progress on the character degree graphs. The graph r ( G ) is defined as follows. The vertex set is the set of prime numbers dividing some character degree, and two vertices p , q are joined by an edge if p q l x ( 1 ) for some x E I T T ( G ) . It is proved by Manz, Staszewski an Willems that for any finite group G, r ( G ) has At most three connected components. Their proof depends on the classification of finite simple groups. For finite solvable group G, if r(G) is connected then the diameter is at most 3, if r ( G ) is disconnected then it has two connected components and they are all regular. Lewis and myself determined the finite solvable groups with disconnected character degree graphs.

Conjecture E (Huppert). If finite solvable group G has a connected degree graph r(G) then the diameter is at most 2. Recently, Lewis provides a counter-example. Now we consider non-solvable groups. First we have the following result which will be soon published. Again the proof depends on the classification of finite simple groups.

Theorem 10( Lewis and White ). Let G be a finite non-solvable group. Then: I. r ( G ) has three connected components if and only if G = S x A where S 2 PSL(2,2")(n2 2) and A is abelian. 11. r(G) has two connected components if and only if there exist normal subgroups N 5 M 5 G so that the following hold: a). M / N PSL(2,q ) , q 2 4 is a prime power. b). Set C / N = CG,,(M/N) then C / N 5 Z ( G / N ) and G / M is abelian.

513

c). If q 2 5, then p does not divide IG : CMI. d). If N > 1 then either M E SL(2,q) or there is a normal subgroup L of G so that M I L E SL(2,q ) , L is elementary abelian of order q 2 , and the action of M I L on I r r ( L ) is the action of SL(2,q ) on the natural module.

e). If p = 2 or q = 5, then either C M < G or N

> 1.

f). If p = 2 and N > 1, then every nonprincipal character in I r r ( L ) extends to its stabilizer in G. It can be easily checked that Janko one Jl has a connected character degree graph with diameter 3. But most simple groups have regular graph. Now let field F be of characteristic p dividing the order of G. Lu Ziqun and I studied the modular version of Huppert 0 - pconjecture and reduce the problem to nilpotent-by-metabelian groups. Also Hanaki provides some necessary or sufficient conditions for N G (P )/ P to be abelian in terms of the structure of the group algebra F G . For any FG-module M with finite dimension let ..... -+ P, -+ P,-l + .... + Po + M -+ 0 be the minimal projective resolution of M , we define the complexity C G ( M )of M to be the smallest integers s 2 0 such that lim(dimFP,)/ns = 0 as n goes to infinity. Carlson proves that the complexity of M is equal to the dimension of the variety V G ( M ), namely, CG ( M ) = dimVG ( M ) where V G ( M )consists of maximal ideals of H * ( G ,F) containing the annihilator of the cohomology ring of M .

If C G ( M )= 0 , then M is projective. If C G ( M )= 1 we say that M is periodic. Theorem 11( Donovan). Let B be a p-block of F G with defect group

D , then p-rank of D = mazCG(M) = m a z C ~ ( M )

Theorem 12(Bessenrodt and Willems). Let G be a finite p-solvable group and M an irreducible FG-module then the complexity C G ( M )is equal

514

to the p-rank of the vertex of M . If p = 2 and a 2-block B contains an irreducible periodic module then the defect group D ( B ) is cyclic, generalized quaternion or semidihedral.

Theorem 13(Zhang). Let G be a finite p-solvable group and B a pblock with defect group D. If M is an irreducible FG-module in B with abelian vertex V and C G ( M )5 p - 1 then V = D unless C G ( M )= p - 1 = 2" and D G / L &! P p - l : (21+2n : p) where DG is the normal closure of D in G and L is the center of DG.