M-Solid Varieties of Algebras (Advances in Mathematics)

M-SOLID VARIETIES OF ALGEBRAS Advances in Mathematics VOLUME 10 Series Editor: J. Szep, Budapest University of Econom...

Author: J. Koppitz | K. Denecke

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M-SOLID VARIETIES OF ALGEBRAS

Advances in Mathematics VOLUME 10 Series Editor: J. Szep, Budapest University of Economics, Hungavy

Advisory Board: S-N. Chow, Georgia Institute of Technology, U S A . G. Erjaee, Shiraz University, Iran W. Fouche, University of South Africa, South Africa P. Grillet, Tulane University, U.S.A H.J. Hoehnlte, Institute of Pure Mathematics of the Academy of Sciences, Germany F. Szidarovszky, University ofAirzona, U.S.A.

P.G. Trotter, University of Tasmania, Australia P. Zecca, Universitd di Firenze, Italy

M-SOLID VARIETIES OF ALGEBRAS

J. KOPPITZ Universitat Potsdam, Germany K. DENECKE Universitat Potsdam, Germany

Q - Springer

Libraryof Congress Control Number: 2005936714

Printed on acid-free paper.

AMS Subject Classifications: 08605, 08B15, 20M07, 16Y60

0 2006 Springer Science+Business Media, Inc. All rights rcscrvcd. This work may not bc translatcd or copicd in wholc or in part without thc writtcn permission of the publisher (Springer Science+Business Media, Tnc., 233 Spring Street, New York, NY 10013, USA), cxccpt for bricf cxccrpts in conncction with rcvicws or scholarly analysis. Usc in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not idcntificd as such, is not to bc takcn as an cxprcssion of opinion as to whcthcr or not thcy arc subjcct to proprietary rights. Printcd in thc Unitcd Statcs of Amcrica.

Contents

Preface 1 Basic Concepts 1.1 Subalgebras and Homomorphic Images . 1.2 Direct and Subdirect Products . . . . . . 1.3 Term Algebras, Identities. Free Algebras 1.4 The Galois Connection ( I d . Mod) . . . .

vii

. . . .

. . . .

. . . .

2 Closure Operators and Lattices 2.1 Closure Operators and Kernel Operators . . . 2.2 Complete Sublattices of a Complete Lattice . 2.3 Galois Connections and Complete Lattices . . 2.4 Galois Closed Subrelations . . . . . . . . . . . 2.5 Conjugate Pairs of Additive Closure Operators

. . . .

. . . . .

. . . .

. . . . .

. . . .

1 1 11 16 24

. . . . .

29 29 31 34 37 41

3 M-Hyperidentities and M-solid Varieties 3.1 M-Hyperidentities . . . . . . . . . . . . . . . . . . . 3.2 The Closure Operators X&, Xg . . . . . . . . . . . 3.3 M-Solid Varieties and their Characterization . . . .

49 49 62 64 3.4 Subvariety Lattices and Monoids of Hypersubstitutions 67 3.5 Derivation of M-Hyperidentities . . . . . . . . . . . 71

4 Hyperidentities and Clone Identities 4.1 Menger Algebras of Rank n . . . . . . . . . . . . . 4.2 The Clone of a Variety . . . . . . . . . . . . . . . .

77 78 84

5 Solid Varieties of Arbitrary Type 5.1 Rectangular Algebras . . . . . . . . . . . . . . . . . 5.2 Solid Chains . . . . . . . . . . . . . . . . . . . . . .

87 87 96

vi

Contents

6 Monoids of Hypersubstitutions 6.1 Basic Definitions . . . . . . . . . . . . . . . . . . 6.2 Injective and Bijective Hypersubstitutions . . . . 6.3 Finite Monoids of Hypersubstitutions of Type ( 2 ) 6.4 The Monoid of all Hypersubstitutions of Type ( 2 ) 6.5 Green's Relations on H y p ( 2 ) . . . . . . . . . . . . 6.6 Idempotents in H y p ( 2 , 2 ) . . . . . . . . . . . . . . 6.7 The Order of Hypersubstitutions of Type ( 2 ' 2 ) . 6.8 Green's Relations in H y p ( n , n ) . . . . . . . . . . 6.9 The Monoid of Hypersubstitutions of Type (n) . . . . . . . 6.10 Left-Seminearrings of Hyper~ubstit~utions

103

. . . . . . . . . .

103 109 114 119 126 138 151 171 181 184

7 M-Solid Varieties of Semigroups 7.1 Basic Concepts 011 M-Solid Varieties of Semigroups 7.2 Regular-solid Varieties of Semigroups . . . . . . . . 7.3 Solid Varieties of Semigroups . . . . . . . . . . . . 7.4 Pre-solid Varieties of Semigroups . . . . . . . . . . 7.5 Locally Finite and Finitely Based M-solid Varieties

197 197 206 252 255 260

8 M-solid Varieties of Semirings 8.1 Necessary Conditions for Solid Varieties of Semirings 8.2 The Minimal Solid Variety of Semirings . . . . . . . 8.3 The Greatest Solid Variety of Semirings . . . . . . 8.4 The Lattice of all Solid Varieties of Semirings . . . 8.5 Generalization of Normalizations . . . . . . . . . . 8.6 All Pre-solid Varieties of Semirings . . . . . . . . .

267 267 269 271 275 283 298

Bibliography

321

Glossary

331

Index

337

Preface

The general study of varieties of algebras with finitary operations was initiated by Garrett Birkhoff in the 1930's. He derived the first significant results in this subject and further developments by Alfred Tarski, and later, for congruence distributive varieties by Bjarni Jbnsson, laid the groundwork for many of the results on lattices of varieties. Varieties are equationally definable classes of algebras of the same finitary type. Identities are used t o classify algebras of the same type into varieties. The relation given by "an algebra satisfies an equation as an identity" defines a Galois connection between the class of all algebras of the same type and the set of all equations of this type. This Galois connection (Id, Mod) consists of the operator I d , which associates to each class of algebras, of the given type the set of all identities satisfied in this class and Mod, which maps each set of equations t o the class of all algebras satisfying these equations as identities. The connection defines two closure operators IdMod and ModId. The fixed points under the operator ModId are the varieties and the fixed points under IdMod are called equational theories. The collection of all varieties of algebras of a given type forms a complete lattice. Birkhoff's variety theorem characterizes varieties as classes of algebras of the same type which are closed under taking of arbitrary subalgebras, homomorphic images and direct products. The equational theories of a given type form another complete lattice which is dually isomorphic t o the lattice of all varieties. Equational theories can be characterized as sets of equations which are closed under arbitrary applications of the five derivation rules for identities. An important area of activity in Universal Algebra has been to try t o classify all varieties of algebras or dually all equational theories, of a given type. Even if the type contains only one binary operation symbol, and even if the binary operation symbol satisfies the associative law, so that the algebras are semigroups, such a project is almost hopelessly complicated.

viii

Preface

An identity is a formula in a first order language with equality where the variables are bound by the universal quantifiers. The associative law V X ~ V X ~ V X ~(x2 ( Z .~2 3 ) M (xl . 2 2 ) ~ 3 ) is satisfied (true) in an algebra with a binary fundamental operation if when any elements of the universe are substituted for x l , x2,x3 and the operation symbol is replaced by the operation of the algebra, the resulting elements from the universe are equal. The fundamental operations the equation

+R,

eR

of a ring R are associative. In

we can replace the binary operation symbol F by either +R or and obtain an identity satisfied in R. This leads us viewing the associative law as a formula of the form eR

This is an example of a formula in a second order language, where now quantification of operation symbols over the set of all fundamental operations of the ring R is allowed. If substitution of any binary term operation from an algebra A for the binary operation symbol F leads to identities satisfied in A we call the formula (*) a hyperidentity satisfied in A. Any substitution of a binary term for the binary operation symbol F is called a hypersubstitution. Hypersubstitutions can be extended to mappings defined on the set of all terms. The concept of a hypersubstitution plays a crucial role in this book and was defined first in [56]. Like ordinary substitutions, hypersubstitutions can be composed and together with an identity hypersubstitution which maps any operation symbol fi to an ni-ary "fundamental" term fi(xl, . . . , xn,) one obtains a monoid IFtyp(r). Submonoids M of IFtyp(r) can be used t o define the weaker concept of an M-hyperidentity. The relation given by "an algebra M-hypersatisfies an equation" defines a second Galois connection between the class of all algebras of the same type and the set of all equations of this type. The operator H M I d associates to each class of algebras of the given type the set of all M-hyperidentities satisfied in this class. The operator HMMod maps each set of equations

Preface

ix

of type T t o the class of all algebras satisfying these equations as M-hyperidentities. The pair (HhIId, HMMod) is a Galois connection between the class of all algebras of a given type and the set of all equations of this type. The fixed points under the closure operator HMModHMId are called M-solid varieties and the fixed points under HMIdHMMod are called M-hyperequational theories. The collection of all M solid varieties of a given type forms a complete sublattice of the lattice of all varieties of this type; and if M I is a submonoid of the monoid M Z of hypersubstitutions, then the complete lattice of all M2-solid varieties is a complete sublattice of the lattice of all MI-solid varieties. This leads us t o consider a Galois connection between submonoids of IFtyp(r) and sublattices of the lattice of all varieties of algebras of the given type. Instead of the very complex lattice L(r) of all varieties of type T one may consider the lattices Snr(r)of all M-solid varieties of type r . There is some hope that these lattices are easier to describe. Instead of the lattice C(7) of all varieties of type T we will also consider the lattices L(V) of all subvarieties of some given varieties V and the lattices SM(V) of its M-solid subvarieties. Dually, we consider the complete lattice of all M-hyperequational theories which are characterized as sets of equations closed under the five derivation rules for identities and one more rule of consequences, the so-called M-hypersubstitution rule. Any variety can be described by only one structure, the clone of the variety. Conversely, every clone can be regarded as the clone of some variety. The clone of a variety V is a multi-based algebra where the different sorts are the sets of n-ary terms over the variety V and where the fundamental operations describe the superposition of terms. There is also a one-based variant of this concept which is called a unitary Menger algebra of rank n ([66]). Hypersubstitutions correspond t o substitutions of the clone of the variety, and identities of the clone of the variety V can be regarded as hyperidentities in the variety V. A variety V is solid if and only if its clone is relatively free. Note that clones are equivalent t o algebraic theories, i.e. particular categories, in the sense of I?. W. Lawvere.

x

Preface

The aim of this book is t o develop the theory of M-solid varieties as a system of mathematical discourse, applicable in several concrete situations. The general theory will be applied to two classes of algebraic structures, to semigroups and t o semirings. Both these varieties and their subvarieties play an important role in Computer Science. The study of rational languages is related to the theory of classes of finite semigroups and monoids. Samuel Eilenberg's correspondence theorem ([42]) connects certain classes of regular languages, so-called varieties of languages, with pseudovarieties of semigroups. Pseudovarieties are classes of finite algebras of the same type which are closed under the formation of homomorphic images, subalgebras and finite direct products. Since the "finite part" of a variety is a pseudovariety (but not conversely), it makes sense to study varieties and M-solid varieties first. For applications of semirings and classes of semirings in Automata Theory and in the theory of Formal Languages see [51].Type (2) algebras seem t o be simple enough to be accessible, yet rich enough to provide an interesting hyperidentity structure. This is especially true for semigroups, where a lot is known about the lattice of all semigroup varieties. In both classes the associative law plays an important role. Semirings are the first class with two binary operation symbols where lattices of M-solid varieties for some monoids M of hypersubstitutions were completely determined. In the first chapter we develop the basic concepts of Universal Algebra, especially the equational theory. In this chapter we also define different complexity measures of terms or trees, some important subsets of the set of all terms of a given type and the superposition operation for terms. The influence of term complexity on the Galois connection between identities and algebras is shown in Chapter 8. A unique feature of this book is the use of Galois connections as a main tool. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, ordertheoretical, categorical and logical theories. This concept, with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators are used throughout the whole book and are introduced in

Preface

xi

Chapter 2. Here three different methods are described t o characterize complete sublattices of a complete lattice; by using a conjugate pair of completely additive closure operators, by special closure or kernel operators defined on the given complete lattice, and by socalled Galois closed subrelations. The theory of conjugate pairs of completely additive closure operators starts with a relation between two given sets and the Galois connection induced by this relation. The two closure operators obtained from this Galois connection define two complete lattices. A subrelation of the given relation defines a new Galois connection and two more complete lattices, which can also be defined as sets of fixed points of two other closure operators forming a conjugate pair and having the property to be completely additive. It turns out that under some conditions the last two lattices are complete sublattices of the first ones. This theory will be used t o characterize lattices of M-solid varieties as complete sublattices of the lattice of all varieties of the given type. The theory of conjugate pairs of completely additive closure operators was used in [16] to characterize lattices of M-solid quasivarieties and in [81] t o characterize lattices of M-solid pseudovarieties. A completely different application of this theory is discussed in [96]. Here the basic Galois connection is the connection (Pol, Inv) between operations and relations defined on the same set A and induced by the relation "an operation f preserves a relation Q". The Galois closed sets under the operator Pollnu are clones. A second Galois connection is given by a group G of permutations on A and the relation that a permutation conjugates a clone onto itself. The Galois closed sets on the clone side are the lattices CG of all clones that are closed under conjugation by all members of some permutation group G. In Chapter 3 we develop the theory of M-hyperidentities and M solid varieties and in Chapter 4 we show the connection between clone theory and the theory of hyperidentities. In Chapter 5 we show that there are non-trivial solid varieties of arbitrary type. For each non-trivial type there are infinite chains of solid varieties. This is surprising since the satisfaction as a hyperidentity is quite a strong requirement. In Chapter 6 we deal with monoid- and semigroup-theoretical prop-

xii

Preface

erties of hypersubstitutions. The Galois connection between subvariety lattices and monoids of hypersubstitutions motivates the study of monoid or semigroup properties of the set of all hypersubstitutions and its submonoids. An example is Green's relation C. If two hypersubstitutions are C-related, then their kernels are equal. We will prove that the kernel of a hypersubstitution is an equational theory. Therefore we have two equal equational theories or two equal varieties. We determine all injective and all bijective hypersubstitutions, study all finite monoids of hypersubstitutions of type (2), determine all idempotent hypersubstitutions of type (2) and of type (2,2), and consider regular elements and the order of hypersubstitutions. To determine the order of all hypersubstitutions of type (2,2) we have to consider several cases, sub- and subsub-cases. This part looks quite technical and includes detailed calculations. The main idea which is used throughout this chapter is the connection between the complexity of a term and the complexity of its image after applying a hypersubstitution. It turns out that the formulas given in [38]are very useful. Moreover, in Chapter 6 we deal with Green's relations on monoids of hypersubstitutions of different types. In Chapter 7 we present results on M-solid varieties of semigroups for various choices of the monoid M. We determine the least and the greatest elements in the lattices of all regular-solid, pre-solid and solid varieties of semigroups. Moreover, we prove a theorem which characterizes all regular-solid varieties of semigroups. From this result we derive characterization theorems for solid and presoild varieties. Using the general theory from Chapter 3 the greatest M-solid variety of semigroups is the M-hyperequational class defined by the associative law, i.e. the class of all algebras of type (2) where the associative law is satisfied as an M-hyperidentity. It is not clear whether such classes are finitely based by identities. This question is discussed in section 7.5. We will also answer the question of which monoids M of hypersubstitutions have the property that the greatest M-solid variety is locally finite. One of the main methods to be used here is the solution of the word problem in a two-generated free algebra over the considered M-solid varieties of semigroups.

Preface

xiii

In the last chapter we determine all solid and all pre-solid varieties of semirings. While there are infinitely many solid varieties of semigroups, the lattice of all solid varieties of semirings is a chain consisting of precisely four elements, and there are altogether 13 pre-solid varieties of semirings. These pre-solid varieties are normalizations and 2-normalizations of the solid ones. Calling a variety k-normal when both sides of any of its nontrivial identities have complexity at least k, we extend several central results on normal varieties to the k-th level and apply the results t o pre-solid varieties of semirings. The material of the last three chapters of this book is based on results which are contained in the habilitation thesis of the second author, and in the Ph.D. theses of Dr. Hippolyte Hounnon and Dr. Thawhat Changphas. The authors would like t o thank their colleagues in the growing "hyperidentity family" around the world for encouragement and helpful discussions. The support given by the Institute of Mathematics and the Faculty of Sciences of University of Potsdam during the summer semester 2004 is gratefully acknowledged.

Prof. Dr. Dr. hc. Klaus Denecke

Potsdam, September 2005

PD Dr. habil. Jorg Koppitz

1 Basic Concepts 1.1 Subalgebras and Homomorphic Images An algebra is a non-empty set A together with a set of finitary 1 operations defined on this set. An n-ary operation on A is for n a function f A : An + A from the n-th Cartesian power of A into A. We let On(A) be the set of all n-ary operations defined on A and

>

00

let O(A) :=

U On(A) be the set of all finitary operations defined

n=l

on A.

Remark 1.1.1 1. Any n-ary operation f on A can be regarded as an ( n + 1)-ary relation defined on A, called the graph of f . This relation is defined by {(al, . . . , a,+l) € An+' I f (al, . . . , a,) = an+l}.

c

2. If B A is a non-empty subset of A, then the restriction f A l B of f A t o B is defined by f A I ~ ( a l ., .. ,a,) := f A ( a l , . . . ,a,) for all a l , . . . , an E B.

3. The definition of an n-ary operation can be extended in the following way to the special case that n = 0, for a nullary operation. We define A0 := (8). A nullary operation is defined as a function f : (0) + A. This means that a nullary operation on A is uniquely determined by the element f (0) E A. For every element a E A there is exactly one mapping fa : (0) + A with fa(@) = a. Therefore a nullary operation may be thought of as selecting an element from the set A. If A = 0 then there are no nullary operations on A. There are two ways to view an algebra, in the non-indexed form as a pair A := (A; FA)consisting of a set A and a set FAof operations defined on A or as an indexed algebra.

Definition 1.1.2 Let A be a non-empty set. Let I be some nonbe a function which assigns t o empty index set, and let

(ft)i,l

1 Basic Concepts

2

ft

defined on A. Then the every element i of I an ni-ary operation pair A = (A; (f:)i,I) is called an (indexed) algebra with the base set or carrier set or universe A, and with (f:)iE1 as the sequence of fundamental operations of A. The sequence r := (ni)iE1 of all arities of the fundamental operations is called the type of A. We use the name Alg(r) for the class of all algebras of type r .

ft

In our definition we do not allow the base set A of an algebra to be the empty set. It is possible to define an empty algebra with the empty set as universe if no nullary operations belong to the type. But mostly we will exclude this case. In the indexed case the fundamental operations form a sequence, not a set, that means, repititions are possible. We will give some examples for our definition. An algebra (G; .) of type r = (2), i.e. having one binary operation, is called a groupoid. The binary operation is simply denoted by . but we will also write z y or just z y . If the binary operation is associative, G is called a semigroup. A semigroup with an additional nullary operation e is called a monoid M = ( M ; e) if e is an identity element with respect t o ., i.e. if for all x E M the equations x . e = e . x = x are satisfied. Monoids are algebras of type r = (2,O). 0

A group

G

=

(G;

a )

,

is a semigroup, which satisfies the axiom

' d a , b ~ G 3 x , y ~ G ( a . x = b a n d ~ . a = b ) (invertibility). A group can also be regarded as an algebra G = (G; ., type (2, 1, 0), where the associative law and the axioms

-I,

e) of

and 'dx E G ( x - e

=

x

=

e.x)

are satisfied.

+,

An algebra R = (R; .) of type r = (2,2) is called a semiring if both these binary operations are associative and the distributive laws 'dx, y, Z E R ( x . ( ~ + z )= x . ~ + x . z )

I.1 Subalgebras and Homomorphic Images and

are satisfied. Rings are semirings with commutative and invertible addition and a commutative ring K = ( K ; is called a field if ( K \ (0); is a group, where the zero element 0 is the neutral element with respect t o addition.

+,

0

)

0

)

An algebra V = (V; A , V ) of type (2,2) is called a lattice, if the following equations are satisfied by its two binary operations, which are usually called meet and join: 'dx, y E V 'dx, y E V 'dx,y,z E V 'dx, y, x E V 'dx E V 'dx E V 'dx,y E V 'dx,y E V If in addition the lattice satisfies the following distributive laws, ' d x , ~E, V ~ (XA(YVZ) = ( X A Y ) V ( X A Z ) ) , 'dx, y, x E V (xV(yAx) = (zVy)A(xVx)), then the lattice is said t o be distributive. Lattices are important both as examples of a kind of algebra, and also in the study of all other kinds of algebras, since any algebra has some lattices associated with it. A lattice can also be regarded as a partially ordered set (V; I),where I is a binary, reflexive, antisymmetric and transitive relation on V. There is a close connection between lattices and partially ordered sets, as the following theorem shows.

Theorem 1.1.3 Let (V; 5) be a partially ordered set in which for all x, y E V both the infimum A{x, y) and the supremum V{x, y) exist. T h e n the infimum and supremum operations make (V; A, V)

4

1 Basic Concepts

a lattice. Conversely, every lattice defines a partially ordered set i n which for all x, y the i n f i m u m A{x, y } and the supremum V{x, y } exist. Proof: Let (V; 5) be a partially ordered set, in which for any twoelement set {x, y ) & V the infimum A{x, y ) and the supremum V{x, y ) exist. We define x A y := A{x, y ) and x V y := V{x, y ) . Then the required identities are easy t o verify. If conversely (V; A, V ) is a lattice, then we define

and see that this gives a partial order relation on V. It is easy to check that A{x, y ) = x A y and V{x, y ) = x V y are satisfied. A lattice C in which for all sets B & L the infimum supremum V B exist is called a complete lattice. Obviously, any finite lattice is complete.

B and the

A bounded lattice (V; A , V , 0 , l ) is an algebra of type (2, 2, 0, 0) which is a lattice, with two additional nullary operations 0 and 1, which satisfy 'dx E V (XAO 'dx E V (xv1

= =

0) and I).

An algebra B = (B;A, V, l , O , 1) is called a Boolean algebra, if (B;A, V, 0 , l ) is a bounded distributive lattice with an additional unary operation satisfying 1

and 'dx E B ( x V 1 x = I ) . Important examples of Boolean algebras are the two-element Boolean algebra ((0, 1); A , V , 0 , l ) with conjunction and disjunction (meet and join operation) as binary, and with negation as unary operations and the power set algebra ( P ( A ); n,U, N , 0,A) of

5

I.1 Subalgebras and Homomorphic Images

A with intersection, union, complementation, the empty set and A as operations. An algebra S = ( S ; of type (2) is called a semilattice, if the operation . is an associative, commutative, and idempotent binary operation on S . This means that a semilattice is a particular kind of semigroup. 0

)

For a given algebra B = ( B ;( f ? ) i E I ) of type r we obtain new algebras using certain algebraic constructions. The first algebraic construction we want to mention, is the formation of subalgebras.

Definition 1.1.4 Let B = ( B ; (f?)i,I) be an algebra of type r . Then an algebra A is called a subalgebra of B, written as A B, if the following conditions are satisfied:

c

(i)

A = ( A ; ( f , i ' ) i , ~ ) is an algebra of type r ;

(ii) A

B;

(iii) 'v'i E I, the graph of f ) is a subset of the graph of f?. The subalgebra property can be checked in the following way:

Lemma 1.1.5 (Subalgebra Criterion) Let B = ( B ; ( f ? ) i E I ) be a n algebra of type r and let A C B be a subset of B for which f ) = f? I A for all i E I . T h e n A = ( A , ( f ) ) i E I ) is a subalgebra of B = ( B , ( f ? ) i E I ) i&f A is closed with respect t o all the operations f? for i E I ; that is, i f f?(Ani) C A for all i E I . ( W e say that f? preserves the subset A of B . ) Clearly, the subalgebra relation is transitive, that is, if A then A C.

c

CB CC

If B = ( B ;( f ? ) i , ~ ) is an algebra of type r and if {Aj I j E J) is a family of subalgebras of B with the non-empty intersection A := Aj of its universes, then it is easy t o see that A is the j, J

universe of a subalgebra of B which is called the intersection of the family {Aj I j E J), denoted by A. This allows us to consider

n

.it J

the subalgebra

( X ) B:= n{A I A

CB

and X

CA}

6

1 Basic Concepts

of B generated by a subset X C B of the universe. The set X is called a generating system of this algebra. The process of subalgebra generation satisfies three very important properties, called the closure properties. This makes the subalgebra generation process an example of a closure operator, which we will consider in more detail in chapter 2.

Theorem 1.1.6 Let B be a n algebra. For all subsets X and Y of B , the following closure properties hold: (i) X C (X)u, (ii) X Y + (X)u (iii) (X)u = ( ( X ) u ) a ,

c

c (Y)u,

( extensivity) ; (monotonicity); (idempotency).

If the intersection of two subalgebras of B , is non-empty, then it is again a subalgebra of B , and we would like t o use this to define a binary operation on the set S u b ( B ) of all subalgebras of B . Therefore a binary operation A on S u b ( B ) is defined by

Unfortunately, this operation is not defined on all of S u b ( B ) , because it does not deal with the case where Al n A2 is the empty set. But this problem can be solved, either by allowing the empty set to be considered as an algebra or by adjoining some new element to the set S u b ( B ) and defining Al A A2 t o be this new element, whenever Al n A2 = 0. The union Al U A2 of the universes of two subalgebras of B is in general not a subalgebra of B . But we can map the pair (Al,A2) to the subalgebra of B which is generated by the union Al U A2:

Altogether we obtain:

Theorem 1.1.7 For every algebra B , the algebra ( S u b ( B ) ;A , V ) is a lattice, called the subalgebra lattice of B . Our second important algebraic construction is the formation of

homomorphic images.

7

I.1 Subalgebras and Homomorphic Images

Definition 1.1.8 Let A

(A; (f,")iEI) and B = (B; (f?)iEI) be algebras of the same type r. Then a function h : A + B is called a homomorphism h : A + B of A into B if for all i E I we have =

for all a l , . . . , ant E A. In the special case that ni = 0, this equation means that h(fk(0)) = fiB(0). Therefore the element designated by the nullary operation in A must be mapped t o the corresponding element f: in B.

fk

If h is surjective (onto), then B is called homomorphic image of A. If the function h is bijective, that is both one-to-one (injective) and "onto" (surjective), then the homomorphism h : A + B is called an isomorphism from A onto B. An injective homomorphism from A into B is also called an embedding of A into B.

A homomorphism h : A

A of an algebra A into itself is called an endomorphism of A, and an isomorphism h : A + A from A +

onto A is called an automorphism of A. It is easy t o see that the image B1 = h(A1) of a subalgebra Al of A under the homomorphism h is a subalgebra of B and that the preimage hpl(B') = A' of a subalgebra B' of h(A) C B is a subalgebra of A. If X

c A is a subset of the universe of an algebra A of type 7 , if

(X)Ais the subalgebra of A generated by X and if h : A

+B

is a

homomorphism, then

If especially X is a generating system of surjective, then h ( X ) generates B since

A and if h : A

+

B is

Let A, B and C be algebras of the same type, and let hl : A + B and h2 : B + C be homomorphisms, then h2 o hl : A + C is also a homomorphism, and when both hl and h2 are surjective, injective or bijective, then the composition has the same property.

8

1 Basic Concepts

Clearly, the identity mapping idA on the set A is always an automorphism of the algebra A. Since the composition operation o is associative, the set End(A) of all endomorphisms of an algebra A forms a monoid ( E n d ( A )0, ; idA),the endomorphism monoid of A. Since the inverse mapping of an automorphism cp : A + A defines again an automorphism, the set Aut(A) of all automorphisms of A forms a group ( A u t ( A )o, ; idA),which we call the automorphism group of A. Any homomorphism h : A + B goes out from the algebra A into the algebra B. We ask for an "internal" description of the homomorphic image inside B. Every function h : A + B from a set A onto a set B defines a partition of A into classes of elements having the same image. Partitions of a set define equivalence relations on that set where two elements are related to each other if and only if they belong to the same block of the partition. If A and B are the universes of two algebras A and B E Alg(r) and if h : A + B is a surjective homomorphism, then h can be compatible with an equivalence relation Q on A in the following sense:

c

Definition 1.1.9 Let A be a set, let 8 A x A be an equivalence relation on A, and let f be an n-ary operation from On(A). Then f is said to be compatible with 8,or t o preserve 8,if for all a1, . . . , an,bl, . . . , b, E A ,

Definition 1.1.10 Let A = ( A ; ( f t ) i , I ) be an algebra of type 7 . An equivalence relation 8 on A is called a congruence relation on A if all the fundamental operations f: are compatible with 8.We denote by Con(A)the set of all congruence relations of the algebra A. For every algebra A = ( A ; (f:)i,I) the trivial equivalence relations AA := { ( a , a ) I a E A} and V A = A x A are congruence relations. An algebra which has no congruence relations except AA and VA is called simple.

I.1 Subalgebras and Homomorphic Images

9

For two congruence relations 81, Q2 E Con(A) the intersection Q1 nQ2 is again a congruence relation on A. This defines a binary operation

Simple examples show that the union Q1 U Q2 of two equivalence relations Q1, Q2 on the set A needs not to be an equivalence relation on A. To have a second binary operation on Con(A) we define at first the congruence relation defined by a binary relation on A. Definition 1.1.11 Let A be an algebra, and let 8 be a binary relation on A. We define the congruence relation (Q)con(A) on A generated by Q t o be the intersection of all congruence relations 8' on A which contain 8: (Q)con(Aj :

=

n{Q'I

8' t Con(A) and Q

C Q').

Again it can be seen that (Q)Con(A) has the three important properties of a closure operator:

Now we define the second binary operation on Con(A) by

v:

Con(A) x Con(A) + Con(A) with (81~82)H (81U ~ ~ ) C O ~ ( A )

and obtain: T h e o r e m 1.1.12 For every algebra A the algebra (Con(A);A, V) is a lattice which is called the congruence lattice of A. If Q is a congruence relation on A, then we can partition the set A into blocks with respect to Q and obtain the quotient set A/Q. In a natural way, for each i t I, we define an ni-ary operation on the quotient set by

ftl*

fi"I*

: (A/Q)ni+ A/Q

1 Basic Concepts with

Of course, we have t o verify that our operations are well-defined, that is, that they are independent on the representatives chosen. But this is exactly what the compatibility property of a congruence relation means and we obtain a new algebra A/Q := (AIQ;( f t / * ) i , I ) , which is called the quotient algebra (or factor algebra) of A by 8. Actually, for every congruence relation Q the algebra A / 8 is a homomorphic image of A under the natural homomorphism defined by

n a t ( Q ): A

+ A/Q w i t h

a

H

[a]*for every a E A .

It is easy to check that n a t ( 8 ) is really a surjective homomorphism. So, for any congruence relation 8 E C o n ( A ) we obtain a homomorphism and it arises the question whether homomorphisms define congruence relations on A . This is also the case since we have:

Lemma 1.1.13 The kernel

ker h := { ( a ,b) E A2 I h ( a ) = h ( b ) } of any homomorphism h : A

+B

is a congruence relation o n A .

Suppose we have a homomorphism h : A + B. We have seen that ker h is a congruence on A, so we can form the quotient algebra A l k e r h, along with the natural homomorphism n a t ( k e r h ) : A + A l k e r h which maps the algebra A onto this quotient algebra. Now we have two homomorphic images of A: the original h(A)and the new quotient A l k e r h . What connection is there between these two homomorphic images? The answer to this question is given by the well-known Homomorphic Image Theorem

Theorem 1.1.14 (Homomorphic Image Theorem) Let h : A + B be a surjective homomorphism. T h e n there exists a unique isomorphism f from A l k e r h onto B with f o n a t ( k e r h ) = h, that is, the following diagram commutes.

1.2 Direct and Subdirect Products

n n t (ker.\h )

Later on we need the sublattice Coni,,(A) congruence relations on A .

f

of all fully invariant

Definition 1.1.15 Let A be an algebra of type 7. A congruence relation 8 E C o n ( A ) is called fully invariant if for all endomorphisms cp : A + A we have (a, b) E 8 + (cp(a),cp(b)) E 8 for all a , b E A. The congruence relations

AA and VA are always fully invariant.

Proposition 1.1.16 The set Coni,,(A) of all fully invariant congruence relations of A forms a sublattice of C o n ( A ).

1.2

Direct and Subdirect Products

In this section we shall examine another important construction, the formation of product algebras. Subalgebras or homomorphic images of a given algebra have cardinalities no larger than the cardinality of the given algebra. The formation of products, however, can lead t o algebras with bigger cardinalities than those we started with. At first we consider the direct product of a family of algebras.

Definition 1.2.1 Let be a family of algebras of type r. Slj of the Slj is defined as an algebra with The direct product

n

I €J

the carrier set

1 Basic Concepts

12 and the operations

for

a,, . . ., antin P;that

is,

If for all j E J, Aj = A, then we usually write AJ instead of 4.If J = 0,then A0 is defined t o be the one-element (trivial)

n

jtJ

algebra of type r. If J = {I, . . . , n),then the direct product can be written as A1 x . . x A,. The projections of the direct product

n A j are the mappings jt J

pk :

n

Aj

+ Ale defined

by (aj)jtJ

ah-.

jtJ

It is easy to check that the projections of the direct product are in fact surjective homomorphisms. We recall the definition of the product (composition) Q1 o Q2 of two binary relations Q1, 82 on any set A: Q10 Q2

:= {(a, b)

Two binary relations Q1 o Q2 = Q2 o Q1.

I

3c E A ((a, c ) E Q2 A (c, b) E Q1)).

Q1,Q2

on A are called permutable,

if

We now consider a direct product of two factors. In this case we have two projection mappings, pl and p2, each of which has a kernel which is a congruence relation on the product, since pl, p2 are homomorphisms. These two kernels have special properties.

Lemma 1.2.2 Let A1, A2 be two algebras of type T and let Al x A2 be their direct product. Then:

(ii)

ker pl o ker p2

=

ker p2 o ker pl;

1.2 Direct and Subdirect Products

This lemma motivates the following definition:

Definition 1.2.3 A congruence 8 on A is called a factor congruence if there is a congruence 8' on A such that 8A8' = aA 8V8' = The pair (8,8') is called a pair of factor congruences on A.

vA.

Theorem 1.2.4 If (8,Q') i s a pair of factor congruences, t h e n A i s isomorphic with the direct product A/8 x A/8' (A A/8 x A/V) u n d e r the isomorphism given by a H ([ale, [ale!).

"

Definition 1.2.5 An algebra A is called directly indecomposable, if whenever A B1 x B2, either lBll = 1 or IB21 = 1.

"

It is well-known (see e.g. [37]) that there is up to isomorphism only one non-trivial directly indecomposable Boolean algebra, namely the two-element Boolean algebra ((0, 1); A , V , 1 , 0 , 1 ) . Cardinality considerations show that a countably infinite Boolean algebra cannot be isomorphic to a direct product of directly indecomposable algebras. But for finite algebras we have:

Theorem 1.2.6 Every finite algebra A i s isomorphic t o a direct product of directly indecomposable algebras. This can be proved by induction on the cardinality of A. Our consideration shows that directly indecomposable algebras are not the "general building blocks" in the study of universal algebra. Therefore, we define another kind of products.

Definition 1.2.7 Let (Aj)j,J be a family of algebras of type T.A subalgebra B c Aj of the direct product of the algebras Aj is

n

.I€ J

called a subdirect product of the algebras Aj, if for every projection mapping pk : Slj + Ak we have jeJ

n

pk(B)

=

Ak.

Examples for subdirect products are the diagonal AA = {(a, a ) I a E A) as well as any direct product. The lattice given by the Hasse diagram below

is the direct product of the two-element lattice C2 eb

and the three-element lattice C3 l

The sublattice C

C C2 x

3

C3 which is described by the diagram l

( 43 )

is obviously a subdirect product of C2 and Cj. It can easily be checked that for the subdirect product B of the family ( A j ) j Ethe J projection mappings

pk :

n

jE J

=

Aj Aj + Ak jE J

1.2 Direct and Subdirect Products satisfy the equation

n k e r ( p j I B)

15 =

AB. It turns out that this

jtJ

property of the kernels of the projection mappings can be used t o characterize subdirect products, in the sense that any set of congruences on an algebra with these properties can be used t o express the algebra as a subdirect product.

Theorem 1.2.8 Let A be a n algebra. Let { Q j I j E J } be a family of congruence relations o n A, which satisfy the equation n Q j = .I€ J

AA. T h e n A is isomorphic t o a subdirect product of the -algebras A/Bj,for j t J . I n particular, the mapping ~ ( a := ) ( [ a ] Q i ) j tdeJ fines a n embedding cp : A + ( A / Q j whose ), image p(A) is a

n

j€ J

subdirect product of the algebras A/Qj. We remark that the converse of this theorem is also true. If A is isomorphic t o a subdirect product of a family (Aj)j, of algebras, then there exists a family of congruence relations on A whose intersection is the relation AA. Algebras which cannot be expressed as a subdirect product of other smaller algebras, except in trivial ways, are called subdirectly irreducible.

Definition 1.2.9 An algebra A of type r is called subdirectly irreducible, if every family { Q j I j E J) of congruences on A, none of which is equal to AA, has an intersection which is different from AA. In this case, the conditions of Theorem 1.2.8 are not satisfied, and no representation of A as a subdirect product is possible. It is easy t o see that an algebra A is subdirectly irreducible if and only if AA has exactly one upper neighbour or cover in the lattice Con(A)of all congruence relations on A. Then the congruence lattice has the form shown in the diagram below.

1 Basic Concepts

16

Subdirect products do have the right property to act as the "general building blocks" in the study of universal algebra, as the following result of G. Birkhoff ( [ 8 ] )shows.

Theorem 1.2.10 Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

1.3

Term Algebras, Identities, Free Algebras

We need an appropriate language if we want to describe classes of algebras of the same type by logical expressions. This formal language is built up by variables from an n-element set X, = { x l , . . . , x,), n 1. The set X, is called an alphabet. We also need a set { fili E I ) of operation symbols, indexed by the set I. The sets X, and {fili E I ) have to be disjoint. To every operation symbol fi we assign a natural number ni I , called the arity of fi. The sequence r = (ni)iE1is called the type of the language. Now we define the terms of our type r , the "words" of our language.

>

>

>

Definition 1.3.1 Let n I. The n-ary terms of type r are defined in the following inductive way: (i) Every variable xi E X, is an n-ary term. (ii) If t l , . . . , tni are n-ary terms and fi is an ni-ary operation symbol, then fi(tl, . . . , tni) is an n-ary term. (iii) The set W,(X,) = W,(xl,. . . , x,) of all n-ary terms is the smallest set which contains xl, . . . , x, and is closed under finite application of (ii). It follows immediately from the definition that every n-ary term is also k-ary, for k > n. Our definition does not allow nullary terms. This could be changed by adding a fourth condition to the inductive definition, stipulating that every nullary operation symbol of our type is an n-ary term. We could also extend our language t o include a third set of symbols, to be used as constants or nullary terms.

17

1.3 Term Algebras, Identities, Free Algebras

Our definition of terms is inductive, based on the number op(t) of occurrences of operation symbols in a term. The operation symbol count op(t) is also inductively defined by (i) op(xi) := 0, if xi EX,, (ii) op(fi(t1,. . . ,t,%)):=

C

j=1

op(tj)

+ 1.

To determine op(t) is one of the methods t o measure the complexity of a term. Another common complexity measure is the depth of a term defined by the following steps: (i) depth(xi) := 0, if xi E X,, (ii) depth(fi(t1,. . . , t,,)) := max{depth(tl), . . . , depth(t,,)}

+ 1.

In a similar way the mindepth of a term can be defined if we replace the second step of the definition of depth by

+

mindepth(fi(tl,. . . , t,,)) := min{depth(tl), . . . , depth(t,%)} 1. Terms can be illustrated by tree diagrams, also called semantic trees. The semantic tree of the term t is defined as follows: (i) If t = xi, then the semantic tree of t consists only of one vertex which is labelled with xi, and this vertex is called the root of the tree. (ii) If t = fi(tl, . . . , t,%)then the semantic tree of t has as its root a vertex labelled with fi, and has ni edges which are incident with the vertex fi; each of these edges is incident with the root of the tree corresponding t o one of the terms tl, . . . , t,% (ordered by 1 2 ni, starting from the left).

< <

0

.

<

As an example we show the diagram of the term

18 where

1 Basic Concepts f2

is a binary and f l is a unary operation symbol. 21

23

22

21

22

In the diagram mindepth(t) is the length of the shortest path from the root to a leaf and depth(t) is the length of the longest path from the root t o a leaf. Other complexity measures are the length of the shortest path from the root to a leaf labelled by a variable zj and the length of the longest path from the root t o a leaf labelled by xj. These numbers depthj (t) and mindepthj(t) are inductively defined by (i) depthj(t) = mindepthj(t) := 0, if t in t ,

=

xi or x j does not occur

(ii) depthj (fi(t17. . . , k t ) ):= max{depthj (tl), . . . , depthj (tni)} 1. mindepthj (fi(t1, . . . , t,,)) := min{depthj (tl), . . . , depthj (t,,)) I if zj occurs in fi(tl, . . . , t,,).

+ +

We also use the following subset of W,(X,):

>

Definition 1.3.2 Let n 1 and let s : (1,. . . , n) + ( 1 , . . . , n ) be a mapping on { I , . . . , n). Then n-ary full terms of type r are defined as follows: (i) For every function s the term

f i ( ~ , ( ~. ). .,

, x,(,)) is full.

(ii) If tl, . . . , t,, are full terms, then fi(tl , . . . , t,,) is full.

1.3 Term Algebras, Identities, Free Algebras

19

If s is the identity mapping then the terms defined in this way are called strongly full and if s is a permutation, the terms are called permutationally full. By W;(X,), W,SF(Xn) and by WFF(Xn), respectively we denote the sets of all full, strongly full, and permutationally full n-ary terms of type r. Clearly,

w,S F (Xn) C w,PF(xn>C w,F(x,)

C W,(Xn).

Let r be a fixed type. Let X be the union of all the sets X, of variables, so X = {xl, 2 2 , . . .). We denote by W,(X) the set of all terms of type r over the countably infinite alphabet X :

This set W,(X) can be used as the universe of an algebra, of type r when for every i E I we define an ni-ary operation f , on W,(X), with

fi. : W,(X)n2 + W,(X)

with

( t l , . . . , tni) H f i ( t l , . . . , tn,).

Definition 1.3.3 The algebra F,(X) := (W,(X); is called the term algebra, or the absolutely free algebra, or the anarchic algebra of type r over the set X . The term algebra F,(X) is generated by the set X and has the socalled "absolute freeness" property, meaning that for every algebra A E Alg(r) and every mapping f : X + A, there exists a unique homomorphism f : F,(X) + A which extends the mapping f and such that f o y = f , where 9 : X + F,(X) is the embedding of X into F,(X). This can be illustrated by the following diagram:

It should be clear that the absolutely free algebras F,(X) and F,(Y) are isomorphic if X and Y have the same cardinality. Therefore, we can use different symbols for our variables. Instead of

20

1 Basic Concepts

x, y , . . . we may also use X I , 2 2 , . . . In the case of the finite alphabet X, we get the algebra FT(X,) of all n-ary terms of type r. We have seen that our operation symbols were used t o define the operations f , on W,(X,). There is another way to define operations on this set.

Definition 1.3.4 Let W,(X,) be the set of all n-ary terms of type 7 . Then the ( n 1)-ary superposition operation Sn (for terms) is inductively defined by the following steps:

+

(i) If x j E X, is a variable and tl, . . . , t, n; S n ( x j , t l , .. . ,t,) := tj, for 1 j

< <

E W, (X,), then

Instead of S n ( t ,tl, . . . , t,) we also write t(tl, . . . , t,). Selecting the variable terms xl, . . . , x, for the nullary operations, we form the algebra n

-

clone r := (W,(X,);

S n , x l , .. . ,x,),

which is an example of a unitary Menger algebra of rank n and will be considered later. An algebra with similar properties can be obtained if we define a superposition operation for n-ary operations on a set A. We consider the set O(A) of all finitary operations on A.

>

Definition 1.3.5 Let On(A),n 1, be the set of all n-ary operations defined on the set A. Then the (n + 1)-ary superposition operation (for operations) : On(A),+l + On(A) is defined by A A g f , . . . , g t ) ( a 1 , . . . ,a,) := f (g1 ( a , . . . 7 a,),. . . , g:(al,. . . , a,)) for every (al, . . . , a,) E An. Here g p , . . . , g,A , as well as f A are n-ary. This can be generalized t o an operation SnlA

SnlA(fA,

1.3 Term Algebras, Identities, Free Algebras

The set O(A) is closed under these operations. Particular operations on On(A) are the projections eyl", mapping each n-tuple of elements from A t o the i-th component, that is, with e:lA (al, . . . , an) := a,. By a clone of operations we want to understand a subset of O(A) which is closed under all operations S2A,m, n E W+ := N \ (0) and contains all projections. O(A) itself is called the full clone of operations defined on A. Instead of SzAwe will write for short s">A

Definition 1.3.6 Let C c O(A) be a set of operations on a set A. Then the clone generated b y C, denoted by (C), is the smallest subset of O(A) which contains C, is closed under composition, and contains all the projections e:"' : An + A for arbitrary n 1 and l
>

Terms are formal expressions on our formal language of type r. In order to formulate equations which are true or false in a given algebra A we have t o evaluate the variables in the terms forming the equations by elements of the concrete set A, and we have t o interpret the operation symbols by concrete operations on this set. This process corresponds to the formation of a rational function from a polynomial f (x) =

n

C aixi over a ring R if we substitute for

i=O x all elements of an overring S

> R.

Definition 1.3.7 Let A be an algebra of type r and let t be an n-ary term of type r over Xn. Then t induces an n-ary operation tA on A, called the term operation induced b y the term t on the algebra A, via the following steps: (i) If t

= xj E

X,, then t A

-

xjA

-

ejn,A

ttt

(ii) If t = fi(tl,. . . , tni) is an n-ary term of type 7 , and t f , . . . , are the term operations which are induced by tl, . . . , tni, then tA = s n i , A ( f A t k . . . , i f t ) .

We denote by w,(x,)* the set of all n-ary term operations of the algebra A, and by w,(x)* the set of all (finitary) term operations

22

1 Basic Concepts

ftli

= ({ E I)),that is, w,(x)~ on A. It is easy t o see that w,(x)~ is the clone generated by the fundamental operations of A. We also use the notation T(A) for this clone and we call T(A) the clone of term operations of A. Further we consider the set Tn(A) := On(A) n T(A) of all n-ary term operations on A. Several properties of the fundamental operations of an algebra A are also valid for term operations. For instance, for a homomorphism h : A + B and an arbitrary n-ary term t of the corresponding type we have: h(tA(al,.. . , a n ) ) = tB(h(al),. . . , h(an)).

Similarly, for a congruence relation 8 of A the implication from Definition 1.1.9 is valid for arbitrary n-ary term operations of A. The universes of subalgebras are not only preserved by all fundamental operations of A but also by all its term operations.

Definition 1.3.8 An equation s = t of terms s , t from W,(X) is said to be an identity in the algebra A of type r if sA = tA, that is, if the term operations induced by s and t on the algebra A are equal. In this case we also say that the equation s = t is satisfied or modelled by the algebra A, and we write A I= s = t. If the equation s = t is satisfied by every algebra A of a class K of algebras of the same type r, we write K s = t. Let I d A be the set of all identities satisfied in the algebra A. If K Alg(r) is a class of algebras, then I d K denotes the set of all equations which are identities in every algebra of K .

c

The equation sA = tA means that for every mapping f : X + A, we have f (s) = f ( t ) ,where f is the uniquely determined extension of f . Therefore s = t is an identity in A iff (s,t) t krr f for all mappings f : X + A. That is, the pair (s,t) must belong to the intersection of the kernels of all these mappings f . Thus an identity s = t holds in an algebra A (or in a class K of algebras), iff (s,t ) is in the intersection of the kernels of f , for every map f : X + A (for every algebra A in K ) . For a class K of algebras of type r the set I d K of all identities satisfied in every algebra from K can be regarded as a binary relation on the set W,(X). But even we have:

Proposition 1.3.9 Let K C Alg(r) be a class of algebras of type r and let I d K be the set of all identities satisfied in each algebra

1.3 Term Algebras, Identities, Free Algebras

A

23

E K . Then I d K is a congruence relation on the absolutely free

algebra FT( X ) of type r . Proof: Clearly, I d K is at least an equivalence relation on W T ( X ) . For the congruence property, suppose that sl = t l ,. . . , sna = tna are identities satisfied in K . This means that sf = t f , . . . , stt = for every algebra A from K . Let f i be an ni-ary operation symbol. A A t A , . . . ,t f t ) . Then our assumption means that f,A ( sA, , . . . , sni) - fi ( By the inductive step of the definition of a term operation induced by a term, this means that [ f i ( s l ., . . , sni)lA= [ f i ( t l.,. . , tn,)lA,and the definition of satisfaction gives f i ( s l , . . . , sn,) = f i ( t l , . . . , tni)E I d A . Thus f i ( s l , . . . , snt) = f i ( t l , . . . , tnt)E I d K , as required for a congruence.

ttt

,

The congruence relation I d K has one more interesting property, it is fully invariant (see Definition 1.1.15).

Theorem 1.3.10 For every class K of algebras of type 7 the set I d K of all identities satisfied in every algebra of K is a fully invariant congruence relation on the absolutely free algebra F T ( X ) . Proof: We have t o show that I d K is preserved by an arbitrary endomorphism y of F T ( X ) . So we take ( s , t ) E I d K =: C, and show that ( p ( s ) p, ( t ) ) is also in C. For this we use the property, that C = I d K is equal to the intersection of the kernels of the homomorphisms f , for all maps f : X i A and all algebras A in K . But for any such A and map f , the map f o p is also a homomorphism from F T ( X )into A, and is the extension of some map g from X into A. Thus our pair ( s ,t ) from C must also be in the kernel of this new homomorphism f o y. This means precisely that the pair ( y ( s ), p ( t ) ) must be in kes- f . Since this is true for all algebras A and maps f : X i A, we have ( p ( s ) p, ( t ) ) in C . This shows that C is a fully invariant congruence. Definition 1.3.11 Let X be a non-empty set of variables, and K be a class of algebras of type r. The algebra F K ( X ) := F T ( X ) / I d K is called the K-free algebra over X or the free algebra relative to K generated by X = X / I d K . Since F K ( Y )is generated by the set Y = Y / I ~ = K {[Y]ldKIY E Y}, instead of F K ( Y )we should actually write F K ( Y ) ;but for nota-

24

1 Basic Concepts

tional convenience this is not usually done. The algebra F K ( X )also satisfies a relative "freeness" property corresponding t o the absolutely freeness property:

Theorem 1.3.12 For any algebra A E K C Alg(r) and every mapping f : X + A , there exists a unique homomorphism f : F K ( X )+ A which extends f .

1.4

The Galois Connection (Id, Mod)

The satisfaction of an equation by an algebra gives us a fundamental relation between the two sets Alg(7) and W , ( X ) 2 . For any subset C W , ( X ) 2 and any subclass K A l g ( r ) we consider

c

c

ModC IdK

= =

{ AE A l g ( r ) I 'ds = t

E

C , ( Ak s

= t ) ) and

{ s = t E M ~ , ( X I )'dA ~ E K , ( Ak s

= t)).

Then the pair ( I d ,Mod) where Id maps from the power set of W , ( X ) 2 into the power set of Alg(7) and Mod maps conversely from the power set of Alg(7) into the power set of W , ( X ) 2 sa'tisfies the following conditions.

Theorem 1.4.1 (i) For all subsets C and C' of W , ( X ) x W , ( X ) , and for all subclasses K and K' of A l g ( r ) , we have C 5 C1+ ModC ModC1 and K C K' + I d K IdK'.

>

>

(ii) For all subsets C of W , ( X ) x W , ( X ) and all subclasses K of

A l g ( r ) , we have C

c IdModC

and K

c ModIdK.

(iii) The maps IdMod and ModId are closure operators on W , ( X ) x W , ( X ) and on Alg(7), respectively. (iv) The sets closed under ModId are exactly the sets of the form ModC, for some C C W , ( X ) x W , ( X ) , and the sets closed under IdMod are exactly the sets of the form I d K , for some K C Alg(r).

1.4 The Galois Connection (Id, Mod)

25

A pair of mappings which satisfies (i) and (ii) is called a Galois connection.

Definition 1.4.2 A class K C Alg(7) is called an equational class, or is said to be equationally definable, if there is a set C of equations such that K = ModC. A set C C W,(X) x W,(X) is called an equational theory if there is a class K C Alg(r) such that C = I d K . (iv), we conclude that the equational classes From Theorem 1.4.1 are exactly the fixed points with respect t o the closure operator ModId, and dually, the equational theories are exactly the closed sets, or fixed points, with respect to the closure operator IdMod. The collections of such closed sets form complete lattices.

Theorem 1.4.3 The collection of all equational classes of type T forms a complete lattice C(T), and the collection of all equational theories of type 7 forms a complete lattice &(7). These lattices are dually isomorphic: there exists a bijection 9 : C(7) + &(7),satisfying ~ ( KvIK2) = cp(K1)A cp(K2) and cp(K1A K2) = cp(K1) v cp(K2). The proof of this theorem can be found in [37].The Galois connection (Id, Mod) gives us an equational description of classes of algebras. Introducing operators H, S , P, which correspond to the algebraic constructions of homomorphic images, subalgebras and direct products, we find an equivalent algebraic approach.

Definition 1.4.4 We define for any class K C Alg(7) the following operators on the class Alg(r) of all algebras of type T. S ( K ) is the class of all subalgebras of algebras from K , H ( K ) is the class of all homomorphic images of algebras from K , P ( K ) is the class of all direct products of families of algebras from K, I ( K ) is the class of all algebras which are isomorphic t o algebras from K , P s ( K ) is the class of all subdirect products of families of algebras from K .

26

1 Basic Concepts

Definition 1.4.5 A class K C Alg(r) is called a variety if K is closed under the operators H, S and P; that is, if H ( K ) C K , S(K) K andP(K) C K. For any class K of algebras of type r , the class V(K):=HSP(K) is the least variety which contains K . If K = {A) consists only of one algebra we write V(A) := HSP({A)). With these definitions we can formulate the main theorem of equational theory, also called Birkhofl's Theorem. T h e o r e m 1.4.6 (Main Theorem of Equational Theory) A class K of algebras of type r is equationally definable if and only if it is a variety. (For a proof see e.g. [lo].) There are two important ways to describe completely all members of a variety. The first way uses all subdirectly irreducible algebras of the variety and the fact that every algebra of a variety V is isomorphic t o a subdirect product of subdirectly irreducible algebras from V. The second way uses all relatively free algebras F v ( X n ) for arbitrary natural numbers n 1 and the fact that

>

V

= HSP({Fv

(X,)

I n E N,n > I)).

Here the question arises whether the free algebras F v ( X n ) are finite. A variety V is said to be locally finite if every finitely generated algebra in V is finite. This is the case iff all free algebras F v ( X n ) are finite. We mentioned already that all equational classes of algebras of the same type r form a complete lattice. Because of Theorem 1.4.6 we can also speak of the lattice C ( r ) of all varieties of type r . (These lattices are in fact algebraic.) The greatest element of the lattice C ( r ) is the variety Alg(7) consisting of all algebras of type 7. Clearly, Alg(r) = Mod{x = x). The least element in the lattice C ( r ) is the trivial variety T consisting exactly of all one-element algebras of type r; we have T = Mod{x = y}. Dually, the greatest element in the lattice I ( r )of all equational theories of type r is the

1.4 The Galois Connection (Id, Mod)

27

equational theory generated by {x = y}, and the least element in I ( r ) is the equational theory generated by {x = x}. Clearly, the first one consists of all equations of type r .

A subclass W of a variety V which is also a variety is called a subvariety of V. The variety V is a minimal, or equationally complete variety if V is not trivial but the only subvariety of V not equal t o V is the trivial variety. It can be shown that every nontrivial V E C(7) contains a minimal subvariety (see e.g. [37]). If V is a given variety of type r, then the collection of all subvarieties of V forms a complete lattice C(V) := {WI W E C ( r ) and W V). This lattice is called the subvariety lattice of V. One of the aims of this book is to present new methods t o study the lattice L(r) or lattices of the form L(V).

c

Equational theories were defined as sets C of equations satisfying IdModC = C. From Theorem 1.4.l(iv), we obtain that C is an equational theory of type r iff there is a class K of algebras of type r with C = I d K . Sets of equations of the form I d K are fully invariant congruence relations on the absolutely free algebra F T ( X ) of type r (Theorem 1.3.10). The properties of a fully invariant congruence relation define the algebraic consequence relation, a formal way t o derive from a given set of equations satisfied as identities in a class of algebras new equations which are also satisfied as identities. We write C t s = t , read as "C yields s = t," if there is a formal deduction of s = t starting with identities in C, using the following five rules of consequence or derivation or deduction rules:

(4) {tj = ti 1 1 5 j 5 ni) t fi(tl, . . . , tn,) = fi(t;, . . . , t i i ) , for every operation symbol fi ( i E I) (the replacement rule).

28

I Basic Concepts

(5) Let s, t , r E W,(X) and let 5, t" be the terms obtained from s, t by replacing every occurrence of a given variable x E X by r. Then {s = t} t 5 = t". (This is called the substitution rule.) The rules ( I ) - (5) reflect the properties of a fully invariant congruence relation: the first three are the properties of an equivalence relation, the fourth describes the congruence property and the fifth the fully invariant property. Equational theories C are precisely sets of equations which are closed with respect t o finite application of the rules ( I ) - (5).

+

If we write C s = t for a set C C W,(X)2 and an equation s = t t W,(X)2 we mean that s = t is satisfied as an identity in every algebra A of type T in which all equations from C are satisfied as identities. The connection between these approaches is given by the Completeness and Consistency Theorem of equational logic: For C C W, (X) x W, ( X ) and s = t t W, ( X ) x W, ( X ) , we have

The "+"-direction of this theorem is usually called completeness, since it means that any equation which is "true" is derivable; the "en -direction is called consistency, in the sense that every derivable equation is "true".

2 Closure Operators and Lattices The collection of all varieties of a given type forms a complete lattice. These lattices play an important role in Universal Algebra and in applications, but their study is difficult. Thus we look for new approaches or tools to use in their study. A useful method is t o try t o study some smaller parts of the large and complex lattice. Such smaller parts should have the same algebraic structure, so we are interested in the study of complete sublattices of a complete lattice.

2.1

Closure Operators and Kernel Operators

In the previous chapter we have seen two examples of operators with closure properties. The generation of subalgebras of a given algebra A from subsets X of the carrier set A of the algebra A defines an operator

(

)A :

?(A) +?(A)

on the power set of A, which is extensive, monotone and idempotent. The generation of congruence relations by a binary relation defines an operator

(

) C ~ :A

?(A2)

+

?(A2)

which also has these three properties. We define:

Definition 2.1.1 Let A be a set. A mapping C : ?(A) + ?(A) A the is called a closure operator on A, if for all subsets X, Y following properties are satisfied:

c

(ii) X

c Y + C(X) c C(Y)

(monotonicity) ,

30

2 Closure Operators and Lattices

(iii) C ( X ) = C ( C ( X ) )

(idempotency).

Subsets of A of the form C ( X ) are called closed (with respect to the operator C ) and C(X) is said to be the closed set generated by X. An operator K : P ( A ) + P ( A ) is said t o be a kernel operator on A, if it is monotone and idempotent and if instead of (i) the condition (i') X

(intensivity)

2 K(X)

is satisfied. Closure and kernel operators are closely related t o complete lattices. Indeed, if C : P ( A ) + P ( A ) is a closure operator, then the set Cc := {X I X

c A and C(X) = X )

(1)

is a complete lattice with respect to the operations

defined by

for arbitrary sets X , Y E Cc since X n Y E LC whenever X, Y E LC and

v : LC x LC + LC defined by

Actually,

A X j and V Xj for arbitrary families (Xj)j,J of elejEJ

jEJ

ments from LC belong t o LC. Therefore, every closure opera,tor C on A defines a subset of ?(A) which is a complete lattice. Conversely, if C & P ( A ) is a complete lattice, then by

for every Y & A a closure operator CL : P ( A ) + P ( A ) is defined. The closed sets with respect to CL are exactly the elements of C. There is a one-to-one correspondence between complete lattices in P ( A ) and closure operators on A. Altogether we have the following well-known theorem.

31

2.2 Complete Sublattices of a Complete Lattice

Theorem 2.1.2 Let C be a complete lattice in P ( A ) . Let C be a closure operator o n the set A. T h e n Cc defined by ( 1 ) is a complete lattice where the operations are defined by ( 2 ) and (3) and CL, defined by (4) is a closure operator o n A. Moreover, the closed sets with respect t o CL are exactly the elements of C and the elements of Cc are exactly the closed sets of C . B o t h equations

CL,

=

C and LC,

=C

are satisfied. There is a similar connection between kernel operators on A and complete lattices in P ( A ) . In this case instead of (4) we have to define

KL(Y) :=

V { Ht C I H c Y).

(4')

The image CL(Y) is defined as the infimum (with respect to C ) of all elements in C containing Y and KL(Y) is the supremum of all elements from C which are contained in Y. The partially ordered set (?(A); c ) itself is a complete lattice with respect to intersection and union of arbitrary families of subsets of A. But as the examples Sub(A) and Con(A) show, the complete lattices Cc are in general not complete sublattices of ( P ( A ) ;C ) since the join operation is not the set-theoretical union. In the next section we want to give a condition which characterizes complete sublattices of a complete lattice.

2.2

Complete Sublattices of a Complete Lattice

In this section we will describe a method to produce complete sublattices of a given complete lattice. We will do so by consideration of the fixed points of a certain kind of closure operators defined on the complete lattice. In 2.1 we showed that the fixed points of a closure operator C : ?(A) + P ( A ) on the power set lattice form a complete lattice in ?(A) and that conversely every complete lattice in P ( A ) defines a closure operator on A. This can be generalized t o arbitrary complete lattices C instead of P ( A ) . For any complete lattice C and

32

2 Closure Operators and Lattices

any closure operator p : C

+ C7we

get the complete lattice

S, := F i x (p) := {T I cp (T) = T} of all fixed points of p; and for any complete lattice S the closure operator ps defined by

C C we have

L

cps(T) :=

A{T' t s I T < T') for T t C. C

<

C

(Here denotes the partial order of C7A is the infimum and V the supremum with respect to 5.) Moreover, for any complete lattice S in C and any closure operator p, we have S,, = S and ps,, = p. The set S, is also called a closure system. There is also a dual 1-1 correspondence between kernel operators and complete lattices in C. For a kernel operator $ : C + C and a complete lattice C, we set

S+

:=

{T E C I $(T) = T) ; L

$s(T) := V{T' E S I T' < T) for T E C. Then we have Fix($) = S$ and $sy= $, for any complete lattice S and any kernel operator $ on C. These results date back t o A. Tarski ([97]). Theorem 2.2.1 Let C be a complete lattice. (i) If p is a closure operator o n C which satisfies

for every index set J , then the set of all fixed points under p,

S,

=

{T E C I p(T) = T),

is a complete sublattice of C and cp(C) = {p(T) I T E C} s,.

=

(ii) Conversely, if S is a complete sublattice of C, then the function cps which is defined by

is a closure operator o n C with ps(C) = S , and ps satisfies condition (*). Moreover, S,, = S and cps,, = p .

2.2 Complete Sublattices of a Complete Lattice (iii) If $ is a kernel operator o n C which satisfies

for every index set J , then the set of all fixed points under $,

is a complete sublattice of C and $(C)

= S$.

(iv) Conversely, if S is a complete sublattice of C then the function which is defined by L

$s(T) :=

V{T' t S 1 T' 5 T} for T t C,

is a kernel operator on C with $S(C) = S, and $ satisfies the condition (**). Moreover, S+, = S and qs, = $. Proof: (i) Let cp be a closure operator on C which satisfies the condition (*). We have t o prove that the set of all fixed points under cp is a complete sublattice of C, that is, that for any index set J. both

L

It is clear that p(A{Tj t S, each j t J we have T, L

from this we obtain

=

Ij

E J))

t J ) . For

> cp(l\{T, E S, I j E J)),and I j E J) > cp(A{T, t S, I j t J ) ) .

cp(Tj)

A{T, E S,

> i { T , E S, 1 j L

L

L

Altogether this gives equality, and A{Tj E S, I j E J) E S,. L

The fact that cp satisfies the join condition (*) gives V{Tj E S, I j t J) t S,. Thus we have a sublattice of C. Clearly, S, cp(C). Since cp is idempotent, cp(T) is in S, for all T E C. This shows that cpK>= s,.

c

(ii) It is easy t o see that cps is a closure operator. We need only show that this closure operator satisfies condition (*) and that cps(C) = S . We prove the latter fact first. Since S is a complete sublattice of

34

2 Closure Operators and Lattices

L, we have cps(L) C S. For the opposite inclusion, we see that for any T E S C

V ~ ( T )=

~I\{T't s I T 5 TI} = T.

Thus S C cps (L), and altogether we have S = cps (L). Since for each j t J we have T, cps(Tj) and cps(T,) E S, the set

<

is an upper bound of the set {Tj I j E J}. Therefore

The right hand side is an element of S . By application of cps on both sides we get

C

C

> T',L we have cps(V{Tj I j E J)) > cps(T,) for C E J . Thus also cps(V{T, I j E J)) > V{cps(Tj) I j E J), giving

Since V{Tj I j E J)

all j the required equality.

(iii) and (iv) can be proved similar to (i) and (ii). The equations are also easy to realize.

2.3

Galois Connections and Complete Lattices

In chapter I we considered the Galois connection (Id, Mod) between identities and model classes. We generalize this example and consider Galois connections as pairs of mappings with special properties between the power sets of two sets.

Definition 2.3.1 A Galois connection between the sets A and B is a pair (p, L ) of mappings between the power sets P ( A ) and P(B), p : P ( A ) + P ( B ) and

L

: P ( B ) + P(A),

2.3 Galois Connections and Complete Lattices such that for all T, T' tions are satisfied:

35

C A and all S, St C B the following condi-

(i) T c T t + - p ( T ) 2 p ( T t ) (ii) T C L ~ ( T ) and

S C S t + - ~ ( S ) 2L(S');

and

S

~L(S).

Galois connections are also related t o closure operators, as the following proposition shows.

Theorem 2.3.2 Let the pair (p, L ) with p : P ( A ) + P ( B ) and

L

: P(B)

+ P(A)

be a Galois connection between the sets A and B . Then: (i) p ~ = p p and

L ~ = L L:

(ii) ~p and p~ are closure operators o n A and B respectively; (iii) The sets closed under ~p are precisely the sets of the form L(S), for some S C B; the sets closed under p~ are precisely the sets of the form p(T), for some T C A.

Proof: (i) Let T A. By the second Galois connection property, we have T C L ~ ( T By ) . the first property, applying p t o this, gives . we also have p(T) C p ~ ( p ( T ) )by , the second p ( T ) p ~ p ( T )But Galois connection property applied to the set p(T). This gives us p ~ p ( T= ) p(T). The second claim can be proved similarly. (ii) The extensivity of ~p and p~ follows from the second Galois connection property. From the first property we see that

>

since p(T) and p(Tt) are subsets of B; and in the analogous way we get from S C St the inclusion ~ L ( S C )p ( S t ) . Applying p t o the equation L ~ = L L from (i) gives us the idempotency of p ~ and , similarly for ~ p . (iii) This is straightforward to verify. Galois connections are " induced " by relations. A relation between the sets A and B is simply a subset of A x B. Any relation R

36

2 Closure Operators and Lattices

between A and B induces a Galois connection, as follows. We can define the mappings

) a Galois connection. Indeed, We verify that the pair (pR,L ~ is if T C TI and if y E pR(T1),that is, if for all x E TI we have (z,y) E R, then also for all z E T we have (x, y) E R and this means, y E pR(T). This shows that pR(T1) C pR(T). Similarly one proves the second condition from Definition 2.3.1 (i). Consider L R I U R ( ~= ) {X E A I VY E P R ( ~ ) ( ( X , Y ) E R ) ) = {x E A I 'dy E {z E B I 'dx E T ( ( x , z ) E R ) ) ( ( x , y ) E R)}. It is easy to see that T C L R ~ R ( T and ) S ~RLR(S).

If conversely (p, L ) is a Galois connection between A and B then we construct the relation R(p,L) :=

U {T X p ( T ) ) C 4 X B TLA

and the Galois connection (pn, p , L ) , LR(,,,,, ) defined by P R ( ~ ,(T) ~ ) :=

{Y E

1 'dx E T((x7 Y)

E R(~,L))}

and

(S) :=

L~(p,L)

{X

E

A 1 VY E S((X, Y)

E R(~,L))).

It turns out that (pn(,,,,,, Ln(p,,))= (p, L).Therefore, there is a oneto-one correspondence between relations between A and B and Galois connections between A and B. We need also the following well-known proposition (see e.g. [37], exercise 2.4.2):

Lemma 2.3.3 Let R C A x B be a relation between the sets A and B and let (p, L ) be the Galois connection between A and B induced by R. T h e n for any families {T, C A I j E J) and {Sj B I j E J), the following equalities hold:

2.4 Galois Closed Subrelations

37

Galois Closed Subrelations

2.4

In section 2.2 we developed a method to produce complete sublattices of a given complete lattice. If R C A x B is a relation between two sets A and B and if (p, L) is the Galois connection induced by R, then the fixed points of the closure operators p~ and ~p form two complete lattices which are dually isomorphic. Now we consider a subrelation R' of the initial relation R , from which we obtain a new Galois connection and two new complete lattices. We describe a property of the subrelation R' which is sufficient t o guarantee that the new complete lattices will be complete sublattices of the original lattices. This property is called the Galois closed subrelation property. Moreover, we show that any complete sublattices of our original lattices arise in this way.

Definition 2.4.1 Let R and R' be relations between sets A and B, and 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: 1) R'

cR

and

2) 'v'T C A, 'v'S C B and L (S) = T).

(pt(T) = S and L'(S)= T

+ p(T) = S

The following equivalent characterizations of Galois closed subrelations can easily be derived (see e.g. [45], [4])

c

Proposition 2.4.2 Let R' R be relations between sets A and B . T h e n the following are equivalent: (i) R' is a Galois closed subrelation of R; (ii) For any T

c A,

if L ' ~ ' ( T=) T then p(T) = pt(T), and for =)S then L(S) = L'(S); any S C B , if ~ ' L ' ( S

(iii) For all T C A and for all S C B the equations L ' ~ ' ( T= ) L pt(T) and ~ ' L ' ( S =)p L'(S)are satisfied.

Proof: (i) + (ii) Define S to be the set pt(T). Then from L ' ~ ' ( T=) T, i.e. L'(S)= T and p' (T) = S by the definition of a Galois closed subrelation we obtain p(T) = S and L(S)= T, i.e. p ( T ) = pt(T)

38

2 Closure Operators and Lattices

and L(S)= L'(S). The claim for subsets S of B can be proved similarly. (ii) + (iii) From p'~'p'(T) = pt(T) and L'~'L'(S) = L'(S)(Proposition 2.3.2) using (ii) we obtain ~ L ' ( S=) ~ ' L ' ( Sand ) L ~ ' ( T=) L'~'(T). (iii) + (i) If pt(T) = S and L'(S)= T, then there follows:

~ ' L ' ( S= ) pt(T), ~ L ' ( S=) p ( T ) ,

L ' ~ ' ( T )= L'(S) and L ~ ' ( T= ) L(S).

Then by condition (iii) we get p(T) = pt(T) = S and L(S)= L'(S)= T. This shows that R' is a Galois closed subrelation of R. Let 'Ft,, and F ' t, be the complete lattices induced by the Galois connection (p, L). We show that any Galois closed subrelation R' of the relation R yields a lattice of closed subsets of A which is a complete sublattice of the corresponding lattice E L , for R. Conversely, we also show that any complete sublattice of the lattice E L , occurs as the lattice of closed sets induced from some Galois closed subrelation of R. Dual results of course hold for the set B.

Theorem 2.4.3 Let R C A x B be a relation between sets A and B , with induced Galois connection (p, L).Let E,, be the corresponding lattice of closed subsets of A.

c A x B is a Galois closed subrelation of R, then the class URl := ' F t L 1 , t is a complete sublattice of 'Ft,.

(i) I f R'

(ii) If U is a complete sublattice of EL,, t h e n the relation

is a Galois closed subrelation of R. (iii) For a n y Galois closed subrelation R' of R and a n y complete ' t,, we have sublattice U of F

URzn= U and RuR, = R'

2.4 Galois Closed Subrelations

39

Proof: (i) We begin by verifying that any subset of A which is closed under the operator ~ ' p 'is also closed under ~ p so , that the lattice E,I,I is at least a subset of EL,.Indeed, if T E E,I,I,so that L ' ~ ' ( T=) T, then by Proposition 2.4.2, we have L

p (T) = L p' (T) = ~ ' p(T) ' =T

and therefore, E,I,I C EL,. Since by 2.1 the infimum in a complete lattice which is included in the power set lattice of a set is the set-theoretical intersection we have l\ {T, I j E J) = 0 Tj =

l\ {Tj I j

XLU

t

% ~ ! /

opkration of

J} and this means,

. I ~ J

is closed under the infimum

xL,.

Now we consider the join. By repeated use of Lemma 2.3.3 and Proposition 2.4.2, we see that V {T, I j E J) = LIP'( U Tj) = jEJ

%/LJ

j E J) and therefore K,I,I is closed under the supremum operation of EL,. (ii) Now let U be any complete sublattice of EL,.We consider the relation {T x p ( T ) 1 T E U), Ru :=

U

which we will prove is a Galois closed subrelation of R. First, for each non-empty T E U we have p(T) = {s E B I V t E T((t,s) E R)), so that T x p ( T ) R. Therefore Ru R. To show that the second condition of the definition of a Galois closed subrelation is met, we let (p', L') be the Galois connection between sets A and B induced by Ru, and assume that p' (T) = S and L'(S)= T for some T A and S B. Our goal is to prove that

c

c

c

c

p(T) = S and L(S)= T. Let T E U. By definition we have

This means that pf(T) is the greatest subset of B with T x pf(T) C Ru. Now from the definition of Ru we have T x p(T) C Ru.

2 Closure Operators and Lattices

40

Therefore, p ( T ) C pr(T). The opposite inclusion also holds since Ru C R. Altogether we have pf(T) = p(T). If pf(T) = S, then L(S)= L ~ ' ( T=)L ~ ( T=)T since T E U C EL,. (iii) Now we must show that for any complete sublattice U of 'FI,,, and any Galois closed subrelation R' of R, we have UR, = U and RuR, = R'. We know that UR, = EL"/,the lattice of subsets of A closed under the closure operator ~ ' p 'induced from the relation Ru. This means that T E URu iff L ' ~ ' ( T=) T. Let T E U. Then as before we have and this shows T E 'FI,I,I Therefore U c UR,. Now let T E URu and let S be the set pr(T). Then we have L'(S)= T, and since Ru is a Galois closed subrelation of R we conclude that p(T) = S and L(S)= T If T = 0 then T t U, and for T #

0 for each t

E T we define

Dt := n{T' t U I t t T' and S c p(Tf)).

U Dt. But now T = L ~ ( T= )L P (U Dt) = t€T U{Dt I t E T} E U. This shows the other direction URu C U.

We can show that T =

t€T

Let R' be a Galois closed subrelation of R, and set := 'FIL~,t= {T C A I L ' ~ ' ( T =) T), and U x p(T) 1 T E UR/}. RuR, :=

U{T

We will show that Ru,, = R'. First, if (t, s) E R' then s E pf({t)). Setting S := pr({t)), we have s E S and L'(S)= ~'p'({t)).NOWtaking T := ~'p'({t)), we have L ' ~ ' ( T=) T, SO T t URt, p' (T) = S and L'(S)= T. Therefore p(T) = S and L(S)= T. Since t t ~'p'({t)) = T and s E S = p ( T ) , we get (t, s) E T x p ( T ) and T E URl. Hence (t, s) E RuR,, and we have shown that R' C RuR,. To show the opposite inclusion, let T t URI, and let S Then we can prove

=

p(T).

p' (T) = p(T) = S and L'(S)= L(S)= L p(T) = T. Therefore T x p(T) c R', and Ru,, c R'. Altogether, we have RuR, = R'. This completes the proof of (iii), and of our theorem.

2.5 Conjugate Pairs of Additive Closure Operators

2.5

41

Conjugate Pairs of Additive Closure Operators

In section 1 we saw that any closure operator y defined on a set A gives us a complete lattice, the lattice IFt, of all y-closed subsets of A. In this lattice, the meet operation is the operation of intersection. The join operation however is not usually just the union; we have

for every B C IFt,. One situation when we do have the join operation equal t o union is the following.

Definition 2.5.1 A closure operator y defined on a set A is said to be completely additive if y(T) = U ?(a) for all T C A. (Here atT

we write y(a) for y ({a)).) We can show easily that when y is an additive closure operator, the least upper bound operation on the lattice IFt, agrees with U B (see D. Dikranjan and E. Giuli, [41] or M. Reichel, [91]). Indeed, we always have U B C y(U B) because of the extensivity of y. Conversely, if a E U B then a E B for some set B E B, and since B E 'Ft, wehave ?(a) C U B a n d y ( U B ) = U y(a) U UB=UB.

UEUB

c

UEUB

This means that U B is y-closed and V B = U B. In other words, when y is an completely additive closure operator on A, the corresponding closure system forms a complete sublattice of the lattice ( P ( A ) ;0, of all subsets of A. Now we study a situation where we have two completely additive closure operators which are closely connected. These results can be found also in [36] and in [19], respectively.

u)

Definition 2.5.2 Let yl be a closure operator defined on the set A and let 7 2 be a closure operator defined on the set B. Let R C A x B be a relation between A and B. Then yl and 7 2 are called conjugate with respect t o R if for all t E A and all s E B we have y ~ ( t x) {s) c R iff {t) x y,(s) c R. When the two operators are completely additive, we can extend this definition given in terms of individual elements t o sets of elements. Thus when (yl, 7,) is a pair of additive closure operators,

42

2 Closure Operators and Lattices

yl on A and 7 2 on B, and they are conjugate with respect t o a relation R C A x B , then for all X C A and all Y C B we have X x y2(Y) C R if and only if y l ( X ) x Y C R. Examples of conjugate pairs of additive closure operators will be given in the next chapter. In this section we develop the general theory of such operators. We assume that we have two sets A and B , and that R is a relation between A and B. This relation induces a Galois connection (p, L) between A and B , for which the two maps p~ and L,U are closure operators. Moreover, the pair ( p ~~, p is) always conjugate with respect t o the original relation R. But p~ and L,LL need not be completely additive in general.

Definition 2.5.3 Let y := (yl, y2) be a conjugate pair of completely additive closure operators, with respect to a relation R C A x B. Let R, be the following relation between A and B: R, := {(t,s) E A x B 1 yl(t) x {s)

C R).

Now we have two relations and Galois connections between A and B. The relation R induces a Galois connection (p, L ) between A and B and the new relation R, induces a second Galois connection, which we shall denote by (p,, L,). The following theorem gives some properties relating the two Galois connections.

Theorem 2.5.4 Let y = (yl, y2) be a conjugate pair of completely additive closure operators with respect t o R C A x B . T h e n for all T C A and S C B , the following properties hold:

2.5 Conjugate Pairs of Additive Closure Operators

Proof: We will prove only (i)-(v), the proofs of the other propositions are dual. (i) By definition, E R,)} P,(T) = {b E B I 'da E T ((0) = {b E B 1 'da E T (yl(a) x {b) R)) = {bEBI'daEyl(T) ((ad) ER)) =p(y1(T)). (ii) Since yl is a closure operator, we have T yl(T); and thus, since p reverses inclusions, p(T) p(yl(T)). Using (i) we obtain P,(T) C 4 9 .

c

c

>

(iii) Extensivity of y2 implies p,(T) C y2(p,(T)). Now let S C p, (T). Then for all s E S and for all t E T, (t, s) E R,, and by definition of R, we get {t} x y2(s) C R. Idempotency of y2 gives {t} x y2(y2(s)) R and thus y2(s) C p,(T) for all s E S. By additivity of y2 we get y2(S) = U y2(s) p,(T); and taking SES

S

p,(T) we obtain y2(p,(T)) equality P, (TI = 7 2 (P, (T)). =

C p,

c

(T). Altogether we have the

The next theorem gives for sets T C A which are closed under ~ p , ) four equivalent characterizations. i.e. with L ( ~ ( T=) T

Theorem 2.5.5 (Main Theorem for Conjugate Pairs of Additive Closure Operators) Let R be a relation between sets A and B, with corresponding Galois connection (p,L). Let y = (yl, y2) be a conjugate pair of completely additive closure operators with respect to the ) the following relation R . Then for all sets T C A with L ( ~ ( T=) T

44

2 Closure Operators and Lattices

propositions (i) - (iv) are equivalent; and dually, for all sets S C B with ~ ( L ( S=) )S , propositions (i') - (iv') are equivalent:

(iii7) L (S)= L~ (S),

Proof: We prove the equivalence of (i), (ii), (iii) and (iv); the equivalence of the four dual statements can be proved dually. (i) + (ii) We always have T C yl(T), since yl is a closure operator. Since ~p is a closure operator we also have yl(T) C ~ p ( y ~ ( T=) ) Lr(pr(T)) = T, by Theorem 2.5.4 (v'). (ii) + (iii) We have p(T) 2.5.4 (i).

=

p(yl(T)) = p,(T) by (ii) and Theorem

(iii) + (iv) We have 7 2 (p(T)) = 7 2 (p, (T))= p, (T),using Theorem 2.5.4 (iii). (iv) + (i) Since the ~,p,-closed sets are exactly the sets of the form

L ~ ( Swe ) , have to find a set S C B with T = L ~ ( S )But . we ham ) L ( ~ ( T= ) )T, by Theorem 2.5.4 (i7)and ly(p(T)) = ~ ( 7 2 ( p ( T ) )= our assumption that T is ~p-closed. Before using this Main Theorem t o produce our complete sublattices, we need the following additional properties.

Theorem 2.5.6 Let R be a relation between sets A and B , with Galois connection (p, L). Let y = (yl,y2) be a conjugate pair of

2.5 Conjugate Pairs of Additive Closure Operators

45

completely additive closure operators with respect to R. T h e n for all sets T C A and S C B , the following properties hold:

Proof: We prove only (i7)and (ii'); the others are dual. ). p~ is a closure operator (i') Suppose that y2(S) C ~ ( L ( S ) Since 2 P(L(YZ(S))) = P,(L,(S)), by Our we have P(@)) = P(L(P(L(S)))) assumption and by Theorem 2.5.4 (v). Also S y2(S), and hence ) ) p,(~,(s)), ) again by Theorem 2.5.4 we have ~ ( L ( SC)p)( ~ ( y ~ ( S = (v). For the converse we have y2(S) C p ( ~ ( y ~ ( S )=) )p,(~,(s)) = ~ ( L ( Susing ) ) , the extensivity of p ~ Theorem , 2.5.4 (v) and our assumption.

c

c

c

). S y2(S) implies y 2 ( p ( ~ ( S ) )C) (ii') Let y2(S) ~ ( L ( S ) Then Y ~ ( P ( L ( Y ~ ( S )We ) ) ) .also have 7 2 ( ~ ( ~ ( 7 2 ( S ) ) ) )= ~~(P(L,(S))) by , by Theorem Theorem 2.5.4 (i'), and y 2 ( p ( ~ ( y 2 ( S ) )= ) ) p ( ~ (S)) ) )p ( ~ ( y ~ ( S )C) )p ( ~ ( p ( ~ ( s ) = ))) 2.5.4 (iv'). In addition, ~ ( L , ( S= ~ ( L ( S )Altogether ). we obtain y 2 ( p ( ~ ( S ) )C) ~ ( L ( S )The ) . opposite inclusion is always true, since y2 is a closure operator. Conversely, S ,u(L(S))implies y2(S) C y 2 ( p ( ~ ( S ) )=) ~ ( L ( Sby ) ) the , extensivity of p ~the , monotonicity of 7 2 and our assumption.

c

Now we are ready t o produce our complete sublattices. We know that from the original relation R and Galois connection (p, L ) we have two (dually isomorphic) complete lattices of closed sets, the lattices F ' I, and F ' I,. We also get two complete lattices of closed sets from the new Galois connection (p,, L,) induced by R,. Our result is that each new complete lattice is in fact a complete sublattice of the corresponding original complete lattice.

Theorem 2.5.7 Let R be a relation between A and B , with induced Galois connection (p, L). Let y = (yl, y2) be a conjugate pair of completely additive closure operators with respect to R. T h e n the

46

2 Closure Operators and Lattices

lattice F ' ,I of sets closed under p,~, is a complete sublattice of ' I,, and dually the lattice F ' ,I is a complete sublattice the lattice F of the lattice EL,. Proof: As a closure system F ' I, is a complete lattice, and we have to prove that it is a complete sublattice of the complete lattice F ' I,. We begin by showing that it is a subset. Let S E F ' I,, so that P&,(S)) = S . Then ~ ( 4 w = P ( ~ & ( S ) ) ) )= P ( ~ ~ L ( Y ~ ( S ) ) ) = ~(L(Y~( =S,)U ) )~ ( L ~=( S )by ) Theorem 2.5.4 (v), and thus SEF ' I,. This shows F',t F ' t,. Since every S in F',t satisfies ~ ( L ( S=)S, ) we can apply Theorem 2.5.5, to get

c

As we remarked after Definition 2.5.1, the fact that 7 2 is an additive closure operator means that the corresponding closure system is a complete sublattice of the lattice (P(B);n, U) of all subsets of B; that is, on our lattice F ',t the meet operation agrees with ordinary set-intersection and the join agrees with union. We already know that the meet operation in F ' I, also agrees with intersection, so we only need t o show that F ' I, is closed under the join operation of F ' I,. Let ( S k ) k E J be an indexed family of subsets of B. Then

by Theorem 2.5.4 (iv'); and then using Lemma 2.3.3 we have L(

u sk) n L ( s ~n) =

k€J

=

k€J

k€J

L y ( ~ k )=

usk).

k€J

Thus conjugate pairs of additive closure operators give us a way t o construct complete sublattices of a given closure lattice. We may also define an order relation on the set of all conjugate pairs of additive closure operators: for a = (yl, 7:) and P = (y2,7;) we set

<

When a P, it can be shown that the lattice F ' I, is a sublattice of F ' I, and dually that F ' I, is a sublattice of 'FI,.,,,

2.5 Conjugate Pairs of Additive Closure Operators

The following additional properties may also be verified:

Sr. Arworn in [2] has generalized the theory of conjugate pairs of additive closure operators t o the situation of conjugate pairs of extensive, additive operators. An alternative proof of Theorem 2.5.7 can be given by showing that R, is a Galois closed subrelation of R and then by using of Theorem 2.4.3.

Proposition 2.5.8 Let R C A x B be a relation and let ( p ~ be ) the Galois connection induced by R. Let y := (yl, y2) be a conjugate pair of completely additive closure operators defined o n A and o n B , repectively. Let R, be the relation defined in Definition 2.5.3 and assume that (p,, L,) is the Galois connection induced by R,. T h e n R, is a Galois closed subrelation of R. Proof: By Definition 2.5.3 the relation R, is a subrelation of R. We apply Proposition 2.4.2 (iii). Let T c A and S c B. Then by Theorem 2.5.4 (iii) we have

In a similar way by Theorem 2.5.4 (iii') and (i) we have

This proves that R, is a Galois closed subrelation of R.

3 M-Hyperidentities and M-solid Varieties In this chapter we apply the theory of conjugate pairs of completely additive closure operators t o the lattice of all varieties of a given type. We assume a fixed type r of operation symbols, and consider the sets A = W T ( X ) 2of all identities (pairs of terms) of type r and B = Alg(r) of all algebras of type r . Between these two sets we have the basic relation R of satisfaction: that is, a pair (s = t , A) belongs t o R iff the algebra A satisfies the identity s = t. This relation induces the Galois connection (Id, Mod) between sets of identities and classes of algebras. The closed sets on one side are the equational classes or varieties, and we have the complete lattice C ( r ) of all varieties of type r . Dually, the closed sets on the identity side are the equational theories, and we have the complete lattice &(7)of all equational theories of type r . These two lattices are dually isomorphic, and in general are very large C(7) and &(r) and complex. Thus it is important t o find some means of studying smaller portions of these lattices, such as complete sublattices. Our goal is to introduce two new closure operators on our sets A and B, which we shall show form a conjugate pair of additive closure operators. The results of Section 2.5 then give us complete sublattices of our two lattices. These new operators are based on the concept of hypersatisfaction of an identity by a variety. We begin with the definition of a hypersubstitution, as introduced in [32].

What does it mean for a variety V of some type r to satisfy an identity s = t as a hyperidentity? Informally, it will mean that not only does the identity hold for the ni-ary fundamental operations occurring in it, but also that it will hold for all possible ni-ary terms of the variety as well. That is, for each operation symbol occurring in s = t , we may choose a term of V of the same arity and we

50

3 M-Hyperidentities and M-solid Varieties

require that the resulting identity holds in V. If all possible such choices of terms (of appropriate arity) lead t o identities which hold in V, we say that V hypersatisfies the identity s = t, or that the identity s = t is a hyperidentity of the variety V. As an example of a hyperidentity, let us consider the type (2) identity F ( x l , x l ) = xI, i.e. the idempotent law. Let B = Mod(F(x1, F(x2, ~ 3 ) = ) F(F(x1,x2),xs), F(xI,x1) = X I ) , the variety of bands. This variety has only six non-equivalent binary terms, which can be written as words X I , 2 2 , 21x2, 22x1, ~ 1 x 2 ~ 1 and x2x1x2,i.e. the universe of the free algebra F B ( X 2 ) consists of the six Chsses [XI] I ~ B ,[x2]I ~ B ,[ ~ 1 ~I ~2 B] ,[ ~ 2 ~I ~l B] ,[ ~ 1 ~ 2I ~~ B1 , ] and [ x ~ x ~ xSubstitution ~ ] ~ ~ ~ of. any one of these six terms for the operation symbol F in the identity F ( x l , x l ) = xl results in an identity which holds in B. Thus this identity is also a hyperidentity of B. However, substitution of the term ~ 1 x 2 for ~ 1 F in the associative identity F ( x l , F ( x 2 ,2 3 ) ) = F ( F ( x l , x2),2 3 ) leads t o an identity which does not hold in B , and this identity is not a hyperidentity of V. As another example, we consider the type (2) identity

called the medial law. This identity does not hold as an identity in the variety of all semigroups, and hence is not a hyperidentity for this variety either. But it does hold as a hyperidentity in two other varieties, the variety D of distributive lattices and the variety C of commutative semigroups. To see this, we consider the terms of these varieties. D is a variety of type (2,2), and has only four binary terms, X I , 2 2 , x1 A x2 and XI V 22. Substitution of each of these in turn for the binary operation symbol F in the medial law above leads to the four identities xl = X I , x4 = x4, (xl A x2) A (x3 A x4) = ( x 1 ~ x 3 ) ~ ( x 2 and ~ x 4( X) ~ V X ~ ) V ( X ~ V=X (~ x) ~ v x ~ ) v ( x ~ v x ~ ) . Since all of these hold in the variety D of distributive lattices, the medial law is a hyperidentity in this variety. Similarly, in the variety of commutative semigroups, any binary term can be written in the normal form t(xl, 2 2 ) = ax1 bx2, where

+

a and b are integers. Substitution of such a term for the operation symbol F in the medial law yields

It is easy to see that this identity is satisfied in the variety C of commutative semigroups, making the medial law a hyperidentity in this variety too. Hyperidentities in this form were first introduced by W. Taylor in 1981 ([%I). The model classes (varieties) of sets of hyperidentities were called hypervarieties in [98]. Taylor's theory of hypervarieties is very close t o Neumann's "varieties of varieties" ([69]). Instead of hypervarieties in our approach we consider solid varieties which are classes of algebras of the same type satisfying certain sets of hyperidentities. A slightly different concept of a hypervariety was used by V. D. Belousov ([6], [7]) and J. Aczd ([I]). This version required for a hyperidentity only that all fundamental operations need be substituted for the operation symbols of the same arity. Hyperidentities in this sense were studied in the books of Yu. M. Movsisjan ([67], [68]). We begin with a number of definitions and some notation. First of all, we generalize the definition of the superposition operation Sn introduced in Definition 1.3.4 t o a (heterogeneous) operation s: : W(Xn) X WT(&Jn W,(Xm). +

Definition 3.1.1 Let WT(Xm)and WT(Xn)be the sets of all mary and n-ary terms of type r for m, n 1,m, n E N,repectively. Then the operation S$ : WT(Xn)x WT(Xm)n+ WT(Xm)is defined inductively as follows:

>

With these operations and the variables as nullary operations we form the multi-based (heterogeneous) algebra

52

3 M-Hyperidentities and M-solid Varieties

which is called full clone of terms.

Theorem 3.1.2 The algebra

satisfies the following identities (CL) sg2, s;(YI, X I , . . . , X J , . ;. , s;(YP7X I , . . . , X n ) ) M s$(s[(z,I;;, . . . , & ) , X I , .. . , X n ) , ( m , n , p= l , 2 , . . .),

G)

(C3) % ( Y ,@, . . . , FZ Y , - ( n = 1 , 2 , . . .). Here 2,Yl,. . . , &,X I , . . . , X n , are variables for terms, S$ are operation symbols and e? are symbols for variables. Proof: ( C l ) We will give a proof by induction on the complexity (operation symbol count, see 1.3) of the term which is substituted for 2 . If we substitute for 2 the variable xi E X p 7then S&(xi,S:(SI, t ~ , .. ,. t n ) , . . . 7 S$(sp7t ~ ,. .. , t n ) ) = S$(si7tl,...,tn) = S$(S~(Z s l~, .,. . , s p ) ,t l , . . . , i n ) . If we substitute for 2 the term t = fi(l17 . . . , lni) E W T ( X p )and assume inductively that ( C l ) is satisfied for 11, . . . , lni, then s g ( f i ( 1 1 ~ . . . , 1Sn:~( )S~I , ~ I , . .A), . . . . 7 S:(sp7tl.. . A)) = fi(SL(11,S ; ( S I , ~ I ,..A), . . . . 7 S ; ( S ~ , ~ . I7 .i n. ) ) ,. . ., S & ( l n , , S ; ( ~ l , t ~.,,.t.n ) , . . . 7 S$(sp,t l . . . , t n ) ) ) = fi(S$(S&(Il,s17.. . , sp),t ~ ,. .. , t n ) , . . . 7 Sg(SL(ln,,~ 1 7 ..., ~p)7t17.. .,tn)) = S:(fi(Sg(Il, S I , . . , s p ) ,. . . , S&(lni7 S I , . . , ~ p ) ) t, l , . . . A ) = S $ ( S k ( f i ( l l , .. . , Li),S I , . . 7 ~ p ) t, l , . . . 7 in). This shows that the algebra clone r satisfies ( C l ) . The proofs of (C2) and (C3) are clear. As a special case of the multi-based algebra clone r we obtain the (one-based = homogeneous) unitary Menger algebra n - clone T of rank n, introduced in 1.3. Identifying m, n and p in ( C l ) , (C2), (C3),we get the defining identities of the variety of unitary Menger

algebras of rank n.

A formal definition of a hyperidentity can be given using the concept of a hypersubstitution

Definition 3.1.3 Let a : {fi I i E I) + W,(X) be a mapping assigning t o every ni-ary operation symbol fi of type 7 an ni-ary term, a(fi). Any such mapping a will be called a hypersubstitution (of type 7). By Hyp(r) we denote the set of all hypersubstitutions of type r . Every hypersubstitution of type 7 induces a mapping 6 : W,(X) + W,(X) on the set of all terms of type r, as follows. For any term t E W,(X), the term 6[t] is defined inductively by (1)

&[XI

:= x for any variable x E

X and

Here n is the maximum of the arities of the terms 6[tl],. . . 6[tnz] on the right hand side. The extended hypersubstitutions 6 are permutable with the superposition operations S$, i.e.

Proposition 3.1.4 If 6 is the extension of a hypersubstitution a , then for all m, n E N+

Proof: We will give a proof by induction on the complexity of the term t . If t = xi E X,, then

54

3 M-Hyperidentities and M-solid Varieties

Let T, be a type consisting of operation symbols of the same arity n. From the previous result there follows that extensions 6 of hypersubstitutions of type T, are endomorphisms of the unitary Menger algebra n - clone 7, of rank n. If we consider a hypersubstitution o as a sequence (on),,,+ where o, maps the n-ary operation symbols of the type to n-ary terms, then the extension 6 = is an endomorphism of the heterogeneous algebra clone T . In order to prove some properties of hypersubstitutions we need a preliminary lemma about the composition of hypersubstitutions.

(en),,,+

Lemma 3.1.5 Let aI and o2 be two hypersubstitutions of type 7 . Then o o2 is a hypersubstitution, and

Proof: It is clear that the map

is a hypersubstitution. We verify the claimed equality by induction on the complexity of terms. If t = x , then

(61 0 0 2 ) [XI x = e2[x] = 61[62[x]] = (GI 0 6 2 ) [ x ] . Now let t = fi(t',,. . . t',%) for 2 E I and assume that (elo 02)^ [ti]= (610 62)[ti]for every j E (1, . . . , ni}. Then (61 0 0 2 ) [f&, . . . tLJ1 = s,"'((ei 0 0 2 ) ( f i ) ,(610 0 2 ) It',],. . . , (610 0 2 ) It&]) = S 2 (61[a2( f i ) ], 61 [62[ t i ] ]. ., . ,61[62[ t & ] ] ) = 61[S:' (02 ( f i ) 62 [ti] . . . 7 62 [tLi])] = 6 ~ [ 6 2 [ f i ( t i >. . . , , tL,)]] = (el 0 e2) [t]. A

A

For the proof we used Proposition 3.1.4. From Lemma 3.1.5 we obtain

Lemma 3.1.6 The operation oh : Hyp(7) x Hyp(7) + Hyp(7) defined by al oh 0 2 := el o 02 is associative and together with aid defined b y a i d ( f i )= f i ( x l , .. . , xni) for all i E I the set Hyp(r) forms a monoid E y p ( r ) := ( H y p ( r ); o h , aid).

Proof: It is clear that aid acts as an identity element for o h , SO we need only show the associativity of oh. From Lemma 3.1.5 we know that (61 o a 2 ) = i1 o 62. Then we have A

If a is a hypersubstitution of type kernel

7 , it

is very natural to ask for its

kera := { ( t ,tt)lt,t' E W , ( X ) and 6 [ t ]= 6 [ t t ] ) . By definition, the kernel of a hypersubstitution is an equivalence relation on the set W , ( X ) . Moreover we have:

Theorem 3.1.7 Let a be a hypersubstitution of type r = (ni)iE1for ni > 1 for all i E I . Then kero is a fully invariant congruence relation on the absolutely free algebra F , ( X ) . Proof: Let fi be an ni-ary operation symbol and ( s j ,t j ) E kera for j = 1 , . . . , ni. Then 6 [ s j ]= 6 [ t j ]for j = 1 , . . . , n j . Let n' be the maximum of the arities of the terms sl, . . . s,%,t l , . . . , t,. Applying the operation SE;we obtain

this means ( f i ( s l , . . , s , ~ ) ,f i ( t l , .. . , t n i ) )E kero. Now we show that kera is fully invariant. Let s : X + W , ( X ) be a substitution and let 3 be its extension (which is the endomorphism s : F , ( X ) + F , ( X ) , uniquely determined by ~ ( t=) S(fi(t1,.. . ,t,,)) = f i ( S ( t l ) , . . . , ~ ( t , ~for ) ) composed terms f i ( t l ,. . . , tni) of type r ) . We notice that any endomorphism 3 : F , ( X ) + F , ( X ) is the uniquely determined extension of a substitution s : X + W , ( X ) . Consider a mapping s* : X + W , ( X ) defined by s* ( x ) := i [ s ( x ) for ] every x E X . Since s* is a substitution

56

3 M-Hyperidentities and M-solid Varieties

it can be uniquely extended to an endomorphism 2 : F T ( X ) + F T ( X ) .We show by induction on the depth of a term t that

for every t E W T ( X ) .Indeed, if t = z E X is a variable then 2 ( 6 [ x ] )= 2 ( x ) = s * ( x ) = 6 [ s ( x ) ]= 6 [ s ( x ) ]by the definition of s*. Let t = f i ( t l , . . . , t,%)and suppose that 2 ( 6 [ t j ]= ) 6 [ s ( t j ) for ] j = 1 , . . . , ni. Then

Here we used that the application of s to a superposition Snni(t,t l ,. . . , tni)of terms gives

(See the remark after Definition 1.3.7.) Let now ( t ,t') E kero. Then

6 [t]= i [ t ' ]and 2 ( 6 [ t ]= ) 2(6[tf]) and by (*)

6 [ s ( t )= ] 6[s(tr)], i.e. ( ~ ( ts )( t, f ) )E kera. Consider an n-ary type r, where every operation symbol is n-ary. If we restrict the kernel of a hypersubstitution o t o the set W,(X,) of all n-ary terms of type r,, then we obtain a congruence relation on the unitary Menger algebra n - clone r, of rank n .

Proposition 3.1.8 Let a be a hypersubstitution of type r,. T h e n (X,)2 is a congruence relation o n the algebra n - cloner,. keral WTm Proof: Let ( s ,t ) E kerol WTn (X,)2 and ( s j ,t j ) E kerol W,, (X,)2 for j = 1, . . . , n. Then 6 [ s ] = 6 [ t ] and 6 [ s j ] = 6 [ t j ] for j = 1 . . , n . Applying the operation S n we obtain S n ( 6 [ s ] , 6 [ s l .] ., ,. 6 [ s n ] )= S n ( 6 [ t ] , 6 [ t .l .] ., , i [ t , ] ) and thus by Proposition 3.1.4, 6 [ S n ( ss, l , . . . , s,)] = 6 [ S n ( tt,i , . . . , t,)], this

For arbitrary types r this result can be generalized to the heterogeneous algebra cloner. Using hypersubstitutions we define the weaker concept of an M-hyperidentity.

Definition 3.1.9 Let M be any submonoid of IFtyp(r). An algebra A is said to M-hypersatisfy an identity u = v if for every hypersubstitution a E M , the identity 6[u] = 6[v] holds in A. In this case we say that the identity u = v is an M-hyperidentity of A and write

An identity is called an M-hyperidentity of a variety V if it holds as an M-hyperidentity in every algebra in V. A variety V is called M-solid if every identity of V is an M-hyperidentity of V. When M is the whole monoid 'Ftyp(r), an M-hyperidentity is called a hyperidentity, and an M-solid variety is called a solid variety. Hypersubstitutions can also be applied to algebras, as follows. Given an algebra A = (A; (f:)iEI) and a hypersubstitution a, we define the algebra a ( A ) : = (A; ( ~ ( f ~ ) ~ )This ~ , ~algebra ). is called the derived algebra determined by A and a . Notice that by definition it is of the same type as the algebra A. To check whether an identity is hypersatisfied can be a complex problem since there are infinitely many hypersubstitutions. But working in a given variety V it turns out that not all hypersubstitutions are important. The following definition opens a way to realize this idea.

Definition 3.1.10 Let V be a variety of type r . Two hypersubstitutions a1 and a 2 of type r are called V-equivalent if and only if al(fi) = 02(fi) is an identity in V for all i E I. In this case we write a1 --v a2. Although the relation -v is defined in terms of how hypersubstitutions behave on the fundamental operations, we can quickly extend this definition to describe behaviour on arbitrary terms and on derived algebras.

Theorem 3.1.11 ([86]) Let V be a variety of type r, and let and a 2 E Hyp(r). Then the following are equivalent:

a1

58

3 M-Hyperidentities and M-solid Varieties

(ii) For all t E W , ( X ) the equation il[t] = i2 [t] is an identity in v. (iii) For all algebras

A E V, ol(A) = 02(A).

Proof: (i)+(ii) This can be proved in a straightforward induction on the complexity of the term t. (ii)+(iii) We consider the terms t = fi(xl, . . . , xna) for i E I and use the fact that A s = t iff sA = tA. By (ii) we have el[fi(xl, . . . , xn,)lA = i2[fi(x1, . . . , xni)lA for all i E I and all A E V. Thus a1(A) = a2(A).

+

(iii)+(i) Here we have il[fi(xl, . . . , xni)lA= i2 [fi(21, . . . , xni)lAfor all i E I and all A E V. Thus

Instead of asking which identities from a variety V admit all hypersubstitutions from a given monoid M of hypersubstitutions we can also ask which hypersubstitutions preserve all identities from V.

Definition 3.1.12 Let V be a variety of type r . A hypersubstitution o of type r is called a V-proper hypersubstitution if for every identity s = t of V, the identity i [ s ]= i [ t ] also holds in V. We use P ( V ) for the set of all V-proper hypersubstitutions of type T. Proposition 3.1.13 For any variety V of type is a submonoid of (Hyp(r);O h , aid).

7,

( P ( V ) ;o h , aid)

Proof: The identity hypersubstitution a i d is a V-proper hypersubstitution for any variety V, so it is in P ( V ) . If a1 and a 2 are in P ( V ) , = then for every s = t E IdV, e2[s]= i2[t]E I d V and e1[e2[s]] el[e2[t]] E IdV. This means that (el o 6 2 ) [s] = (el o e2) [t] E I d V and thus (al o h 0 2 ) [s] = ((al o h 0 2 ) [t] E IdV. Therefore, a1 o h 02 E P ( V ) , and P ( V ) is a submonoid of IFtyp(~). A

A

Clearly, a variety V is solid if and only if P ( V ) = Hyp(r). Moreover, any variety V is M-solid for M C P ( V ) , and P ( V ) is the largest M for which V is M-solid.

Proposition 3.1.14 ( [ 8 6 ] )Let V be a variety of type 7. Then the following hold: (i) For all 0 1 , a2 E H y p ( r ) , if a1 --v 0 2 then a1 is a V-proper hypersubstitution iff o2 is a V-proper hypersubstitution. (ii) For all s, t E W, ( X ) and all al,a2 E Hyp(r) , if a1 --v 0 2 then the equation el [s]M [t]is an identity in V iff [s]M e 2 [ t ] is an identity in V.

el

e2

Proof: (i) Let a1 be a V-proper hypersubstitution. Then for all identities s M t in I d V the equation 61[s]M e l [ t ]is an identity in V, and then a1 -v 0 2 gives

v

62 [s]M 61 [s]M 61 [t]= 62 [t].

The other direction can be proved in the same way. (ii) Suppose that e l [ s ]M eil[t]is an identity in V. The equations 61 [s]= 62 [s]and [t]= 62 [t]are identities in V by Theorem 3.1.11. Thus e2[s]M e 2 [ t ]is an identity in V. The converse follows in the same way.

el

It is clear from the definition that the relation -v is always an equivalence relation on Hyp(r). It is sometimes, but not always, a congruence as well. But if the variety V is solid, then -v is a congruence.

Proposition 3.1.15 Let V be a solid variety of type r . Then the relation --v is a congruence on Kyp(7). Proof: Assume that ol -v o2 and a3 -v 04. Then by Theorem 3.1.11 for a l l i E I we get

and

60

3 M-Hyperidentities and M-solid Varieties

By transitivity we get

a1 oh a3

-v

02 oh 04.

The last proof shows that -v is always a right congruence, even if V is not solid. If V is M-solid for a submonoid M of 'Ftyp(7), then the restriction -v IM is a congruence on M. For now, we focus on the important fact that Proposition 3.1.14 essentially means that the submonoid P ( V ) of Hyp(r) is a union of equivalence classes of the relation -v. This is also true when we restrict our attention to a submonoid M of 'Ftyp(7), and to the relation -v IM, as we now show.

Lemma 3.1.16 Let M C Hyp(r), and let V be a variety of type 7. T h e n the P ( V ) n M is a u n i o n of equivalence classes of the restricted relation -vM. Proof: Let a be a hypersubstitution in P ( V ) n M , and let p E M satisfy o p. To see that p is also in P ( V ) , we let s = t be any identity of V. Since o -v p we have i [ s ] = 6 [ s ] and i [ t ]= b[t] both in I d V by Theorem 3.1.11. Since o is V-proper, we also have i [ s ] = i [ t ]in IdV. From this it follows that p[s] = p[t] is in IdV. This proves that any hypersubstitution p from M which is V-equivalent to a hypersubstitution a in P ( V ) M is also in p(v) M.

n

The importance of this result lies in what it tells us about testing whether identities are M-hyperidentities of V. To this end we define a new variant of solidity.

Definition 3.1.17 Let M be a monoid of hypersubstitutions of type r , and let V be a variety of type r . Let q5 = $(V) be a choice function which chooses from M one hypersubstitution from each equivalence class of the relation -vlhi, and let N f ( V ) be the set of hypersubstitutions so chosen. Thus N r ( V ) is a set of distinguished hypersubstitutions from M , which we might call V - n o r m a l form hypersubstitutions We will say that the variety V is Nr(V)-solid if for every identity s = t t I d V and for every hypersubstitution a t N f ( V ) , the identity 6 [ s ]= 6[t]t IdV. Thus a variety V is Nr(V)-solid if the result of applying any one of the distinguished hypersubstitutions from N f ( V ) t o an identity

of V is still an identity of V. We note that the set N r ( V ) is not always a submonoid of E y p ( r ) , as in [3] was pointed out. When we compose two hypersubstitutions CT and p in N f ( V ) , according t o the usual composition in H y p ( ~ )the , result need not be in N,"'(V), although it is equivalent to some element in N r ( V ) . If we define a new product * on N r ( V ) which assigns this equivalent element t o CT * p, we get a groupoid structure on N f ( V ) , but the operation * is not always associative. As the next theorem shows, the fact that N f ( V ) is not a submonoid does not interfere with its usefulness in testing for M-solidity of V.

Theorem 3.1.18 Let M be a monoid of hypersubstitutions of type T and let V be a variety of type T. For any choice function 4, V is M-solid if and only if V is Nr(V)-solid. Proof: It is clear that if V is M-solid then it is certainly also N,"'(V)-solid. Conversely, suppose that V is N,"'(V)-solid. This means that all the members of the set N r ( V ) are also members of P ( V ) n M . Since by Lemma 3.1.16, P ( V ) 0 M is a union of -VIMclasses, anything equivalent to an element of N f ( V ) is also in P ( V ) n M . But by construction any element of M is equivalent t o an element of N,"'(V). Thus M c P ( V ) , and V is M-solid. Theorem 3.1.18 guarantees that t o test V for M-solidity, it is enough t o apply t o identities of V only one hypersubstitution from each ~ v l n r - c l a s sWe . remark that we could also generalize this result a bit, as follows. If 8 is any equivalence relation on M for which P ( V ) M is a union of equivalence classes, then it would suffice t o test only one hypersubstitution from each equivalence class.

n

Theorem 3.1.18 is very useful in testing for hyperidentities for a particular variety V. But it is important t o note that for M, must hyperidentities, when M is a proper submonoid of E y p ( ~ )we be careful in using this theorem. It is essential that we first fix a particular submonoid M, then consider a choice function 4 on M which chooses from M one element of each -vlM-class. It is in fact more natural in practice t o first partition the whole monoid 'Flyp(r) according t o the relation -v for a particular V, choosing a normal form representative from each class, then to restrict to M by taking only those representatives from M . But as examples show, this pro-

62

3 M-Hyperidentities and M-solid Varieties

cess can lead to a different result. In the next section we will use the theory of conjugate pairs of additive closure operators developed in Chapter 2 t o characterize M-solid varieties.

The Closure Operators

3.2

x&,X g

We want to apply the results of 2.5 to get a characterization of Msolid varieties of algebras. We now introduce some closure operators on the two sets Alg(r) and W, ( X ) 2 .On the equational side, we can use the extensions of our M-hypersubstitutions to map any terms and identities to new ones. That is, we define an operator by

Xg

This extends, additively, to sets of identities, so that for any set C of identities we set xff[E]

=

U{xE[s ~

t I s]~t t E}.

Using derived algebras we define now an operator X& on the set Alg(r), first on individual algebras and then on classes K of algebras, by

If M = Hyp(r) we will simply write XE and xA,respectively. For the proof that the operators X& form a conjugate pair we need the following lemma.

xg,

Lemma 3.2.1 For any term t E W , ( X ) we have

Proof: We will give a proof by induction on the complexity of the term t. If t = xi is an n a r y variable, then 8[xilA= x! = =

44

xi

.

3.3 M-Solid Varieties and their Characterization

63

Assume that t = f i ( t l , . . . , t,,) is a composite term and that &[ti]" = t ~ ' for ~ 'all 1 J n i . Then

< <

Proposition 3.2.2 Let r be a fixed type and let M be a n y suband are additive clornonoid of IFtyp(r). T h e two operators sure operators and are conjugate with respect t o the relation of satisfaction.

XE

Proof: Since M is a submonoid of IFtyp(r), it contains the identity hypersubstitution a i d , and the extension of this hypersubstitution maps any term t to itself. This shows that the operator is extensive. The property of monotonicity is clear from the definition. The idempotency follows from the fact that M is a monoid: for any a and p in M , the composition a oh p is also in M ; so for any identity s = t we have 6 [ b [ s ] = ] 6 [ p [ t ] in ] x E [ s = t ] . Thus X E, is a closure operator. The proof for is similar. Both these closure operators are additive by definition, so it remains only to show that they are conjugate with respect t o satisfaction. For this we need t o show that for any algebra A and any identity s = t of type r , we have

XE

Xif

For any a E M , the definition of satisfaction means that A satisfies 6 [ s ] = 6[t]iff the induced term operations satisfy 6 [ s I A = 6 [ t I A . Similarly, a ( A ) satisfies s = t iff so(A) = t o ( A ) .But 6 [ s I A = ~ ~ ( ~ 1 , and similarly for t . This shows that o(A)satisfies s = t iff A satisfies i [ s ]= i [ t ]and , completes our proof.

3.3

M-Solid Varieties and their Charact erizat ion

Let M be any submonoid of IFtyp(r). Since M contains the identity hypersubstitution, any M-hyperidentity of a variety V is an

3 M-Hyperidentities and M-solid Varieties

64

identity of V. This means that the relation of M-hypersatisfaction, s = t, (or A s = t for M = Hyp(r)), written as A M-~YP

~ Y P

defined between Alg(r) and W,(X)2, is a subrelation of the relation of satisfaction from which we induced our Galois connection (Id, Mod). The new Galois connection induced by the relation of M-hypersatisfaction is (HMId,HhfMod), defined on classes K and sets C as follows: H M I d K = {s = t E W,(X)2 I s = t is an M-hyperidentity in A for all A in K ) , HhfModC = {A E Alg(r) I all identities in C are M hyperidentities of A). Sets of equations of the form HhfIdK are called M-hyperequational theories and classes of algebras of the same type having the form HMModC are called M-hyperequational classes. For M = Hyp we speak of hyperequational theories, H I d K , and hyperequational classes HModC. It turns out that the Galois closed classes of algebras under this connection are precisely the M-solid varieties of type 7, which then form a complete sublattice of the lattice of all varieties of type r . Thus studying M-solid and solid varieties gives a way to study complete sublattices of the lattice of all varieties of a given type. Since we now have a conjugate pair (x&,Xg)of additive closure operators defined on Alg(r) and on W,(X)2, respectively, and the Galois connection (Id, Mod) between Alg(r) and W,(X)2 as well as a second Galois connection (HMId,HhfMod) between Alg(r) and I= we can apply the W,(X)2 which is induced by the relation M-~YP

results of 2.5.

Lemma 3.3.1 Let

(x&,xff) be the

conjugate pair of additive closure operators with respect to the relation Alg(r) x W,(X)2 s = t ifl s = t is satisfied as an identity in A and defined by A let for a submonoid M Hyp(r) be the relation defined by

c

M-~YP

A

I= M-~YP

s

=t

ifl A satisfies s

=t

as an M-hyperidentity. Then

3.3 M-Solid Varieties and their Characterization

C A l g ( r ) and for

for all K hold:

(i) Hhf ModC (ii) Hhf ModC

=

all C

C W , ( X ) 2 the following

65

properties

M o d X g [C],

c ModC,

[HhfModC]= Hhf ModC,

(iii)

(iv) X f f [ I d H ~ M o d C =]IdHMModC,

(i') H M I d K = I ~ ~ & [ K ] , (ii') H M I d K C I d K , (iii')

xff [ H M I d K ]= H M I d K ,

(iv') X & [ModHhf I d K ] = ModHhf I d K ,

(v') HMIdHMil/fodC= Idil/fodXfr[C]. This lemma corresponds to Theorem 2.5.4 and needs no proof. Using these properties one can prove the following characterization of M-solid varieties corresponding to Theorem 2.5.5.

Theorem 3.3.2 Let M be a monoid of hypersubstitutions of type T . For any variety V of type T , the following conditions are equiualent: (i) V

=

HMModHMIdV.

[V]= V .

(ii)

(iii) IdV

=

HMIdV.

(iv) X f f [ I d V ]= I d V .

And dually, for any equational theory C of type r, the following conditions are equivalent: (i') C (ii')

=

HMIdHhfModC.

xff [C]= C .

3 M-Hyperidentities and M-solid Varieties

66 (iii') ModC

=

HhfModC.

(iv') X&[ModC] = ModC. Condition (iii) of this theorem says that V is an M-solid variety, since every identity of V is an M-hyperidentity. In analogy with the (Id, Mod) case, a variety which satisfies condition (i) is called an M-hyperequational class. Thus our M-solid varieties are precisely the M-hyperequational classes. Dual results hold for Mhyperequational theories, using the second half of the theorem. We will apply this if we determine the greatest M-solid variety of semigroups for certain monoids M of hypersubstitutions. In this case we have to look for the M-hyperequational class of the associative identity. The theorem corresponding to Theorem 2.5.6 is also quite useful for applications.

Theorem 3.3.3 Let M be a monoid of hypersubstitutions of type 7. Then for all sets K Alg(7) and C W , ( X ) 2 the following properties hold:

c

c

c

=

HhfModHhfIdK

e

c

=

HhfIdHhfModC

e

(i) x&[K] ModIdK e ModIdK X & [ ~ o d ~=dModIdK. ~ ] (ii) Xg[C] IdModC e IdModC Xg[IdModC] = IdModC.

Proposition (ii) of this theorem can be applied if the variety V = ModC is given by an equational basis C. If we want to check whether V is M-solid, it is enough to apply all hypersubstitutions from M t o the equational basis C. If the arising equations are satisfied as M-hyperidentities, then the variety is M-solid. From Proposition 2.5.8 there follows that the relation of M-hypersatisfaction is a Galois closed subrelation of that of satisfaction as identities. From this result we obtain:

Theorem 3.3.4 Let M be a monoid of hypersubstitutions of type 7. Then the class Shf(r)of all M-solid varieties of type T forms a complete sublattice of the lattice C(T) of all varieties of type T. Dually, the class of all M-hyperequational theories forms a complete

3.4 Subvariety Lattices and Monoids of Hypersubstitutions

67

sublattice of the lattice of all equational theories of type 7 . Also, if M l ,M2 are monoids of hypersubstitutions and if M I is a submonoid of M 2 , t h e n for the lattices of M-solid varieties we have that SM2 ( r ) is a complete sublattice of SMl( r ) . Proof: For the proof we apply Theorem 2.5.7. If M1 C M2, then xiI1[V]C x&2[V]for every variety V of type 7 . If V is M2-solid, [V]= V , then because of the extensivity of X;;ffl we have i.e. if

( r )C and therefore xiI1[V]= V and V is MI-solid. This shows SM2 SMl( 7 ) . Again by application of Theorem 2.5.7 one proves that Shr,( 7 )is a complete sublattice of Snrl( 7 ) .

3.4

Subvariety Lattices and Monoids of Hypersubstitutions

If V is a variety of type r and C(V)is the lattice of all subvarieties of V , then Shf( V ) := S M ( r n ) C(V) is the lattice of all M-solid subvarieties of V . Now we apply the general Galois theory to the set Hyp(r)of all hypersubstitutions of type r and t o the set L(V) of all subvarieties of the variety V of type r and its power sets P(Hyp(r))and P ( L ( V ) )For . a variety W and a hypersubstitution a we set a ( W ):= { a ( A )I A E W ) .We define the following relation R C Hyp(r) x L(V) between Hyp(r) and L(V) :

R : = { ( o , W )I o E Hyp(r) and W E L(V) and a ( W )C W ) . This means (a,W ) E R iff o is a W-proper hypersubstitution. Clearly, if a1 -w a 2 then (al,W) E R iff (a2,W) E R. This relation R induces a Galois connection (q,8) between Hyp(r) and L(V).For any M C Hyp(r) and for any L C L(V) we define , ) E R ) ) and Q ( L ):= {a E q ( M ) := {W E L(V) I Ya E M ( ( a W Hyp(r) I YW E L ( ( a W , ) E R ) ) ) .Using the general theory we obtain a Galois connection between Hyp(r) and L(V) and therefore the following propositions which can be found in [22]:

3 M-Hyperidentities and M-solid Varieties

68

Proposition 3.4.1 Let V be a variety of type T . For the mappings q : P ( H y p ( r ) ) + P ( L ( V ) ) and Q : P ( L ( V ) ) + P ( H y p ( r ) ) we have: ( i ) If M C M' H y p ( ~ then ) q(M) L ( V ) then Q ( L ) Q ( L t ) ,

>

> q ( M t ) ,and if L C L'

C

(ii) for any M E P ( H y p ( 7 ) )and for any L E P ( L ( V ) ) ,we have Q ( q ( M ) )and L C q(Q(L)), M

c

(iii) if for any M E P ( H y p ( 7 ) ) and any L E P ( L ( V ) ) we set L := q(_B(L))and M := B ( q ( M ) )then the mappings defined b y M + M and L + are closure operators. (This means, we have the properties M C M , M C M' i M C M , M = M and dually for L. Thus we call a subset M of H y p ( r ) closed if M = M and dually, subsets L c L ( V ) are called closed if L = L.) (iv) for any L E P ( L ( V ) ) ,the set L is closed ijf there is a set M C H y p ( r ) such that L = q ( M ) , and for any M E P ( H y p ( 7 ) ) , the set M is closed ijf there is a set L C L ( V ) such that M = B(L); in particular, we have Q ( q ( Q ( L ) )=) Q ( L ) and v ( Q ( v ( M ) )=) v ( M L ( v ) for any M , M' E P ( H y p ( 7 ) )we have q ( M U M ' ) = q ( M ) n q ( M t ) and for any L, L' E P ( L ( V ) )we get Q ( L uL') = Q ( L n ) Q(Lt), (vi) let L , L' E P ( L ( V ) ) then ( L ,L') E kerB ijf L = L' and let M , M' E P ( H y p ( r ) )then ( M ,M ' ) E kerq ijf M = M'. Part (vi) of the proposition means that all members of each kerQclass have the same kernel. Thus we can define a map 8 on ~) Dually, we define a map P ( L ( V ) ) / k e r Qby H ( [ L ] ~ ~:=~ [Q(L)lkerV. f j on P ( H y p ( ~ ) ) / k e rby q v([M]~,,,):= [q(M)Ikere. It follows from the properties of a Galois connection that these two maps are bijections. Corollary 3.4.2 The maps 8 and f j are bijections between P ( L ( V ) ) / k e r Qand P ( H y p ( r ) ) / k e r q .

3.4 Subvariety Lattices and Monoids of Hypersubstitutions

69

Now we consider the restriction of this Galois connection t o certain special kinds of sets. This is motivated by the result ([34])that any submonoid M of IFtyp(r) determines a complete sublattice of the lattice C(V), the sublattice SM(V)of all M-solid subvarieties of the variety V. So it is very natural to restrict our Galois mappings 8 and q to submonoids M of IFtyp(r) and to sublattices C of C(V), respectively.

Lemma 3.4.3 For any subset L of L(V), the image Q(L) is the universe of a submonoid of 'Flyp(r); and for any subset M of Hyp(r), the image q ( M ) is the universe of a sublattice of C(V). Proof: Let L be a subset of L(V). For any variety W , the set of all W-proper hypersubstitutions forms a monoid (Proposition 3.1.13), a submonoid of IFtyp(r). The image Q(L) is the intersection of the universes of these monoids for every W E L, and thus a monoid. Now let M be a subset of Hyp(r). By definition, q ( M ) consists of W , for all a E M . In Theothose W in L(V) for which o ( W ) rem 3.3.2 and in Theorem 3.3.4 (see also [34]) it was shown that when M is a submonoid, this is equivalent to Xif [w]= W , and also that X& is a closure operator on classes of algebras, with the set of all subvarieties W of V with X & [ ~ = ] W forming a (complete) sublattice Shf(V) of the lattice C(V) consisting of all M-solid subvarieties of V. Thus we have q ( M ) = Snr(V), the universe of a sublattice of C(V), when M is a submonoid of IFtyp(r). But then for any arbitrary subset M of Hyp(r), we have q ( M ) = q(Q(q(M))) with Q(q(M))the universe of a submonoid from the first part of this proof, so again q ( M ) is the universe of a sublattice.

c

Let S(Hyp(7))be the submonoid lattice of the monoid Hyp(r), and let C(C(V)) be the lattice of all sublattices of C(V). We now define two mappings P and a as the restrictions of Q t o L(C(V)) and q to S ( H y p ( r ) ) respectively. Then the previous lemma shows that ,b' is a mapping from L(C(V)) to S ( H y p ( r ) ) and a is a mapping from S(Hyp(7)) to L(C(V)).

Lemma 3.4.4 For any C , K E L(C(V)) and for any M , N E S ( H y p ( r ) ) , we have P(C) A P(K) = P(C V K) and a(M V N) = a(M)A a ( N ) .

70

3 M-Hyperidentities and M-solid Varieties

Proof: This follows from one of the properties of a Galois connection and the previous lemma. In the same way as

8 and 7 we define mappings P,a with

: S ( H y p ( r ) ) / k e r a+ L ( C ( V ) ) / k e r P and : L ( C ( V ) ) / k e r P+ S ( H y p ( r ) ) / k e r a .

p

Corollary 3.4.5

a and

p are bijections.

It is clear that the maps a and p do not preserve joins, and hence are not lattice homomorphisms. Also, from the previous lemma we have P ( C ) r \ p ( K )= ~ ( C V K for ) , any C and K. It is also always true that P(C) V P ( K ) is contained in P(C A K ) , but there are examples which show that this inclusion can be strict. Thus P is not a lattice anti-homomorphism.

Corollary 3.4.6 The intersection of closed submonoids from S ( H y p ( r ) )is closed, and the intersection of closed sublattices from C ( C ( V ) )is also closed. Thus the closed objects in S ( H y p ( r ) ) ,and dually of L ( C ( V ) ) ,form a lattice under inclusion, with meet equal to intersection. Every submonoid M of 'Hyp(r) determines a complete sublattice a ( M ) = S M ( V )of the lattice C ( V )of all subvarieties of the variety V . Considering a set &t of submonoids of H y p ( r ) we define the set CM = { a ( M ) 1 M E M } of complete sublattices of C ( V ) and ask u&r which conditions CM is a sublattice of the lattice C ( C ( V )of all sublattices of C ( V ) . since for submonoids M I ,M 2 of 'Hyp(r) with Ml M2 we have a ( M l ) = SMl( V ) 2 S M 2( V ) = a ( M 2 ) ,one conclusion is that if &t is a chain then CM - is also a chain and thus a sublattice of C ( C ( V ) ) . From our earlier results we get the following proposition:

c

Proposition 3.4.7 Let &t be a sublattice of S ( H y p ( r ) .) T h e n the following conditions are equivalent: forms a sublattice of C ( L ( V ) ) , (i) CM (ii) for any two monoids M 1 , M 2from

4 M 2 ) = snr, ( V ) v Snr2 ( V ) E

M,

we have a(M1) V

3.5 Derivation of M-Hyperidentities

71

Proof: (i) ==+(ii) is clear. (ii) ==+ ( i ) Since by assumption the join of elements of CM is in C M 7we need only check meets. That is, we need to c h e z that am)A u(nt,) E CM - for any two monoids M I ,M 2 E M . This is immediate from Lemma 3.4.4 and the fact that iZ/i is a sublattice.

Dually, for a set C of sublattices of C(V) we can consider monoids = {P(L) I L E C). which are P-images of the lattices in C: We could state and prove a dual theorem which characterizes when &J is a sublattice of the lattice of all submonoids of Hyp(r).

3.5

Derivation of M-Hyperidentities

We defined a set C C W,(X) x W,(X) t o be a hyperequational theory if there is a class K C Alg(r) with C = H I d K . An easy consequence of this definition and the previous lemmas and theorems is the following characterization of hyperequational theories: An equational theory C C W,(X) x W,(X) is a hyperequational theory if and only if XE[C]= C. In Section 1.4 we described the one-to-one correspondence between equational theories and fully invariant congruence relations on the absolutely free algebra F,(X) of type r. This raises the question of what additional properties a fully invariant congruence must have in order to correspond to a hyperequational theory. To answer this, in [32] the concept of a totally invariant congruence relation was introduced. Let A = (A; (f$)i,I) be an algebra of type r.A mapping h : A + A is called a semi-weak endomorphism (introduced by M. Kolibiar and K. Glazek in [48], [63], see also [49], [50]) of A if for every i E I there exists a term ti E W,(X) such that

for all a l , . . . , a n z E A. The algebra I3 = (B;(f?)iEI), where B is the image of A with respect t o h and f: = t t l ~ is, called the semi-weak endomorphic image of A with respect to h.

72

3 M-Hyperidentities and M-solid Varieties

Definition 3.5.1 A congruence relation 6' C A x A of an algebra A is called totally invariant if (a, b) E 6' implies (h(a),h(b)) E 6' for every semi-weak endomorphism h of A. Note that totally invariant congruences are also called solid congruences by Schweigert and Kilgus in [95]. Since we may take ti = fi(xl, . . . , xni), we see that every endomorphism is a semi-weak endomorphism. Thus every totally invariant congruence is also fully invariant. There is a strong connection between semi-weak endomorphisms and hypersubstitutions. Given any semi-weak endomorphism h, with corresponding terms ti, we may define a hypersubstitution a by a : fi H ti, i E I. Then the semi-weak endomorphic image B is a subalgebra of a[A]. Moreover, h is a semi-weak endomorphism of A if and only if h is a homomorphism from A into a[A] for some hypersubstitution a . Conversely, from any hypersubstitution a we can produce a semiweak endomorphism on F,(X), the absolutely free algebra of type r . Indeed, the extension mapping i is a semi-weak endomorphism of the absolutely free algebra F,(X), since for every i E I there exists a term ti E W,(X), namely a (fi), such that

Conversely, every semi-weak endomorphism h of F,(X) can be expressed as a composition of a hypersubstitution i with an ordinary endomorphism q5 of F,(X). To see this, for h a semi-weak endomorphism of F,(X) we consider the mapping q50 : X i W,(X) defined by :c H h(:c). Since F,(X) = (W,(X); ( is a free algebra in the class of all algebras of type r, the mapping can be extended to a unique endomorphism q5 : F,(X) + F,(X), i.e.

fFfx))ieI)

Now we show that h(t) = q5(6[t])for t E W,(X), where a is the hypersubstitution defined by mapping fi t o t i . Proceeding by induction on the complexity of terms, we first note that for t equal to any variable x, we have q5(i[x]) = q50(x) = h(x). Now

3.5 Derivation of M-Hyperidentities

73

c

Theorem 3.5.2 An equational theory C W,(X) x W,(X) of type r is a hyperequational theory if and only if it is a totally invariant congruence of FT ( X ).

c

Proof: Let C W,(X) x W,(X) be a hyperequational theory, so there exists K for which C = H I d K . Then H I d K = I ~ ~ ~ [soK ] , C = XE[C].This means that C is invariant with respect t o hypersubstitutions. Since a hyperequational theory is a fully invariant congruence on F,(X), C is also invariant with respect to endomorphisms. But every semi-weak endomorphism of F,(X) is the composition of a hypersubstitution and an endomorphism, so C is invariant with respect t o semi-weak endomorphisms. Conversely, every totally invariant congruence C is an equational theory since it is fully invariant. The set C is closed with respect t o hypersubstitutions since the extension of a hypersubstitution is a semi-weak endomorphism. Thus C is a hyperequational theory. These considerations can be generalized to submonoids M of 'Ftyp(r). Clearly, an equational theory C is an M-hyperequational theory, that is C = H M I d K for a class K of algebras of type r , iff XfIIC] = C. If M,, denotes the set of all semi-weak endomorphisms which correspond to the hypersubstitutions from M , then a congruence relation 6' C A x A of an algebra A of type r is called Msw-totally invariant if (a, b) E 8 implies (h(a),h(b)) E 8 for every semi-weak endomorphism h E Msw of A. Then an equational theory C C W,(X) x W,(X) of type r is an M-hyperequational theory if and only if it is a M,,-totally invariant congruence of F, ( X ) . In 1.4 we mentioned that by Birkhoff's theorem ([8]), a set C of identities of type r can be represented as the set I d K , for some variety K of type r , if and only if C is closed with respect t o the rules (1) - (5) of inferences given in Section 1.4. The properties of a M,,-totally invariant congruence (or equivalently of

74

3 M-Hyperidentities and M-solid Varieties

an M-hyperequational theory) are reflected by the same inference rules ( 1 ) - ( 5 ) plus one new inference rule ( 6 ) , called the M-hypersubstitution rule. This additional rule was described by Graczyriska and Schweigert in [56]for M = H y p ( 7 ) .

Definition 3.5.3 Let M be a monoid of hypersubstitutions of type 7 . The following are the rules of derivation for Mhyperidentities:

(1)

0 t

t

= t (for any term t E W , ( X ) ) ,

M-~YP

(3) {tl = t 2 , t 2

= t33)

tl

= t3,

M-~YP

for every operation symbol

( 5 ) let t ,t', r E W , ( X ) and let

fi

(i E I ) ,

t", t"' be

the terms obtained from

t , t' by replacing every occurrence of a given variable x by

I-.Then

{t

= t'} t t=

E

X

(the substitution rule),

M%P

( 6 ) {tl = t 2 )

t

6 [ t l ]= 6 [ t 2 ]for , any hypersubstitution o E

M-~YP

M (the M-hypersubstitution rule). We write C

wl

=

t

wl

=

w2 if there is a formal deduction of

M-~YP

w2 starting with identities in C and using the derivation rules ( 1 ) - ( 6 ) . If we apply the usual rules ( 1 ) - (5) for equational logic we will write C t wl = w2. Note that rule ( 6 ) commutes with all the derivation rules ( I ) - ( 5 ) . That is, if C is a set of identities closed with respect t o the rule ( 6 ) , then all consequences of C by the rules ( 1 ) - ( 6 ) are consequences of C by the rules ( 1 ) - ( 5 ) . Therefore we have:

Lemma 3.5.4 ( [ 5 5 ] )For M

=

'Flyp there holds C

k M-~YP

t2 u Xf$]

t tl = t 2 .

tl

=

75

3.5 Derivation of M-Hyperidentities

Proof: +== This direction is clear, since every identity from xff [C] can be derived from the set C by applying the hypersubstitution rule ( 6 )

+Since C C X f f [C],we have C t t l = t2 +X g[C] k M-~YP M-~YP t l = t 2 . Thus we have to prove that every consequence of X f f [ C ] by application of the rules ( I ) - (5) is closed with respect t o the hypersubstitution rule (6). We use induction on the length of a proof. If t l and t 2 are the same terms then X f f [ C ]t 6 [ t l ]= 6 [ t 2 ] for any hypersubstitution a. If t l = t 2 is a consequence of t2 = ttl by application of the symmetry rule then 6 [ t l ]= 6[t2]is a consequence of 6[t2]= 6 [ t l ]by symmetry. If t l = t3 is a result of application of transitivity on t l = t 2 and t 2 = t3 then 6 [ t l ]= 6[t3] follows from 6 [ t l ]= 6[t2]and 6[t2]= 6[t3]applying the transitivity rule. If fi(tl, . . . , tn,) = ffi(ti,. . . , t;,) is a consequence of the identities t j = t i , ( I j n i ) , by the replacement rule (4) then ~ ( f ~ ) ( 6 [. .t, 6 ~ []t n, .i ]= ) a ( f i ) ( 6 [ t i.]. ,. , 6[tLi])is a consequence of 6[ti]= 6 [ t ; ](,1 j n i ) ,by the same rule using ~ ( f i If ) .tl = t2 is a consequence of t l = t 2 by replacing every occurrence of a given variable x E X in t l and t 2 , respectively, by a term r then 6[&]= 6[12]is a consequence of 6 [ t l ]= 6 [ t 2by ] replacing every occurrence of x by 6 [ r ]This . means that t o obtain a proof of t l = t 2 (as a hyperidentity), we can first apply rule (6) and then rules ( I )

< <

< <

-

(5).

We write C

I=

tl

= t 2 if

the implication A

M-~YP

tl

= t2 for

I=

C

+A

M-~YP

any algebra

A

E

Alg(r) is satisfied. Then we

M-~YP

have the following completeness and consistency theorem of M -

hyperequational logic: Theorem 3.5.5 For arbitrary sets C C W , ( X ) x W , ( X ) and any identity t l = t2 E W , ( X ) x W , ( X ) ,

Proof: The proposition C

I=

M-~YP

tl

= t 2 is equivalent to t l = t 2 E

76

3 M-Hyperidentities and M-solid Varieties

HMIdHMModC= IdHhf ModC = ~ d M o d ~ f ~ B[ yCthe ] . completeness theorem of equational logic (described in Section 1.4),this is [C]t tl RZ t 2 ,which in turn is equivalent t o equivalent to

4 Hyperident it ies and Clone Identities An identity is a formula in a first order language with equality where the variables are bound by the universal quantifier. Let us consider the associative law as an example:

The associative law is satisfied (true) in an algebra A = (A; f A ) with a binary fundamental operation f A if when any elements of A are substituted for X I , 2 2 , x3 and the operation symbol . is replaced by the operation f the resulting elements from A are equal. Algebras with a binary associative operation, that is semigroups, are models of the associative law. The associative law can also be viewed as a formula of the form

*,

This is an example of a formula in a second order language, where now quantification of operation symbols over the set of all binary term operations of the algebra A is allowed. The set T2(A)of all binary term operations of the algebra A together with the ternary superposition operation

2,A

2,A

and the binary projections el , e2 forms an algebra 'T2(A) := (T2(A); e?lA,e;lA), which is an example of a unitary Menger algebra of rank 2 (see 1.3). The algebra 'T2(A) is also called the binary clone of A. Now we will show that hyperidentities, consisting of n-ary terms, in the algebra A, that means, sentences in a second order language correspond to identities, that means, sentences in a first order language, in the unitary Menger algebra Tn(A) of rank n. S21A,

78

4.1

4 Hyperidentities and Clone Identities

Menger Algebras of Rank n

First of all we want t o give the definition of unitary Menger algebras of rank n which we already mentioned in 1.3.

+

Definition 4.1.1 An algebra (111;S n , e l , . . . , en) of type r = ( n 1 , 0 , . . . , 0 ) is called a unitary Menger algebra of rank n if it sa,tisfies the axioms ( C l ) ,(C2), (C3):

(C2) Q n ( A j 7 X I ., . , X n ) FZ X j for 1 (C3) Sn(xj, XI,. . . ,A,)

M

< j < n,

X j for 1 5 j 5 n.

+

(Here 3" is an ( n 1)-ary operation symbol, X I , . . . , A, are nullary operation symbols and X o ,X I , . . . , X,+l, Y l ,. . . , Y, are variables.)

+

We notice that an algebra (111;S n ) of type r = ( n 1 ) which satisfies ( C l ) is called a Menger algebra of rank n . In 1.3 we introduced the algebra n - clone7 with W,(X,) as universe, with the superposition operation Sn for n-ary terms as (n+l)ary operation and with x l , . . . , x, as nullary operations.

Proposition 4.1.2 The algebra n algebra of rank n .

-

cloner is a unitary Menger

Proof: We have to prove that ( C l ) , (C2), (C3) are satisfied. ( C l ) We give a proof by induction on the complexity of the term which is substituted for Xo. Substituting for Xo a variable xj, then by the definition of Sn we have S n ( x j ,Sn(t17~ 1 . ., . 7 ~ n ). ., . , Sn(tn7~ 1 . ., . 7 ~ n ) ) = Sn(tj7 ~ 1 ,. .. , s,) = S n ( S n ( x j , t l.,. . , t n ) ,S l , . . . , S,). If we substitute for Xo the term t = f i ( r l , . . . ,r,,) and assume inductively that ( C l ) is satisfied for r l , . . . , r,,, then S n ( f i ( r 1 , . . , r n z )S, n ( t l ,~ 1 ,. .. , ~ 7 3 .) .~, Sn(tn, . ~ 1 7 ..., ~ n ) ) = f i ( S n ( ~Sl n 7 ( t l ,~ 1 ,. . , ~ n ). ., . , Sn(tn,~ 1 7 .. .7 ~ n ) ). ., . 7 S n ( ~ n Sn(t17 i, ~ 1 ,. . .7 ~ n ). ., . 7 Sn(tn7~ 1 ,. . .7 sn))) = S n ( f i ( S n ( r 1 7 t l.,.., t n ) , . . . , S n ( r n i , t l , . . , t,)), ~ 1 ,. . , s,) = S n ( S n ( f i ( r 1 7..,.rn,%), t17... , t n ) , ~ 1 7 ..., s,) = S n ( S n ( t , t ,l . . . , t , ) , S l , . . . , S,).

4.1 Menger Algebras of Rank n

79

(C2) This equation corresponds t o the definition of Sn. (C3) We give a proof by induction on the complexity of the term substituted for Xj. If X j is one of the variables xi E X,, then S n ( x i , x l , . . ,x,) = xi. Assume that for X j a term s = fi(sl,. . . , s,,) is substituted and that Sn(sj,X I , . . . , x,) = sj for 1 j n. Then S n ( f i ( s l , .. . ,s,J,x1,. . . ,x,) = f i ( S n ( s l , x l , .. . ,x,),. . . , s n ( ~ n z , x.1. ,.,x,)) = fi(~l,...,S,J which finishes the proof.

< <

Since ( C l ) , (C2), (C3) are identities, the class of all unitary Menger algebras forms a variety which is denoted by Vnr Now we consider an n-ary type 7, where every operation symbol fi has the same arity n , n 1. It is easy t o see that in this case the set of all strongly full terms W;F(X,) (introduced in 1.3) together with Sn forms a Menger algebra of rank n. Let FvA,(Y) be the free algebra with respect t o VM, freely generated by y = {Y, I i E I), where y is a new alphabet of individual variables indexed by the index set I of the operation symbols fi. (y) and by By & and XI,. . . , A, we denote the operations of FvA, FTn:= { fi(xl, . . . , x,) I i E I) we denote the set of so-called fundamental terms. Then we are able t o prove:

>

Theorem 4.1.3 The algebra n - clone% is free with respect to the variety VM of unitary Menger algebras of rank n, generated by the set FTn.

-

Proof: We prove that n - cloner, is isomorphic with FvA4 (31). We define a mapping cp : WTn(X,) FvA,({Y, I i E I}) inductively as follows:

The homomorphism property can be proved by induction on the complexity of the term to. If to = x j for some 1 j n , then cp(Sn(xj,tl, . . . , t,)) = ~(tj) n ( j ,( 1 ) . . , ( = Sn(cp(to),cp(tl), . . . , v(t,)). Inductivel~,

<

<

80

4 Hyperidentities and Clone Identities

Thus cp is bijective on the generating set of both algebras and therefore cp is an isomorphism. In a similar way one can prove that W z F ( X , ) is closed under S n and that the algebra clonesFr, := (w;~(x,); S n ) is free with respect to the variety of all Menger algebras of rank n . For more background on Menger algebras see e.g. [94] ( [ 6 6 ] )We . have seen that the free algebra n - cloner, is generated by the set FTn = { f i ( x l , .. . , x,) I i E I ) . Therefore any mapping 7 from FTn into WTn(X,) can be uniquely extended to an endomorphism rj of n - cloner,. Such mappings are called clone substitutions. Let Substc be the set of all such clone substitutions. Together with a binary composition operation defined by 71 72 := 7 , o 72 where o is the usual composition of functions and with the identity mapping id on { f i ( x l , . . . , x,) I i E I ) we see that (Sub&; 0 ,i d ) is a monoid. Then we have:

Proposition 4.1.4 The monoids (Hyp(r,); o h , aid) and (substc; a,i d ) are isomorphic. Proof: We define a mapping cp : Substc + Hyp(r,) by q H q o aid. Actually, q o aid E Hyp(r,) and cp is well-defined. The

81

4.1 Menger Algebras of Rank n

mapping cp is surjective since any hypersubstitution a E Hyp(r,) can be obtained as p(q) for q = aoaidl. Since a i d defines a bijection between the sets { fi I i E I) and { f i ( x l , . . . , xni) I i E I} the existence of ozl is clear. The injectivity of 9 is also clear. Because of the freeness of n - cloner, every substitution 7 can be uniquely extended to an endomorphism 7 of n - cloner,. To show that cp is a homomorphism we first verify the equation

Indeed, for terms t

=

fi(21, . . . , x,) we have

Inductively, for t = fi(tl, . . . , t,) we assume that (*) is satisfied for the terms tj, j = 1 , . . . , n and we have

Clearly the monoid (Substc; O, id) is isomorphic with the endomorphism monoid of n - cloner,. Let V be a variety of type r, and let Id,V be the set of all identities of V consisting of n-ary terms. Then we have:

Corollary 4.1.5 Id,V is a congruence on n

-

cloner,.

Proof: We have to prove that from s = t, si = ti E Id,V, i = 1 , . . . , n there follows Sn(s,s l , . . . , s,) = S n ( t , t l , . .. ,t,) E Id,V. At first we prove by induction on the complexity of the term t that for every n E N+ from t l = s1, . . . , t, = s, E Id,V there

82

4 Hyperidentities and Clone Identities

< < < <

i n, follows S n ( t , t l , . .. ,t,) M S n ( t ,~ 1 ,. . , s,). If t = xi, 1 then S n ( t , t l , .. . ,t,) = ti M si = S n ( t ,s l , . . . , s,) E Id,V. Assume now that t = fi (11 , . . . , I,) and that for l j , 1 j n , we have already S n ( l j , t l , . . ,t,) M S n ( l j , s l , . . ,s,) E Id,V. Then S n ( t ,t l ,. . . , t,) = fi(Sn(l17 t l , . . . , t,), . . . , Sn(1,, t l , . . . , t,)) M f i ( S n ( l l , s l, . . . , s,) , . . . , Sn(l,,sl , . . . , S,)) = S n ( t , s l, . . . , S,) E Id,V since Id,V is compatible with the operations of F T n ( X ) . The next step consists in showing

We recall that there is another possibility to define an algebra with the universe WTn(X,). We take for every operation symbol fi the n-ary operation f i : (WTn(X,)), WTn(X,) defined by f i ( t l ,. . . , t,) := f i ( t l , . . . , t,) and obtain the algebra FTn(X,) := (WTn(X,);( f i ) i E 1 )which is called the absolutely free algebra of type r, generated by X,. It is well-known that for a variety V of type 7, the n-ary identities Id,V form a fully invariant congruence relation on FTn(X,). Therefore from t M s E Id,V we obtain S n ( t ,s l , . . . , s,) M S n ( s ,s l , . . . , s,) E Id,V by substitution. Assume now that t M s , tl M s l , . . . , t, M s, E Id,V. Then S n ( t , t l , .. . , t n )M S n ( t ,s l , . . . , s,) M S n ( s ,s l , . . . , s,) E Id,V. The compatibility with the nullary operations of n-clone-r, is also clear.

-

If we consider only n-ary identities of a variety V then the concept of solidity has to be replaced by n-solidity.

Definition 4.1.6 Let V be a variety of type 7,. Then V is called n-solid if every identity from Id,V is satisfied as a hyperidentity in v. Now n-solid varieties can be characterized as follows:

Theorem 4.1.7 A variety V of type r, is n-solid ifl Id,V fully invariant congruence relation on n - clone%.

-

is a

Proof: By Corollary 4.1.5 Id,V is a congruence relation on n cloner,. Let V be n-solid and let cp : n - cloner, n - cloner, be an endomorphism of n - cloner,. Then for s M t E Id,V we have

4.1 Menger Algebras of Rank n p ( s ) = (cp o aid) [s]= ( p o I T i d ) A

A

83

[t]= cp(t)E Id,V since ( p o aid)

A

is the extension of a hypersubstitution. This shows that Id,V is fully invariant. If conversely, Id,V is fully invariant, s = t t Id,V and o E Hyp(r,), then 6 [ s ]= 6 [ t ]E Id,V since the extension of a hypersubstitution is a clone endomorphism. This shows that V is n-solid. By the last result we can form the quotient algebra n cloneV := n - cloner,/Id,V which belongs also t o the variety of unitary Menger algebras of rank n. This algebra is generated by the set { [ f i ( x l ., . . , x ~ ) I i] E ~ I ) .~ The ~ algebra n - cloner, is the quotient algebra of the absolutely free algebra F(,+l,0,...,o)(Y)/+{(c~),(c~),(c~))~ of type ( n 1,07 . . . 7 0) by the fully invariant congruence generated by ( C l ) ,( C 2 ) and ( C 3 ) over a countably infinite alphabet y . It is well-known (see [ l o ] )that fully invariant congruences on n - clone-r, correspond to fully invariant congruences on F(n+l,o,...,o)(y). Moreover, fully invariant congru( y )correspond to equational theories, i.e. to ences on F(n+l,o,...,o) sets of identities of certain varieties and that the quotient algebras are relatively free. This gives the following characterization of hyperidentities in varieties V of type 7, consisting of terms of arity n and of n-solid varieties.

+

Theorem 4.1.8 Let V be a variety of type r, and lets = t E Id,V. Then s = t is a hyperidentity in V ijf s = t is an identity in n - cloneV. Theorem 4.1.9 Let V be a variety of type 7,. Then V is n-solid iff n - cloneV is free with respect to itself, freely generated b y the set { [ f i ( z l ., . . , xn)]Id,V I 2 E I ) , meaning that every mapping from { [ f i ( x i , .. . , xn)]Id,V I 2 E I ) to the universe of n - clonev can be extended to an endomorphism of n - cloneV. Direct proofs of Theorem 4.1.8 and of Theorem 4.1.9 can be found in [25].These results can be generalized to clonesFr,. In this case instead of Hyp(7,) we consider the submonoid HypsF(r,) mapping each n-ary operation symbol fi t o a strongly full nary term. Then the variety V of type 7, is SF-solid if for every IT t HypSF(rn)and every s FZ t t IdsFV := Id,V n ( W c F ( X n ) ) 2 we have 6 [ s ]= 6 [ t ]E IdsFV. Identities s = t t IdsFV which are

84

4 Hyperidentities and Clone Identities

closed under all a E HypsF(r,) are called SF-hyperidentities. Then SF-hyperidentities correspond to identities in clonesFr,. The set IdsFV is a fully invariant congruence of clonesFr, iff V is SF-solid iff the quotient algebra clonesFV := clonesFr,/IdsFV is free with respect to itself.

4.2

The Clone of a Variety

In 3.1 we defined the multi-based (many-sorted, heterogeneous) algebra

and proved that this algebra satisfies the identities ( C l ) , (C2), (C3) (see Theorem 3.1.2). The algebraic theory of many-sorted algebras was developed independently by A. I. Mal'tsev ([64], multi-based algebras), P. J. Higgins ([58], graded algebras) and G. Birkhoff and J. D. Lipson ([9],heterogeneous algebras). For an approach t o heterogeneous algebras see also [102]. Subalgebras are sequences of subsets of the different universes which are preserved by the heterogenous operations, congruence relations are sequences of equivalence relations on the different universes which are preserved by the heterogeneous operations. In this way all universal-algebraic concepts from chapter 1 can be generalized t o the heterogeneous case. The equations ( C l ) , (C2), (C3) are identities in the heterogeneous sense. Therefore cloner is a member of a variety VMs of heterogeneous algebras. Heterogeneous algebras satisfying ( C l ), (C2), (C3) are called Menger systems. Let ~ r s ( ( y , ) n t N be +the ) free algebra with respect t o VMs, freely generated by a sequence (y,),,N+ of alphabets y, := {Y, I i E I,) with U I, = I . We denote n€N+

the operations of FvA4,((y,),,,,+) by S;, for 7-71n, E N+ and by (Xi)i5n,neN+.Let F," := { fi(x1, . . . , 2,) I i E I,) be the set of SOcalled n-ary fundamental terms, n E W+. Then we can generalize Theorem 4.1.3:

Theorem 4.2.1 The algebra clone7 is free with respect to the

4.2 The Clone of a Variety

85

variety VMs of Menger systems, freely generated b y the system (F,"),€N+. Proof:To prove that cloner is isomorphic with Fv,,,, ((y,)ntN+) we define a sequence of mappings

inductively as follows: (i) v n ( z j ) := X j , I

< j < n, n E,'N

Then the homomorphism property of cp, can be proved for every n E W+ as in the proof of Theorem 4.1.3. m. Clone substitutions are sequences (qn),,,+ with qn : F," + W, (X,). Every clone substitution can be uniquely extended to a clone endomorphism ( v , ) , ~ ~ +If. F,Sn denotes the set of all n-ary operation symbols of type 7,then a hypersubstitution can be regarded as a sequence (a,),,,+, a, : F,Sn + W, (X,). Proposition 4.1.4 gives a bijection between the set of all clone substitutions (q,),,,+ and the set of all hypersubstitutions of the form (a,),,,+. If V is a variety of homogeneous (one-based) algebras of type r, then the sequence (Id,V),,,+ turns out t o be a heterogeneous congruence on doner. As in the case of homogeneous algebras we may form the quotient algebra cloneV := cloner/(Id,V),,,+. The heterogeneous algebra cloneV is called the clone of the variety V . For every solid variety V the congruence relation (Id,V),,,+ is even fully invariant, meaning that for every endomorphism (cp,), ,+ of cloner, for every n E N+ and for every s = t E Id,V we have v n ( s ) = vn(t)E Id,V. Altogether,

Theorem 4.2.2 A variety V of type r is solid iff (Id,V),,,+ fully invariant congruence on cloner. For the proof we generalize Theorem 4.1.7

is a

86

4 Hyperidentities and Clone Identities

Theorem 4.2.2 can be generalized to M-solid varieties. In this case (Id,V),,,+ is not fully invariant, but invariant with respect t o all endomorphisms which extend the substitutions corresponding to the hypersubstitutions from M . Corresponding to Theorem 4.1.8 we get:

Theorem 4.2.3 Let V be a variety of type r and l e t s = t E IdV. T h e n s = t i s a hyperidentity i n V i f l s = t i s a n identity in cloneV. We have also:

Theorem 4.2.4 Let V be a variety of type r . T h e n V i s solid iff cloneV i s free with respect t o itself, freely generated by the sequence ([F:]~dn~)nEN+ = ( { [ f i ( ~ 1 ,. .. , z n ) ] l d n V I 2 E In))nEN+. Instead of the sequence (W,(X,)),,,+ one may consider the sequence (W,SF(Xn)),,,+. It turns out that this sequence of sets is closed under the superposition operations S$ and therefore is a heterogeneous algebra and sat((W:F(Xn))nEN+; (S$),,nEN+) isfies ( C l ) . Fore more generalizations and results see [17], [21] [23], POI, P I .

5 Solid Varieties of Arbitrary Type All solid varieties of type r form a complete lattice S ( r ) which is a complete sublattice of the lattice L ( r ) of all varieties of type r . For every submonoid M 'Flyp(r)the lattice S ( 7 ) is a complete sublattice of the lattice Snr(r)of all M-solid varieties of type 7 . If we can find solid varieties of any type r we have also M-solid varieties for type r . In this chapter we will prove that for each type there are infinitely many M-solid varieties.

c

5.1

Rectangular Algebras

The set Alg(r) of all algebras of type r is the largest variety of type r . It is a solid variety, since it is equationally defined by the identity x = x, which is satisfied as a hyperidentity and hence every identity is a hyperidentity. This makes Alg(r) the largest element of S ( 7 ) .On the other end, the smallest element is the trivial variety T of type 7 , which as the model class of the identity z = y is also solid. In this section we will prove that there is a least non-trivial solid variety of type r and will investigate this least non-trivial solid variety in the lattice S ( r ) .These results can be found in [go].

Definition 5.1.1 Let A = ( A ;(f))iE1) be an algebra of type r . Then A is called a projection algebra if every ni-ary fundamental operation is a projection on A, SO ft = e;"", for some 1 j ni. Let P, be the class of all projection algebras of type r and let RAT = V(P,) be the variety generated by the set P,. The elements of RA, are called rectangular algebras.

ft

< <

As an example we consider the type r = (2).Let us write RA2 for the variety of rectangular algebras of type r = (2).In this case we have exactly two projection algebras on A : ( A ; e:'") and ( A ; e;'"), the so-called left-zero and right-zero semigroups on A. Every projection algebra ( A ; is contained in the variety generated by the two-element projection algebra Bi = ({O, 1 ) ; e ~ ' { O ' l 'for ), i t { I , 2). e?lA)

88

5 Solid Varieties of Arbitrary Type

Therefore RA2 must contain these both varieties, and is the smallest variety of type T = (2) with this property. That is, RA2 is the join of V(B1) and V(B2). Also, B1 x B2 is in V(B1) V V(B2), so V(B1 x B2)is contained in V(B1) V V(B2).Conversely, V(Bl x B2) contains both V(B1) and V(B2),and hence contains also their join. This shows that V(B1 x B2) = V(B1) V V(B2). It is easy t o check that an equational basis for RA2 is the set { ( X ~ X ~ )M X ~x ~ ( x ~ M x X~ ~)Q , X: M x l ) and thus RA2 is the wellknown variety of rectangular bands. Rectangular bands have been considered by many authors (cf. [53] for references). We note that another example of rectangular algebras are the diagonal algebras, i.e. rectangular algebras of type T = (n) ([83]). It is easy t o verify that the variety RA2 of rectangular bands is solid. Let Hyp(2) be the set of all hypersubstitutions of type (2). We denote the hypersubstitution which maps the binary operation symbol f to the binary term t by at. Consider the set Hyp(2)l -RA,, which can be determined by using all elements of the two-generated free algebra with respect to RA2. These elements give the four elements a,,, a,,, a,,,, , a,,,, of Hyp(2)l -RA,. Applying these hypersubstitutions t o each of the basic identities {xl (22x3) M (x1x2)x3M 21x3, x: M xl} we get identities in RA2. This proves the solidity of RA2. Our aim now is t o show that the variety RAT is solid, for any type r, and that it must be contained in any non-trivial solid variety of type r . Proposition 5.1.2 Let V be a non-trivial solid variety of type 7. Then V contains all projection algebras of this type, and therefore contains the variety RAT of all rectangular algebras of type 7. ProoE Let B = ( B ; be any projection algebra of type 7. Since V is a non-trivial variety and since V is closed under formation of direct products, V contains algebras of arbitrarily large cardinalities, and we may choose an algebra A = (A; (f:)iEI) from V with 1B1 5 IAl. Consider the hypersubstitution a which assigns to every ni-ary operation symbol fi the ni-ary term zja E WT(X). Then the projection algebra A = (A; ( e ~ ; ' " ) ~is~just ~ ) the derived algebra a(A), which belongs t o the solid variety V. Every surjective mapping h : A + B is a homomorphism of a(A) E V onto the

5.1 Rectangular Algebras

89

projection algebra B. Consequently, B E V. To prove that RAT is solid, we will need an equational description of this variety. We introduce an identity basis EST, which we will eventually show does in fact define RAT. Let F = { f i I i E I} be a set of operation symbols of type r, and let EST be the set of the following identities: ( I D f ) , (AB;) and (Mf,,), where f ,g E F are operations with arity n and rn, respectively, and 1 j n.

< <

The absorption laws can be combined and are equivalent t o the identity

Note that for r = (2) and the binary operation symbol f (which will be skipped as usual) these identities are

which form an identity basis for the variety of rectangular bands. It is easy to check that every projection algebra of type r satisfies all identities in EST. Since RAT is the variety generated by all projection algebras, we have RAT C ModEsT. We first show that the converse is true for finite type r , and then later we introduce more complicated tools for the proof in the general case.

90

5 Solid Varieties of Arbitrary Type

Theorem 5.1.3 ( [ g o ] )Let T be a finite type, with the finite set { f i I i E I ) of operation symbols, and let A E M o d C s T . T h e n A is isomorphic t o a finite direct product of projection algebras. Proof: Let f be an n-ary operation symbol of type 7 from { f i I i E I ) . We are going t o show that there are n algebras A1,. . . , A, such that the following is true: (i)

A is isomorphic to the direct product A1 x . . . x A,

(ii) for every j E ( 1 , . . . , n ) the fundamental operation f projection.

is a

Doing this for each operation symbol of the finite set { fi li E I } and for each already obtained direct factor Aj, after a finite number of such decompositions we achieve direct factors which are projection algebras. In order to prove (i) and (ii), we fix an element a. E A, and consider Aj = { f A ( a o ,. . . , ao, b, ao, . . . , a o ) I b E A } , where b is at the j-th place. Claim I:

Aj is a subalgebra of A:

Claim 2: f A I A j = e t l " ~: Indeed, by ( A B f )we have f A ( f A ( ~ 0 ,. . , a07 bl, ao,. . . , ao), . . . , f A ( a o , . . . , a07 bn, ao,. . . , a o ) ) = f A ( a o , . . . , ao, b j , ao, . . . , ao), i.e. the restriction f A I A j is the j-th projection. Claim 3: Let I3 := A1 x . . . x A,. Then the mapping h : I3 + A defined by ( e l , . . . , c,) H f A ( c l , . . . , c,), ci E Ai, for 1 i n, is an isomorphism: Clearly, h is the restriction of f A : An + A onto B. We prove Claim 3 in three steps.

< <

a) The mapping h is surjective, since by ( A B f )and ( I D f ) we have b = f A ( b , . . . , b) = f A ( f A ( b , a o, . . . , a o ), . . . , f A ( a o , . . . , a 0 , b ) ) for

5.1 Rectangular Algebras

91

every b E A. b) The mapping h is injective: E ~Ai, for 1 i n. For Let f A ( x l, . . . , x,) = fA(yl, . . . , Y , ) , x ~ , Y any x E Aj we have x = f A(ao,. . . , ao, b, ao, . . . , ao) for some b E A and therefore f A(ao,. . . , ao, f A(ao,. . . , ao, b, ao, . . . , ao), ao7. . . , ao) = f A (ao7. . . , ao, b, ao, . . . , ao) = x by (ABj) . Consequently, again by (AB;) we get

< <

This theorem together with the remarks preceeding it give the following:

Corollary 5.1.4 ([go]) For any finite type T, the variety RAT is the model class of EST,i.e., RAT = ModEsT. For a finite type r the variety RAT is finitely axiomatizable (has a finite basis for its identities). In [90]the authors proved the following additional facts about rectangular algebras: (i) An algebra of finite type r is rectangular if and only if it is isomorphic to a finite direct product of projection algebras of type T: RAT = IPfi,(PT), where Pfi, means the formation of finite direct products.

5 Solid Varieties of Arbitrary Type

92

(ii) Every rectangular algebra of type r is isomorphic t o a subalgebra of a direct product of two-element projection algebras. (iii) The subdirectly irreducible algebras in RA, are precisely the two-element projection algebras of type r.

n

ni of all arities of the operation ( i 4 If no is the product no := %I symbols in r then there are up t o isomorphisms exactly no subdirectly irreducible algebras in RAT, and therefore RAT has 2"O subvarieties. Now we prove the solidity of the variety RA,.

Corollary 5.1.5 For any finite type

7,

the variety RA, is solid.

Proof: We have to prove that every identity of RAT is a hyperidentity. Following our general theory (see Theorem 3.3.3) it suffices t o show this for the identities of the generating system E S T .Since the variety RAT is generated by the set P, of all projection algebras of type r , and since every identity from the set EST is a hyperidentity in every algebra from P, (which is easy to check), every identity of Cs, is indeed a hyperidentity of RAT. We note that this proof will carry over to the non-finite case, once we have proved that RAT = ModEST holds in that situation too. For any type r we have RA, C ModCsT. In order t o show the converse, we introduce the idea of a normal form for terms in RA,. In a given variety V of type r, a normal form for terms is a mapping which assigns t o every term t E W,(X) a term N F ( t ) E W,(X) of the same arity such that

V

I= t = N F ( t ) and V I= t = tt u N F ( t ) = N F ( t t ) .

Again, we consider terms of a finite type r first.

Definition 5.1.6 Let 7; be a finite type with operation symbols { f l , . . . , f k ) , k > 1, where the arity of fi is ni, i = 1,2, . . . , k and let no := n l n z . . . nk. Then the no-ary term t, with the tree diagram given by

5.1 Rectangular Algebras

93

is called a general N F - t e r m for RA,. W e can recursively define t h e N F - t e r m (or normal form) t,, as follows: Let s l ( z l , . . . , z m l ) := f l ( z l , . . . , z n l ) ml , := n l , where t h e arity o f sl is nl and where s i + l ( x l , . . . xrni+1) := f i + l ( s i ( x l , . . . xrn2),si(xrn2+1,. . . 7 ~ 2 r n i ) ., . . 7 si ( ~ ( n ~ + ~ - ~ .~. r. 7 nxm2+1 ~ + l)), where t h e arity o f si+l is mi+l := ni+lmi,for 2 = 1 , 2 , . . . , k - 1. T h e n we define t, t o b e s k ( x l , .. . , z,,). 7

W e notice that for t h e types r,*w i t h operation symbols { f l , . . . , fi}, we have tC = " ( z l , . . . , zmt)for t h e terms si, ( 2 = 1 , 2 , . . . , k ) , as defined above. T h u s , i n particular, i f t l ,. . . , tni are N F - t e r m s o f t y p e r;, t h e n f i ( t l , . . . , tni)is an N F - t e r m o f t y p e 7;.

Lemma 5.1.7 Let

7;

be any finite type for k

> I.

Then:

( i ) There is a projection algebra A E P, with fi A -- e,:n> A , for each qi with 1 qi ni,i = 1 , . . . , k such that t$ = e,n o 4 with j = ql (q2 - 1 ) m l (qg - 1)mz . . . (qk - 1)mk-1 (where the mi are defined as above). Every j E { I , . . . , n o ) can be obtained for suitable qi.

+

< <

+

(ii) For any two N F - t e r m s s and st, if RA, st are identical.

+ +

+ s = st, then s and

(iii) Let s be an N F - t e r m and let x be a variable. T h e n RA, s = z if and only if z is the only variable i n s . Proof: ( i ) follows from t h e definition o f t,;.

+

5 Solid Varieties of Arbitrary Type

94

(ii) Let ~ ( x , , , , . . ,xun,) and sf(x,:,,.. . ,xVn,) be two N F terms. By (i) for each j E (1,. . . , n o } there is a projection algebra A E P, C RAT with :t = eyol". Thus s ( . . . ,x u , ) sf( x u, . . . , x , ) implies x u = RAT sA( , . . . , x u , ) = sfA (xu,, . . . , x u ) = xVi. Consequently, s(xu,, . . . , xun,) and sf(x,, , . . . , xun,) are identical.

+

(iii) Let s(x, . . . , x) be an NF-term such that x is the only variable s ( x , . . . x) = x. If occurring in the term. Then by ( I D f ) , RAT conversely RAT s(ul, . . . , u,,) = x then as in the proof of ( i i )we get x as the only variable.

+

+

Now we prove:

Lemma 5.1.8 Let

T = T : be any finite type. For each term t E W,(X) there is a unique NF-term N F ( t ) such that

(Here N F ( t ) is a normal form for t E W , ( X ) in the variety RAT of rectangular algebras of type r . )

Proof: The uniqueness of N F ( t ) follows from Lemma 5.1.7 (ii). We prove the existence of N F ( t ) by induction on both the complexity of the term t and on the number k of operation symbols f l , . . . , fk. If we have no operation symbols, then the corresponding terms are just the variables. In this case the proposition is trivially true since N F ( x ) = t,(x) = x. Let k 1. Assume that the proposition is true for every type T with operation symbols f l , . . . , fi. We are going to show that the existence is satisfied for a type T with operation symbols fl, . . . , fk. In fact, if x E W, ( X ) is a variable then N F (x) = s(x, . . . , x) because of Lemma 5.1.7 (iii). Now, let t E W,(X) be an arbitrary term of the form f i ( t l , . . . , t,,), i E (1,. . . , k), and assume that N F (tl), . . . , N F ( t n i ) already exist. Then by definition N F ( t i ) = fk(ti,, . . . , tink)for some NF-terms tj,, . . . , tin, of type rf with operation symbols {fl,. . . , f k p l ) for 1 j ni.

>

< <

If i = k then by (ABf,) E Cs7 we have RAT + t = f k ( t l , . . . , t n k ) f k ( f k ( t l 1 , . . . ,tlnk),. . . ,fk(tnkl, . . . 7 tnknk)) f k ( t l l 7 t 2 2 , . . . 7 tnknk).

95

5.1 Rectangular Algebras

The latter term is an IVF-term because of the remark after Definition 5.1.6. If i < k , then RAT f i ( t l , . . . ,tn,) f i ( f i c ( t 1 1 , . . . 7 t l n k ) , . . . 7 fic(tn,17 . . . tn,n,)) f i c ( f i ( t 1 1 , . . . 7 t n i l ) , . . . 7 f i ( t i n k , . . . 7 tn,nk)). Since f i ( t i l ,. . . , tn,l), . . . , fi (tlnk7 . . . , tnink) are terms of type 7' with operation symbols f l , . . . , f k P l , the normal form terms of these terms exist and RAT satisfies t = f k (fi (til,. . . , tn,l),. . . , fi (tlnk7 . . . 7 t n , n k ) ) = fic(NF(fi(ti1,. . . 7 t n , ~ ) ). ,. . N F ( f i ( t i n k ., . . 7 t n , n k ) ) ) , the last term being an IVF-term of type r with operation symbols f i , . . . , f k . Therefore N F ( t ) exists. Now we are ready t o prove that ModEST is contained in R A T , for any type r . For any term t E W T ( X )we construct a reduced normal form term N,F(t) in the following way: Step 1: Let rfi, be the finite type which is determined by all operation symbols occurring in t . Then we form the unique IVF-term t' := N F ( t ) in RAV,,. Step 2: For every subterm s = fi(st, . . . , st) of t' we apply ( I D f ) and substitute st for s. This gives us a unique reduced normal form , N,F : W T ( X )+ W T ( X ) term N,F(t) in the variety R A T f a nwhere is a map which assigns to every term its reduced normal form term. Since we need only identities from EST to produce the reduced nort = N r F ( t ) for every term t . For a mal form, we see that EST pair ( s ,t ) E W T ( X ) 2of terms we construct the reduced normal form terms with respect t o the finite type rfinconsisting of all operation symbols occurring in s or t . Then we have:

+

Lemma 5.1.9 Let r be an arbitrary type, and let s, t Then

Proof: +== If N,F(s) = N , F ( t ) , then since EST and EST t = IVrF(t) it follows that EST s RAT s = t .

E

WT(X).

+ s = N,F(s) =

t , and thus

5 Solid Varieties of Arbitrary Type

96

==+Let

RAT I= s = t. Together, Csr k s = N r F ( s ) , CsT k t = IV,F(t) and Csr C IdRA, imply that RAT N r F ( s ) = IV,F(t). Then using the properties of the projection algebras of type r and the definition of a reduced normal form term, we deduce that the reduced normal forms must be identical. The calculation is easy and can be left as an exercise t o the reader. This lemma gives the last piece needed t o complete our proof. If s = t t IdRA, then we obtain IV, F(s) = IV, F ( t ) , and since Csr IV,F(t) = t and Csr k IV,F(s) = s, we get Csr s = t. This shows that IdRA, C IdModCsr, and hence RAT ModCsr. Altogether we have our result:

>

Theorem 5.1.10 For any type r, the set Csr is a basis for all identities of the variety RAT of rectangular algebras. Thus RAT is the least non-trivial solid variety of type r .

5.2

Solid Chains

In the previous section, we showed that for any type there is at least one non-trivial solid variety. In this section, we show that there are infinitely many solid varieties of type r , for any r . Our approach here is to use the Galois connection (HhIId, HMMod): instead of defining varieties by sets of equations and then proving that they are solid, we will begin with hyperequational classes, which by definition are solid. The most results presented in this section can be found in [28].

Definition 5.2.1 Let r = (ni)iE1 be any type, consisting of operation symbols fi of arity ni, for i E I . Choose one symbol f j , of some fixed arity n = n j , and one fixed variable symbol z. We define terms ar, of type r inductively, by: al = fj(x, . . . , x), ar,+~ = f j b r , , . . . ,~ k ) . Then we set Vj = HMod{ak = ak+l). That is, Vk is the largest solid variety in which the identity a k = a k + l is a hyperidentity. By definition, then, each of the varieties Vkis solid, and our goal now is to prove that these varieties are all distinct, giving us an infinite chain of solid varieties. In order to do this, we shall need

97

5.2 Solid Chains

a number of technical lemmas which tell us about the structure of identities in the varieties Vk.We shall be particularly interested in the number of occurrences of the operation symbol f j , and the number of occurrences of any variable w, in terms and identities. For w a variable and t a term, we will denote by opj(t)the number of occurrences of f j in t , and by vb,(t) the number of occurrences of w in t.

Lemma 5.2.2 Let k vbx(ak)= n k .

> 1. Then opj(ak)= 1 + n + - - 0+ n k l , and

These formulas follow easily from the inductive definition of ak.

Lemma 5.2.3 Let a E H y p ( 7 ) with opj(o(fj)) > 1. Then opj (6[ak] ) > o ~(ak). j Proof: We give a proof by induction on the number k. For k = 1 , we have opj ( i [ a l ]= ) opj (i[ f j ( x ,. . . , x ) ] )= opj ( a (f j ) ( x ,. . . , x ) ) I + . . . + n"l- - opj ( a l ) .Then inductively we get o ~(e[ak+l]) j = o ~( e j [ f(ak, j . . . ,~ k ) ] ) = ~ ~ j ( ~ ( f j ) (. ~ . . [ea[ ak k]],) ) > 1 nopj ( 6 ( a k ) )since , opj ( a (f j ) ) > 1 , > l + n ( l + n + . . . + n k - l )= l + n + . . . + n k

>

+

Proof: If o p j ( a ( f j ) = ) 0, then o p j ( i [ a k ]=) ~ p ~ ( i [ a = ~+ 0 ,~ ] ) since in both terms ak and ak+l the only operation symbol is fj. Otherwise, when o p j ( o ( f j )> ) 1, we have both opj(6[ak]) and 0 p ~ ( 6 [ a ~>+1~ ]n) . . . nk-' by Lemma 5.2.3.

+ + +

Lemma 5.2.5 If for a E H y p ( 7 ) the term a(f j ) is not a variable, then v b x ( i [ a k ] )n k .

>

Proof: We proceed by induction on k . For k = 1 , we have vb, ( i [ a l ] ) = vbx(i[fj(x . .,. , x ) ] )= vb,(a( f j ) ( x ,. . . , x ) ) n = n l , since a ( f j )

>

98

5 Solid Varieties of Arbitrary Type

is not just a variable. Inductively, we have ubx(6[ak+l]) = vbx (G[fj ( a k , . . . , a k ) ] ) = vbz(a(fj)(&[ak], . . . , e[ak])) nk+', since each input 6[ak]contains at least n k occurrences of z by the induction hypothesis, and a(fj)must use this input at least n times.

>

>

Lemma 5.2.6 For any a E Hyp(-r)and any k 1, we have either ubx(6[ak])= ubx(6[ak+l]) or both ub, ( i [ a k ] ) and ub, (i[ak+l])

>

k

n. Proof: If a(fj) is a variable, then the result of applying a to any 1, is just x. In this case we get ubx(6[ak]) of the terms a,, for p = ~ b , ( i [ a ~ + ~= ] )1. Otherwise, if a(fj) is not a variable, the claim follows by Lemma 5.2.5.

>

Lemma 5.2.7 Let k > 1. A n y identity u = v which holds in the variety Vk satisfies the condition (A) For all variables w, either ubw(u)= ubw(u) or both ubw(u) and ubw(u) are greater than or equal to n k . Proof: We have IdVk = ModC for C = {6[ak]= 6 [ ~ k + ~I ]a E Hyp(-r)). This means that IdVj is the closure, under the five deduction rules, of the set C. We proceed by induction on the generation of identities from C, using these rules. First of all, it is clear that any identity in C satisfies condition (A): the claim for w = z follows from Lemma 5.2.6, while for any w # z we have no occurrences of w in any identity in C . Now we consider the application of each of the five derivation rules t o identities with property (A). ( I ) For any variable w and any term s , we have vbw(s) = ubw(s), so identities of the form s = s satisfy condition (A).

(2) If s t = s.

= t is an identity which satisfies condition

(A), then so is

(3) Let s = t and t = u be identities which satisfy condition (A), and let w be any variable. There are four cases to consider:

5.2 Solid Chains a) vb,(s) = vb,(t) and vb,(t) = vb,(v), b) vb,(s) = vb,(t) and vb,(t), vb,(v) nk, c) vb,(s), vb,(t) nk and vb,(t) = vb,(v), d) vb,(s), vb,(t) nk and vb,(v) nk. In all cases, we have either vb,(s) = vb,(v) (in case a) or both vb,(s) and vb,(v) n k . Thus rule (3) leads to identities which satisfy condition ( A ) .

>

> > >

>

(4) Let i E I, and let s, FZ t,, for 1 5 r 5 ni be identities of Vkwhich satisfy condition ( A ) . We consider the new identity fi ( s ~. ., . , s,,) FZ fi ( t l ,. . . , t,,). Let w be any variable. If vb, (s,) = vb, (t,) for all 1 < r < ni, then our new identity also has an equal number of occurrences of w on each side. But if there is some 1 < r < ni for which vb,(s,) # vb,(t,), then both of these numbers are > n k ,and hence in the new identity both sides also have nk occurrences of

>

w. Thus in either case the new identity satisfies condition (A).

(5) Let s FZ t be an identity of Vkwhich satisfies condition ( A ) ,and let y be a variable and r be a term. We consider the new identity s = sub(r,y, s ) = sub(r,y, t ) = f , where sub(r,y, s ) denotes the term obtained from s by substituting the term r for each occurrence of the variable y. Let w be any variable. We consider two cases, depending on whether the variables w and y are equal or not. a) If w = y: Then we have vb, ( s ) = vb, ( s ).vb, ( r ), and similarly for vb,(t). If vb,(r) = 0, this gives vb,(s) = vb,(t) = 0. Otherwise, if vb,(r) > 1, we apply condition ( A ) t o the identity s = t and the variable y: either vb,(s) = vb,(t), or both are > n k . Then either vb,(s) = vb,(q, or both are n k , and the identity s = f fits condition ( A ) . b) If w # y: In this case we have vb, ( s ) = vb, ( s )+ vb, (s).vb, ( r ) , and similarly for vb, (t).If vb, ( r ) = 0 , then applying condition ( A ) t o s = t and each of the variables w and y means that 3 = t also satisfies condition ( A ) .A similar argumentation can be used if vb, ( r ) 0.

>

>

>

Lemma 5.2.8 Let k 1. Any identity u = v which holds in the variety Vk satisfies the condition ( B ) either opj ( u ) = opj ( v ) , or both opj ( u ) and opj ( v ) are 1 n . . . + nk-1.

> + +

5 Solid Varieties of Arbitrary Type

100

Proof: With C = { i [ a k ]FZ i [ a k + l ]I a E H y p ( r ) } with a k as defined in Definition 5.2.1, we proceed by induction on the generation of identities from C , using the deduction rules. It follows from Lemma 5.2.4 that any identity in C satisfies condition ( B ) and we have t o consider the application of each of the five deduction rules to identities satisfying property ( B ) .

( 1 ) For any term s , we have o p j ( s ) = o p j ( s ) ,so identities of the form s = s satisfy condition ( B ) . ( 2 ) If s = s.

FZ

t is an identity which satisfies condition ( B ) , then so is

t

and t = v be identities which satisfy condition ( B ) . We consider four cases: a) opj ( s ) = opj ( t )and opj ( t )= opj ( v ) , b) o p j ( s ) = o p j ( t ) and o p j ( t ) ,o p j ( u ) 1 n - - - n k p l , c) opj ( s ) , opj ( t ) 1 n . . nkpland opj ( t )= opj ( u ) , n"'. d) o p j ( s ) ,o p j ( t ) ,o p j ( v ) 1 n In all cases, we have either opj ( s ) = opj ( v ) (in case a) or both opj ( s ) and o p j ( v ) 1 n . . n k - l . Thus rule ( 3 ) leads t o identities which satisfy condition ( B ).

( 3 ) Let s

=t

> + + + > + + + > + + + > + + +

( 4 ) Let i E I, and let s, = t,, for 1 < r < ni be identities of Vj which satisfy condition ( B ) .We consider the new identity f i ( s l , . . . , s,,) = f i ( t l ,. . . , t,,). If opj(s,) = opj(t,) for a11 1 < r < ni, then our new identity also has an equal number of occurrences of f j on each side. But if there is some 1 5 r 5 ni for which opj (s,) # opj (t,), then both of these numbers are 1 + n + . . + n k p l ,and hence in the new identity both sides also have > 1 + n + . . + nk-I occurrences of f j . Thus in either case the new identity satisfies condition ( B ) .

>

( 5 ) Let s FZ t be an identity of Vkwhich satisfies condition ( B ) ,and let y be a variable and r be a term. We consider the new identity 3 = sub(r, y, s ) = sub(r, y, t ) = f. We note that the number of occurrences of the symbol f j in the new term 3 is the number in the term s , and any new ones introduced when y is replaced by r ; the number of such new occurrences is the number of occurrences of y in s , multiplied by the number of fj's in r . This gives us opj(3)

101

5.2 Solid Chains

+

opj ( s ) ub,(s) . opj ( r ) , and similarly for opj (q. If opj ( r ) = 0 , we have opj ( s ) = o p j ( s ) and opj ( t )= opj (t),so from condition ( B ) on the identity s = t we have either o p j ( s ) = o p j ( t ) or both are > 1 n . nk-l. If opj ( r ) # 0 , we see that opj ( 3 ) and opj (t)are equal if both o p j ( s ) = o p j ( t ) and vb,(s) = vb,(t) hold. Otherwise, by condition (A) of Lemma 5.2.7 on vb,(s) and condition ( B ) on s = t , and using the fact that n" 1 n . nkpl, we must have both o p j ( g ) and o p j ( t ) 1 n nkpl. =

+ + +

+ + + > + + +

Now we can prove our result:

Theorem 5.2.9 T h e varieties Vk, for k of distinct solid varieties of type r.

> 1, form a n infinite chain

Proof: It is clear that the identity a k + l = a k + z is a consequence (by one application of deduction rule (4)) of the identity a k = ak+l. Hence we always have Vk Vk+l, and the varieties Vk form a chain. But it follows from Lemma 5.2.8 that the identity a k = a k + l does not hold in the variety Vk+l since the two terms contain different numbers of occurrence of f j , but the term a k has only 1 n . . - nkpl occurrences of the symbol f j . Thus the varieties Vk in our chain are all distinct.

c

+ + +

6 Monoids of Hypersubstit ut ions In this chapter we want to investigate the structural properties of monoids of hypersubstitutions. The Galois connection between subvariety lattices and monoids of hypersubstitutions considered in 3.4 motivates t o study the monoid properties of E y p ( r ) and its submonoids. We are interested in finite monoids, in injective and bijective hypersubstitutions, in idempotent hypersubstitutions, in regular elements, we want t o determine the order of hypersubstitutions and consider Green's relations. Finally we show that together with an addition of hypersubstitutions the set of all hypersubstitutions of a given type forms a 1ejLseminearring. First of all we want t o recall some basic facts from semigroup theory.

6.1

Basic Definitions

We want t o start with some remarks on the order of an element a of a monoid M. The order of a is defined, as in group theory, to be the order of the subsemigroup (a) = {a, a 2 , .. .). If there are no repetitions in the list a , a2,. . ., that is, if

then the semigroup generated by a is isomorphic to the semigroup (N+;+). In this case we say that a has infinite order. If there are repetitions among the powers of a , then the set {n E N

1 312' E $(an

=

an', n

# n')}

is non-empty and so has a least element. Let us denote this least element by m and call it the index (or pre-period) of the element a. Then the set {X E W+ I am+z = am) is non-empty, and so it too has a least element r which is called the period of a. Then the following properties are satisfied (see [59]):

6 Monoids of Hypersubstitutions

104

(ii) for a11 u, v E w+,am+u- am+"iff u to v modulo r ) ,

(iv) Ka := {am,am+', . . . , am+'-')

-

v(mod r ) (u is congruent

is a cyclic subgroup of (a).

An element a of a monoid M or of a semigroup S is called idempotent if a 2 = a. Therefore, the order of an idempotent element is 1. Let E(S) ( E ( M ) ) be the set of all idempotent elements of the semigroup S (of the monoid M ) . If a semigroup S with at least two elements contains an element 0 such that, for all x E S,

we say that 0 is a zero element (or a zero) of S, and that S is a semigroup with zero. Clearly, any zero element is idempotent. An element a is called regular if there is an element z in M (in S) such that a = a m . Let Reg(M) (Reg(S)) denote the set of all regular elements of M (of S ) . Clearly, every idempotent element of M (S) is regular and if x E M (S) is regular such that x = xyx for some y E M (S),then xy and yx are regular elements of M (of S ) . A monoid (semigroup) is called regular if all its elements are regular. It is not difficult t o see that the full transformation semigroup 'FtA consisting of all mappings f : A + A defined on the set A is regular and that the semigroup PAof all partial mappings defined on A is also regular. A non-empty subset A of the universe S of a semigroup S is called a left ideal if SA A, a right ideal if A S A and a (two-sided) ideal if it is both, a left and a right ideal. Here SA is the complex product of S and A, that is the set {sa I s E S,a E A). Ideals for monoids are defined in the same way. The smallest ideal containing an element a E S is called the principal ideal I ( a ) generated by a. It consists of all elements of the form paq for some p and q in S . An ideal is called proper if it is different from S . A semigroup without zero is called simple if it has no proper ideals. A semigroup S with zero is called 0-simple if

c

c

6.1 Basic Definitions (i) (0) and S are its only ideals, (ii) SS # S.

Green's relations are special equivalence relations which can be defined on any semigroup or monoid, using the idea of mutual divisibility of elements. They are defined in the following way. For any monoid M and any elements a , b of M , we say aCb iff there are c and d in S such that ca = b and db = a. Dually, a R b iff there are c and d in S such that ac = b and bd = a. It follows easily from these definitions that C is always a right congruence, while R is always a left congruence. The relation IFt is defined as the intersection of R and L, and the relation 2) is the join C V R . It is easy to see that C V R = R o C = C o R, where o here refers t o the usual composition of relations. Finally, the relation J is defined by a J b iff there exist elements c, d, p and q such that a = cbd and b = paq. For Green's relations we will use the following notation, (a, b) E R and a R b and similarly for the other relations. For more information about Green's relations in general, we refer the reader to [59]. Using Green's relation R (L) the set of all regular elements of a semigroup S can be described as follows (see e.g.[59]):

Theorem 6.1.1 Let x be an element of the universe of a semigroup S . T h e n the following are equivalent: (i) x is regular. (ii)

[XI,

contains a n idempotent element.

(iii) [xIL contains a n idempotent element. Then for the set R e g ( S ) of all regular elements of the semigroup S we have

Let a , /3 be elements of HA. We denote by I m a the set of all images of a. For Green's relations we have: (a) (a,p) E R if and only if I m a

=

Imp

106

6 Monoids of Hypersubstitutions

(b) (a,p) E 4 . (c)

C if and only if kera

(a,p)E D

if and only if IIrnal

=

=

kerp (kera is the kernel of

IIrnPI.

A local subsemigroup of a semigroup S is any subsemigroup of the form eSe, where e is an idempotent of S. It is known (see [59]) that if two idempotents e and f are D-related, then the corresponding local subsemigroups eSe and f Sf are isomorphic. If a is a hypersubstitution of type r, then its extension i is a transformation on WT(X). Therefore in this case we can use (a), (b), (c) and (d). Let 7, be an n-ary type with only n-ary operation symbols. By Proposition 4.1.4 and the freeness of n - elonern for any type r, the monoid IFtyp(rn) is isomorphic to the endomorphism monoid of n - cloner, and the extended hypersubstitutions 6 are endomorphisms of n - cloner,. As a consequence, for any hypersubstitution a the set I m i is the universe of a subalgebra of n - elonern. By Theorem 4.2.1 this subalgebra is generated by the set {a(fi)I i E I) (regarded as graded set of terms). Let (T) be the subalgebra of n - cloner, generated by the set T of terms. Now RHyp(T,) denotes Green's relation on Hyp(r,) and REnd(n-cloneT,) is Green's relation on E n d ( n - cloner,). We will skip the subscript if there is no doubt t o which monoid we refer. Then (a) gives

Corollary 6.1.2 Let al, a 2 be two hypersubstitutions of type rn. Then ( ~ 1 , ~ E 2 )R H ~ ~ ( ~((01 , ) (fi) 1 E I)) = ( ( a 2 (fi) I i E I ) ) . Further we have:

Proof: We will give a direct proof of this proposition. Indeed, we have + 30, a' (a1oh a = 02 and a 2 oh a' = 0,) (al,a2)E RHyp(Tn) + (al oh a)^= 6, o 6 = i2and e2o 6' = i1 (61,6 2 ) E R~nd(n-clone~,) . Conversely, if (el,i2) E REnd(n-cloneTn), then I m i l = I m i 2 and

*

107

6.1 Basic Definitions

I}) = ({a2(fi) I i E I)) for the clones generated by the sets {al(fi) I i E I) and {a2(fi) I i E I}, respectively. But this means, al(fi) E ({a2(fi) I i E I}) = Im62 and 0 2 ( f i ) E ({ol(fi) I i E I)) = I m e l , respectively and then there are sequences (ti)iEIand (ti)iEIof terms where ti, ti are ni-ary such that ({al(fi)

I

i

E

i E I . There are hypersubstitue,[ti] = al(fi) and i l [ t i ] = a2(fi), tions a, a' with a(fi)= ti, at(fi)= ti, i E I and then 62[a(fi)] = ( 0 2 oh 0)(fi) = 01 (fi) and 61[a'(f i ) ] = (a1 oh a') ( f i ) = a 2 ( fi) for all i E I and thus (al,a2)E RHyp(T,).

Clearly, the "+"-direction is also satisfied for the other Green's relations. We notice that the opposite is not satisfied for Green's relation CHyp(Tn). TO consider a counterexample, for any term t E WT(X) we need the following notation:

le ftmost(t)

-

rightmost(t)

-

the first variable (from the left) which occurs in t, the last variable which occurs in t.

Now we assume that F is a variable over the two-element alphabet {f,g}. For an arbitrary binary term t of type T = (2,2) we define two semigroup terms Lp(t) and Rp(t) over the alphabet {f,g} inductively as follows: (i) if t

=

F ( x i , t2), t 2 E W(2,2)(X2),xi E X2,then Lp(t) := F ;

(ii) if t

=

F(tl,zi),tl

E W(2,2)(X2), zi E

X2,then Rp(t) := F;

If t is visualized by a tree, then Lp(t) is the left path from the root to the leaf which is labelled by the leftmost variable in t and Rp(t) is the right path from the root to the leaf which is labelled by the rightmost variable in t. For example, if t has the form

6 Monoids of Hypersubstitutions

then Lp(t) = g f and Rp(t) = g f f . We denote by op(Lp(t)) and op(Rp(t)) the numbers of operation symbols occurring in the semigroup words Lp(t) and Rp(t), respectively. Now we consider the type r = (2) with the binary operation symbol f and the hypersubstitutions ol,o2, which map f to the terms f ( ~ 1~, 1 and ) f (f (f ( ~ 1X,l ) , f ( ~ 1X,l ) ) , f (f ( ~ 1X,l ) , f ( ~ 1z, l , respectively. 2) Obviously, (01 0 2 ) # C ~ ~ p ( but kero1 = u s , t) I 6 f ( X I , X l ) [sl = 6 f ( X l , X l ) [ t l l = {(s,t) I le f tmost(s) = le ftmost(t) and op(Lp(s)) =0~(L~(t))l= keri2. From the last equation and from (b) we obtain 61CEnd(2-,l,n,(2))62. From (b) we have

By Theorem 3.1.7 for each hypersubstitution a of type 7 the relation kera (which is by definition equal to ker6) is a fully invariant congruence relation on the absolutely free algebra F T ( X ) . Therefore hypersubstitutions a1 and 0 2 which are C-related t o each other define the same variety Modkeral = Modkera2. Usually we will write R instead of RHyp(T) and similar for the other Green's relations. Let t E WT(X) be a term of type r. Throughout this section we will also use the following notation:

6.2 Injective and Bijective Hypersubstitutions

ops ( t ) f irstops ( t )

6.2

-

the set of all variables occurring in t , the number of all variables occurring in t , the number of occurrences of the variable xj in t , the set of all operation symbols occurring in t , the first operation symbol (from the left) occurring in t , the number of occurrences of the operation symbol f in the term t , the number of occurrences of operation symbols before the first occurrence of a variable letter, the number of occurrences of operation symbols after the last occurrence of a variable letter.

Injective and Bijective Hypersubstitutions

We are interested in all hypersubstitutions a for which their extension i is injective or even bijective. Injectivity of i is equivalent with kera = Aw,(x). To formulate our results we need the concept of a regular hypersubstitution.

Definition 6.2.1 A hypersubstitution a of type r is said to be regular if var(a(f i ) ) = { x l ,. . . , xni) for all i E I . Let Reg(r) be the set of all regular hypersubstitutions of type T . (For T = ( n ) we will write Reg ( n ).) It is easy t o see that the set Reg(r)of all regular hypersubstitutions of type r forms a submonoid of 'Ftyp(r).Then for type r = (n) we have:

>

Proposition 6.2.2 ([27])If a E Hyp(n),n 2 and i [ t ]= i [ t t ] for two terms t , t' E W(,)( X ) , then t = t' or a is not regular.

> >

Proof: Assume that a is regular. Since n 2, the hypersubstitution a maps the n-ary operation symbol f t o a term which uses at least two variables and therefore depth(a(f )) 1. We will give a proof by

110

6 Monoids of Hypersubstitutions

induction on the depth o f t . At first we consider the case that t = xi is a variable. Then 6 [xi]= xi = i[t']. Since for t' = f ( t i ,. . . , t;) we have

depth(6[tf]) = depth(a(f)(6[t',], . . . , 6 [ t L ] )> ) 1 the term t' is also a variable and t' = xi, i.e. t = t'. Assume now that t = f ( t l ,. . . , t,) is a compound term and that from 6 [ t j ]= 6[ti] follows t j = ti for j = 1 , .. . , n. Then we have

Since a ( f ) uses all variables X I , . . , x,, this is true only if 6 [ t j ]= &[ti]for j = 1 , . . . , n. Now we can use the hypothesis of the induction and conclude that t j = ti for j = 1,. . . , n and this means, t = f ( t l ,. . . , t n ) = f ( t i ,. . . , t;) = t'. It is clear that Proposition 6.2.2 does not hold if we have more than one operation symbol. An easy consequence of Proposition 6.2.2 is the following corollary:

Corollary 6.2.3 Let a be a hypersubstitution of type r 2. Then the extension 6 is injective ijf a E Reg(n).

=

( n ) ,n

>

Proof: Assume that 6 is injective and o @ Reg(n) and that var(a(f )) = { x k l ,. . . , xkl) where { x k l ., . . , xk,} is a proper subset of (21,.. . ,x,). If t = f ( t l ,. . . , t,), t' = f ( t i , .. . , t;) and tkl = t , . . . , t = tkl, but t j # ti for at least one j E { I , . . . , n) \ { k l , . . . , k l ) , then 6 [ t k l= ] 6[tk,],. . . , 6 [ t k l= ] 6[tkl], and 6 [ t ]= S n ( o ( f )6, [ t l ]. ,. . ,6[t,])= S n ( a ( f )6[tl,], , . . . , 6 [ t ; ] )= 6 [ t f ] , but t # t'. This shows that o E Reg(n) must hold if 6 is injective. If conversely, a E Reg(n) and i [ t ]= i [ t f ]then , by Proposition 6.2.2 we can show that t = t' and hence i is in.jective. Now we consider arbitrary types r = (ni)iE1.In general, a hypersubstitution a maps different operation symbols t o different terms. If we ask for pairs ( t ,t') E kera with t # t', not only the variables occurring in t and in t' but also the operation symbols are important.

6.2 Injective and Bijective Hypersubstitutions

111

Definition 6.2.4 Let Hypreg(r) = Reg(r) n { a E Hyp(r) I f irstops(a(f i ) ) = fi for all i E I } . (Clearly, the set Hypreg(r) forms a submonoid of IFtyp(r).) Then we can prove:

Theorem 6.2.5 Let 7 = (ni)i,I,ni > 1, be an arbitrary type and assume that a E Hypreg(r). If a maps no unary operation symbol to a variable, then for arbitary terms t , t' E W , ( X ) we have 6 [ t ]= 6 [ t f ]ijf t = t'. Proof: Clearly, from t = t' follows 6 [ t ] = 6 [ t f ]Assume . that 6 [ t ]= 6 [ t f ]We . will use induction on the complexity of the term t . Let t = xi E X be a variable. From the regularity of a and from xi = &[xi]= 6 [ t f ]there follows t' = xi (since depth(6[tf]) = 0) and thus t = t'. Therefore, we may assume that t = f i ( t l ,. . . , t,,) and t' = fi ( t i ,. . . , tLj) and that 6 [ t ]= 6 [ t r ]From . the last equality follows f irstops(6[t])= f irstops(6[t']) and because of 6 [ t ]= Snt( a ( f i )", t l ] , . . . , "t,,]) and 6 [ t f ]= Sni ( a (f j ) , A[t;],. . . , B [ t L ] ) we obtain firstops(a(fi)) = firstops(a(fj)) and then i = j since o E Hypreg(7). We assume that from 6 [ t j ]= 6[ti]follows ti = ti ,j = 1, . . . , ni. Since o is regular, from 6 [t]= 6 [ t f we ] obtain 6[ti] = 6[ti],j = 1, . . . , ni and can apply the hypothesis. Altogether, t = t'. By B i j ( r ) we denote the set of all a E Hyp(r) such that 6 : W , ( X ) i W , ( X ) is a bijection on W , ( X ) .

Lemma 6.2.6 ( B i j ( r ) o;h , aid)forms a submonoid of IFtyp(r). Proof: Clearly, aid E B i j ( r ) since i i d [ t ]= t for all t E W , ( X ) . We have t o show that ol o h o2 E B i j ( r ) for ol,o2 E B i j ( r ) , i.e. (al o h 0 2 )is a bijection. But this is clear since (al o h 0 2 ) 61 0 e2 and the composition of bijections is a bijection. ^=

Let B denote the set of all bijections on the set { f i I i E I) preserving the arity. As usual we denote by S, the set of all permutations o n t h e s e t { I , . . . , n ) for 1 5 n E w+. Weset A : = U S, and

P

:= { p E A'

I

l
p ( i ) E S,, for i E I } . The following theorem characterizes B i j ( r ) for any type r .

112

6 Monoids of Hypersubstitutions

>

Theorem 6.2.7 Let T = (ni)iE1 be a type with ni 1 for all i E I . For each a E H y p ( r ) the following statements are equivalent: (i) a E B i j ( r ) . (ii) There are elements h E I3 and p E h ( f i )( ~ p ( i ) ( l ). ,. . , ~ p ( i ) ( n i ) for ) all E I .

P such that a ( f i ) =

Proof: (ii)+ (i) We show by induction on the depth of terms that 6 is injective and surjective.

Injectiuity: Let s, t E W , ( X ) with i [ s ] = i [ t ] .Suppose that depth(s) = 0. Then s is a variable, i [ s ] = s and i [ t ]= s. If depth(t) > 1, then there were a n i E I witht = f i ( t l , .. . ,t,,) and by our presumption 6 [ t ]could not be a variable. Hence, depth(t) = 0 and thus 6 [ t ]= t . Consequently there holds s = 6 [ s ]= 6 [ t ]= t . Suppose now that from i [ s ' ] = i [ t f ]there follows s' = t' for s f ,t' E W, ( X ) with depth(sf) n . Assume that depth(s) = n + 1. Then depth(t) > 1 and there are i,j E I with s = f i ( s l , . . . , sni) and t = f j ( t l ,. . . , t n i ) . Now we have , . . . xp(i)(ni)), ~ [ S I .]. . 7 G[snz]) and i [ s ] = Sni ( h ( f i )(xpii)(l) = Sn3( h ( f j () x p i j ) ( l .)., . 7 X p ( j ) ( n j ) ) ,e[tl], . . . 7 6 [ t n j ] ). 6[t] From 6 [ s ]= 6 [ t ]there follows h ( f i ) = h ( f j ) and thus fi = f j , i.e. i = j since h is a bijection. Hence

<

< <

k ni. By our hypothesis we and therefore 6 [ s k ]= 6 [ t k ]for 1 get sk = tk for 1 k ni. Consequently, s = f i ( s l , . . . , sna)= f j ( t l , .. . , tni)= t .

< <

Surjectivity: For w E X we have i [ w ]= w. Suppose that for any s E W , ( X ) with depth(s) n there is an s E W , ( X ) with i [ s ]= s. Let now t E W , ( X ) with depth(t) = n + 1. Then there is an i- E I with t = fi(tl, . . . , tni)and by our hypothesis there are tl,. . . , tnaE W , ( X ) such that i[&] = tk for 1 k ni. Further there is an element j E I with h ( f j )-= fi and ni = n j . Now we consider the term t:= f j ( t P i j ~ - l.i.l .j ,7t p ( j j - ~ ( n ,There ) ) . holds

<

< <

6.2 Injective and Bijective Hypersubstitutions

113

(i)* (ii) Since 6 is surjective for each j E I there is a term s j E W, ( X ) of minimal depth with i [ s j ]= f j ( z l ,. . . , z n j ) Obviously, depth(sj) = 0 is impossible. Thus there are an element k E I and terms r l , . . . ,r,, E W , ( X ) with sj = f k ( r l , . . . ,r,,). Then G[sj] = 6 [ f k ( r 1 7 . . . , r n k ) ] = S n k ( O ( f k )i > [rl]> 7 '['fLk]) = f 3 ( ~ 1 > , X,j)' ' ' '

' ' '

This is only possible if a ( f k ) E X or a ( f k ) = f i ( a l , . . . , a n 7 ) nj. with all . . . , a,? E X,,, I{al, . . . , a,?} 1 = n j , and thus nk But a ( f k ) E X is impossible since otherwise there were an i E {I,. . . , n k } with a ( f k ) = xi and i [ s j ]= i [ r i ]where depth(sj) > depth(ri). This contradicts the minimality of s j and shows that nj, for all j E I there is a number k ( j ) E I with nkij) E X,,,?, with I{a17... ,a,,?}\ = nj such that a ( f k i i ) )= a .a f j ( a l , .. . , a,?). Assume that nk(j) > nj for some j E I . Then there is an x t X,,,j, \ u a r ( a ( f i i j ) ) )i.e. , x is not essential in a ( f k i j )and ) thus 6 is no bijection on W , ( X ) (see [29]),a contradiction. Thus nk(j) = nj and o ( f k ( j ) ) = f j ( z T ? ( l ).,. . , ~l;.~(,,))for %Ime K j E Sn,. Assume that there are j , I E I with 1 # k ( j ) such that f j is the first operation symbol in a (f l ) . We put t := i[ f l ( x l ,. . . , x,,)]. Then t = f j (tl, . . . , trr,) for some t l , . . . , tn3 E W,( X ). Since 6 is surjective, there are sl, . . . , s,, E W , ( X ) with i [ s i ]= ti for 1 5 i 5 nj. Then i [ f k ( j ) (s,;l(l) . . . , s,;l(,?))I

>

>

=

s n j ( ~ ( f ia[s,-lill], ij))7 ...7

=

sn' ( f j ( z T ? ( l ) >

= =

fj(tl7...7tn,) t.

' ' ' 7

a [ sx3- 1 n j )1 )

z T J ( n t ) )t7r ; l ( l ) >'

' ' 7

tT;l(n?))

Since f k i i ) ( ~ , - l i l ) , . . . , ~ , ; l ( , ~ ) ) # f r ( r l ., . . , z,,), 6 is no injective, a contradiction. Altogether this shows that the mapping h : { f i I i E I ) + { f i I i E I ) where h ( f ) is the first operation symbol in a (f ) is a bijection on { fi I i E I } preserving the arity. Further, let p EA ' with p(i) := ;.ri for i E I . Then p E P.Consequently, we have ~ ( f i=) h ( f i ) ( x p ( i ) ( .l ). ., 7 xp(i)(n,) ) for all i E I .

114

6.3

6 Monoids of Hypersubstitutions

Finite Monoids of Hypersubstitutions of Type (2)

In this section we consider hypersubstitutions of type ( 2 ) . Let W ( { x } ) ( W ( { y ) ) )be the set of all terms built up only from the binary operation symbol f and the variable x ( y ) and let A be an additional term with f ( x ,A) = f ( A , x ) = x. We set

Again we denote by at for t E W ( { x ,y } ) the hypersubstitution which maps f to t. Then we prove:

Lemma 6.3.1 If u , v E W + ( { x ) ) then of(,,,) oh o f ( x , v )E { o f( x , u )7 o f ( x , v )) (and if u,v E W + ( { Y ) )then of(,,,) E { o f(u,,) 7 o f ( w 1). )

The proof of the second proposition is similar. For a subset L of I/V+({x))we set M x ( L ) : = {aid}U {af(,,u) I u E L } and ML(L) = M,(L) U {a,, a,} and for a sub( u E set J C W + ( { y ) ) we put M,(J) : = {aid) U and Mi ( J ) = MY ( J ) U {a,, a,} . For any L and any J the J) sets Mx ( L ), MI ( L ), M, ( J ), Mb ( J ) form submonoids of Eyp(2). Let F, (F,) be the set of all finite subsets of W + ( { x } )(of W + ( { y ) ) ) and L E F, ( J E F,). Then M , ( L ) , M k ( L ) , M , ( J ) , M & ( J )are finite monoids of hypersubstitutiorls of type (2). Further we need the finite submonoids of 'Ftyp(2)with the universes:

6.3 Finite Monoids of Hypersubstitutions of Type (2)

115

By Sfin(Hyp(2)) we denote the set of all finite submonoids of WYp(2).

Theorem 6.3.2 Sfin(Hyp(2)) = {MT,MD) U {Mi I 1 5) u {M&) I L E F,) u {M;(L) I L E F,) u {M,(J) u {M/,(J) I J E F,).

J E

4)

Proof: Because of Lemma 6.3.1 all these monoids are finite. We have to prove that there are no other finite submonoids of IFtyp(2). Assume at first that vb(a(f)) 5 2 for each a E M . Then M E {MT, MD, MI, M2, M3, M4, M5, Mx(A), M&), M m > M&)> M&> A H > M W > M,(Y), Kg, A H > Mh(y)) and M E Sfi,(Hyp(2)). (Note that for one-element sets L = {u} instead of M,(L) we write M,(u).) Suppose now that there exists a hypersubstitution a E M with vb(a(f)) > 2 and that M does not belong to the listed monoids. Then there are the following possibilities: a) Both z and y occur in a (f ), b) a ( f ) = f ( u , v ) with u , v E W({x)); u

# x,

d) there are a term u E W+({x)) with a (f ) = f (z,u) and a hypersubstitution a' E M with a' # MA(W+({z)), e) there are a term u E W+({y)) with a (f ) = f (u, y) and a hypersubstitution a' E M with a' # Mh(W+({y)). If a) or b) or c) is satisfied then we define recursively the following sequence of terms: to : = a ( f ) , ti+, : = &[ti]for i E N. It , N, and thus the set is easy t o check that vb(ti) < ~ b ( t ~ + ~i ) E {ti I i E N) of these terms is infinite. Since M is a monoid, for every i E N the hypersubstitution ai with ai(f)= ti is an element of M and M is infinite, a contradiction t o our assumption. If d) or e) is satisfied then a' E {of(,,,) I u E W({y))) in case d) and a' E {af(,,,) I u E W({z))) in case e), respectively. Otherwise we obtain a contradiction as in a), b), or c). We consider the following sequence of terms: to : = a ( f ) , tzi+l : = it[tzi]and t2i+2 : = 6 [ h i + l ] i E N. Then vb(tzi) < ~ b ( t ~ for ~ +every ~ ) i E N.

116

6 Monoids of Hypersubstitutions

Therefore the set {tzi I i E N} is infinite. Since for every i E N the hypersubstitution ai with a i ( f ) = t2i is an element of M, the monoid M is infinite, a contradiction. We want t o get more insight into the structure of Sfin(Hyp(2)). We are also interested in the lattice of all submonoids of Eyp(2). One cannot expect that Sfin(Hyp(2)) forms a lattice. But we will determine all lattices of submonoids of 'Ftyp(2) consisting only of finite monoids.

Lemma 6.3.3 Each of the sets

forms a sublattice of the submonoid lattice of E y p ( 2 ) Proof: L1 is the set of all submonoids of M 5 and thus it is a sublattice of the submonoid lattice of Eyp(2). Let M,, M b E L,. Then there holds: (i) M , = M,(A) or there exists a set I, E F, with M , or M a = Mk(I,),

= Mx(Ia)

(ii) M b = M,(A) or there exists a set Ib E Ex with M b = M x ( I b ) or M b = M k ( I b ) . Clearly, we have M , ( A ) V M x(Ib) = Mk ( I b ) ,M y( A ) V Mk ( I b )= ML (Ib) and M , (I,) V M , ( A ) = M , (I,),ML (I,) V M , ( A ) = M I ) . By Lemma 6.3.1, M,(Ia) V M,(Ib) = M,(I, U

Ib)7 Mx(Ia)vML(Ib)= ML(IauIb),M;(Ia)vMx(Ib)= ML(I,uIb) and Mk (I,) V Mk ( I b )= Mk ( I , U Ib),where M x ( I , U Ib),Mk ( I , U Ib)E L,. Therefore, M , V M b E L,. Moreover, M,(A) A M k ( I b )= Mk(Ia)AMy(A) = M y ( A ) M , y ( A )AMx(Ib) = Mx(Ia)A M y ( A )= Mz(Q))> M,(Ia) A Mz(Ib) = Mz(Ia) A M;(Ib) = ML(Ia)A Mx(Ib)= M x ( I an I,) and ML(I,) A ML(Ib)= ML(I, n Ib)where M x ( I an I,), M k ( I , n I,) E L,. Therefore M , A M b E L,. SO, L, forms a sublattice of the submonoid lattice of Eyp(2).In the same

6.3 Finite Monoids of Hypersubstitutions of Type ( 2 )

117

manner one can prove that L, is a sublattice of the submonoid lattice of IFtyp(2). Now we show that the sublattices of the submonoid lattice of 'Flyp(2) consisting of finite monoids are exactly the sublattices of L1, L, and L,.

Theorem 6.3.4 Let H be a set of finite submonoids of 'Flyp(2). Then the following is equivalent: (i) H forms a sublattice of the submonoid lattice of 'Flyp(2). (ii) H forms a sublattice of the submonoid lattice of L1, of L,, or of L,.

Proof: (ii)

+ (i) is clear because of Lemma 6.3.3.

(i) + (ii) Assume that H @ L,, H @ L,, and H @ L1. Then there are M a , M b , M cE H with M a # L,, M b # L,, M e # L1. From Ma # Lx it follows Ma E ( L y\ { M x ( X )M , y( A ) ,M & ( X )M , T } )U { M DM , l , M 2 ,M 3 ,M q ,M 5 ) . The condition M , # L1 implies M , E L, \ L1 or M , E L, \ L1. Suppose that M , E L, \ L1. Then there exists a hypersubstitution a E Mc and a term u E W ( { x } ) \ { x ) with 6 [ f( x ,y ) ] = f ( x ,u ) .Moreover M a E { M o ,M I ) or there exists a hypersubstitution a1 E Ma and a term v E W ( { y ) ) with a l ( f )= f ( v ,y). Then we have 6 [ a l ( f )=] 6 [ f( v ,y ) ] = f ( x , w) with x , w E W ( { y ) )and w # y. Moreover we have 6[of(,,,)(f)] = 6 [ f( y ,x ) ]= f ( y , w) with w E W ( { y ) ) \ { y ) . Clearly, for each finite # M. submonoid M of IFtyp(2) we have a oh 01 # M and a oh Since a oh 01 E Ma V Mc or a oh of(,,,) E Ma V Mc, respectively, we obtain M a V M c $ Sfin(Hyp(2)) and thus M a V M c # H C Sfin(Hyp(2)), a contradiction. In the same manner one shows that M , E L, \ L1 is impossible. As a corollary of Theorem 6.3.4 we see that the set Sfin(Hyp(2)) of all finite submonoids of 'Flyp(2) does not form a sublattice of the submonoid lattice of 'Flyp(2). As an example for a sublattice of the submonoid lattice of IFtyp(2) we will describe the lattice L1 by its Hasse diagram.

6 Monoids of Hypersubstitutions

The

following isomorphisms

are

M 3 , M k ( x ) N M i ( y ) , Mk(A) M,(8 M,(Y).

"

N

easy

to

check: M 1

M3, MZ(x)

Mx(A)

" N

The monoids M D ,M , ( A ) ,M y( A ) ,M , ( x ) ,M y( y ) are the atoms in L1 and are also atoms in the submonoid lattice of Nyp(2).Because of Q ~ ( z , uO)h Q f ( z , u ) - Q ~ ( Z , U )for any u E W + ( { z )every ) two-element monoid M z ( u )is an atom in the submonoid lattice of 'Flyp(2).Now we show that any atom in the submonoid lattice of 'Flyp(2) is a finite monoid.

Lemma 6.3.5 The atoms of the submonoid lattice of 'Flyp(2) are finite submonoids of 'Flyp(2). Proof: Let M be an atom of the submonoid lattice of 'Flyp(2).Assume that M @ Sfin(Hyp(2)). Because of M @ L1 there exists a hypersubstitution a E M with v b ( a ( f ) )= p 3 (see proof of Theorem 6.3.2).Since there are only finitely many terms t E W ( { x y, } ) with vb(t) p there is a hypersubstitution a0 E M and a natural number q > p such that v b ( o o ( f )= ) q. Let M o be the submonoid of 'Flyp(2) generated by ao. Clearly, for any hypersubstitution a1 E Mo we have a1 = aid or v b ( a l ( f ) )> p. Therefore a @ Mo, that means Mo c M and M cannot be an atom of the submonoid lattice of 'Flyp(2).

>

<

6.4 The Monoid of all Hypersubstitutions of Type (2)

119

As a consequence we get:

Theorem 6.3.6 The submonoid lattice of 'Hyp(2) has, up to isomorphisms, exactly two atoms, namely MD and Mx(X)(each of them is two-element). Proof: We apply Theorem 6.3.2 and Lemma 6.3.5. If there is a set I C W+({x}) with M = M x ( I ) and 1 1 1 2 then M cannot be an atom since for any term u E I the monoid M has the proper submonoid M, (u). Using the definitions of L1, L,, L, we see that atoms have the form M x ( u ) ( M y ( v ) ) for u E W+({z)) (for v E W+({y))) or M D . All atoms of the form M x ( u ) and M y ( v ) are isomorphic t o each other, but not to M D .

>

As a corollary we have

Theorem 6.3.7 The submonoid lattice of 'Flyp(2) is not atomic. Proof: Let M be the submonoid of 'Flyp(2) generated by the hypersubstitution a with a ( f ) = f (z,f (z,y)). Now we define recursively the following sequence of hypersubstitutions: 00 : = 0,

0i+1( f )

It is easy to check that M atom included in M .

6.4

=

=

f (x7f (x70 i (f ))) , i

E

N.

I

{ai i E N) U {aid)but there is no

The Monoid of all Hypersubstitutions of Type (2)

Any hypersubstitution o of type (2) is determined by the binary term to which it maps the fundamental operation symbol f and conversely, any term t E W({x, y)) defines a hypersubstitution at E Hyp(2). We remark that we have in fact defined a mapping from the universe of the free groupoid F(2j ({x, y}) := (W(2j({x, y)); f )of all terms of type (2) on the two-letter alphabet t o the universe of the monoid

6 Monoids of Hypersubstitutions

120

Nyp(2). This mapping is clearly a bijection, but it is not a homomorphism, since it does not preserve structure. That is, F ( 2 ) ({x, y}) is a free groupoid under the usual binary operation f which composes two terms u and v into the term f (u, v), and as we have seen, 'Flyp(2) is a monoid under the operation o h of composition of hypersubstitutions. But in general of(,,,) is not equal to a , o h a v , meaning that we do not have a homomorphism. We may use this bijection to induce a new operation * on W(2)({x,y)), in the obvious way, but this gives an entirely new structure on W(2)({z,y)). In 6.3 we have shown that 'Flyp(2) has a finite submonoid M consisting of the six elements a i d , (the identity element of the monoid), a,, a,, of(,,,), and of(,,,). The first five of these are easily seen t o be idempotents, while the element of(,,,) satisfies of(,,,) o h of(,,,) = a i d and hence has order two, that means, the submonoid of 'Flyp(2) generated by of(,,,) consists of two elements. The elements a, and a, play a special role in Hyp(2), just as the corresponding terms x and y play a special role as the projection terms. For any a in Hyp(2), we have a, o h a E {a,, ay),ay o h a E {a,, ay), a o h a, = a, and a o h ay = ay. From this we can conclude that a, and a, are both right zero elements of the monoid Nyp(2), that they are the only right zero elements, and that there are no left zero elements. Moreover, the set {a,, a,} forms a subsemigroup of 'Ftyp(2), and is an ideal of 'Ftyp(2). In fact, it must be contained in any ideal of 'Flyp(2), so it is the kernel of 'Flyp(2), the intersection of all ideals of 'Flyp(2). This shows that 'Flyp(2) is not simple.

Lemma 6.4.1

(i) If a,

oh

at

=

aid, then either a,

=

at

= aid

Proof: (i) It is clear that s and t cannot equal z or y alone, so suppose that s = f (a, b) and t = f (c, d). Then a, o h at = a i d gives

This forces S2(a,6, [c],6, [dl) = x and S2(b, 6,[c],6,[dl) = y. From this we must have a = c = x or a = y and d = x; and b = c = x or b = y and d = x. These four cases then lead t o only the possibilities

121

6.4 The Monoid of all Hypersubstitutions of Type (2) listed in the lemma. (ii) This proof is similar to that of (i).

By this lemma we can decompose the elements a i d and of(,,,) into a product of two elements in essentially only trivial ways. It also tells us that the elements a i d and of(,,,) are the only elements of Hyp(2) with either a left or a right inverse in 'Ftyp(2). Now we describe the idempotent elements of 'Ftyp(2), using the I u E W({x))), E, = following notation. We let E, = {af { ~ f ( v , y ) I 7, E ~ ( { Y I )and ) , E = E x UEy. Corollary 6.4.2 For any two elements f (x, u) and f (x, w) of W({X)), we have of(,,,) O h o f ( ~ , w )Of(,,,) for U, E W(2)( X ) . This corollary, and a dual proposition for elements of E,, means that both &, and &, are left zero bands of idempotents in 'Ftyp(2). This has several important corollaries. Corollary 6.4.3 (i) A n y non-empty subset of Ex or of Ey is a subsemigroup of 'Hyp(2). (ii) 'Hyp(2) has uncountably m a n y subsemigroups and submonoids. (iii) 'Ftyp(2) has submonoids which are not finitely generated. In fact, for any subset I of W ({x)), the set M, (I)= {af(,,,) U{aid) is a submonoid of 'Ftyp(2).

I u E I)

By the Galois connection from 3.4 each submonoid of 'Ftyp(2) corresponds t o a complete sublattice of the lattice of all varieties of type (2). Here we may consider the sublattice of all Mx(I)-solid varieties. Especially we are interested in M,(I)-solid varieties of semigroups.

+

Theorem 6.4.4 Let L be any subset of W({x)), and let T = {I vb(u) I u E L). Let V be any variety of semigroups, and let L Z = Mod{xy FZ x) be the variety of left zero semigroups. T h e n V is Mx(I)-solid i f l either LZ C V C Mod{x" xk2 I k E T ) or V C Mod{x" yk I k E T ) .

122

6 Monoids of Hypersubstitutions

The proof can be found in [35]. In order to determine whether there are more idempotents in 'Hyp(2), we need t o know more about the result of composing two hypersubstitutions. The next two lemmas will compare the number of variables of the term corresponding t o the product o f ( c , d )O h o f ( u p ) with the number of variables of the terms f (c, d) and f (u, v). But first we need some additional notation.

Definition 6.4.5 Let f (c, d) E W({x, Y)) with ub(f (c, d)) at least 2. We define four related new terms, as follows: (i) f (c, d) is the term formed from f (c, d) by interchanging all occurrences of the letters x and y, i.e. f (c, d) = S2(f(c, d), y, x) and f (c, d) = S2(f (c, d), y, x) with the ternary superposition operation S2. (ii) f (c, d)' is the term defined inductively by x' f (a, b)' = f (b', a').

=

x, y'

=

y, and

(iii) Cx[f(c, d)] is the term obtained from f (c, d) by replacing every occurrence of the letter y by an x. (iv) C, [f (c, d)] is the term obtained from f (c, d) by replacing every occurrence of the letter x by an y. We notice that Cx[t](C,[t]) can be written as et[f(x, x)](Zt[f (y, y)]).

Lemma 6.4.6 Let ~ f ( ~ , ~d h) o f ( ~=, ow. ~ ) Then vb(w) > vb(f (c, d)), unless f (c, d) and f (u, v) match one of the following ten possibilities:

6.4 The Monoid of all Hypersubstitutions of Type (2)

123

PIY30f: Let o f ( ~ , d ) Oh of(,,u) ow. We want to compare vb(w) with vb(f (c, d)). Writing a = a f ( , , d ) , we have w = f (S2(c, 6[u],6[v]),S2(d,6[u],6[v])).We now consider a number of cases, depending on whether vb(u) = 1 and vb(v) = 1 (i.e., are variables z or y) or not. Case 1. If both u, v E {x, y), then i [ u ] = u and 6[v] = v. This gives four possible subcases: (i) u = v = z: Then we have of(,,d) oh of(x,x) - o ~ , [ f ( ~ , d ) which l, is (El). (ii) u = v = y: This gives (E2). (iii) u = x and v = y: Then of(,,,) = a i d , and we have (E3). ~ ) , we have (E4). (iv) u = y and v = z: Then of(,,,) = ~ f ( ~ , and Case 2. If u = x and v is not in {x,y}, then w = f (S2(c,z, 6[v]),S2(d,z, 6[v])).If c and d are both terms in z only, then for any term v we get w = f (c, d) again, as in (E5). Otherwise, suppose that (at least) one of c or d uses the letter y. Then the length of w depends on the length of the subterm 6[v]. As long as 6[v] has length greater than one, we will have w longer than the term f (c, d). But it is not possible here to have 6[v] of length one, since v is not a single variable in this case. Case 3. If u = y and v is not in {x, y), then w = f (S2(c,y, 6[v]),S2(d,y, 6[v])).If c and d are both terms in the letter z only, then we get w = f (c, d), which is (E6). Otherwise, if (at least) one of c or d uses the letter y, v not a single variable means that w is longer than f (c, d). Case 4. If v

=

x and u is not in {x,y}, then w

=

f (S2(c,6[u],x ) , S2(d,i [ u ] ,2))). If c and d are both in W({y}), then we have w = f (c, d) , and (E7). Otherwise, by an argument similar

6 Monoids of Hypersubstitutions

124

to that of the previous two cases, we get w longer than f (c, d) . Case 5. If v = y and u is not in {x, y), then as in the previous case we get either (E8) or a longer term than f (c, d). Case 6. If neither of u or u is in {x, y}, then we note that 6[u] and i [ u ] cannot be in {x, y) either, unless we have c and d both in{x, y). So except in these special cases, the term w is longer than f (c, d). The four special cases involve f (c, d) equal to one of the four terms f (x, y) , f (y, x) , f (x, x) and f (y, y) . The first of these gives a, = a i d Oh O~(U,V ) of(u,t~), so w = f (u, u) and we have (E9). The second case has f (c, d) = f (y, x) so w = f (u, u)', giving (E10). For f (c, d) = f (x, x), we have w = f (6[u],6[u]). Here we see that w is longer than f (c, d) = f (x, x). Finally, if f (c, d) = f (y, y), we argue in a similar manner that we get either (E7) or (E8) again, or a longer term than w.

Lemma 6.4.7 Let a

not be equal to any of a,, a,, a i d or 2. If f (c, d) uses both these af(y,z).Let u be a term with ub(u) letters (in this case we will also say that f (c, d) is essentially binary) the term w corresponding to the composition a oh a, is longer than = Of(,,$

>

U.

Proof: We proceed by induction on the complexity of the term u. For the base case, let u = f (s,t ) be a term with ub(u) = 2, so that both s and t are in {x, y). Then we have 6[s] = s and i [ t ] = t , so 3 ( since c and d that w = f (S2(c, S, t), S2(d,s, t)). Here vb(w) are not both in {x, y) by assumption), so it is longer than u. Now suppose that u = f ( r , z ) . If for both r and z we have ub(r),ub(z) 2, we can assume by induction that vb(6[r]) > vb(r) and vb(6[x]) > vb(x). Then we have w = f (S2(c,6[r],6[z]),S2(d, 6[r],6[x])). When f (c, d) is essentially binary, by induction we see that vb(w) > vb(f (r,z)). But it is possible to have u = f (r,z) where one (but not the other) of r or z is a single letter. In this case one of 6[r] or 6[z] is also a single letter, but the other is longer (ub is greater), by induction. Since f (c, d) uses both letters x and y, we can conclude again that vb(w) = vb(f (S2(c,6[r],6[x]),S2(d,6[r],6[x]))) > vb(f (r,2)).

>

>

Note that Lemma 6.4.7 is more restricted than Lemma 6.4.6, since

6.4 The Monoid of all Hypersubstitutions of Type ( 2 )

125

it deals only with a product in which the leftmost factor uses a term which uses both letters. If the leftmost factor uses a term in one letter only, the term corresponding to the product can be either shorter, longer, or of the same length as the term from the rightmost factor. For example, we have

Now we can show that the idempotents we have found so far, those in M5 or Ex or E,, are exactly all idempotents of IFtyp(2).

Theorem 6.4.8 A hypersubstitution a E H y p ( 2 ) is a n idempotent i f f o is in ExU E, U{ox, a,, aid). Proof: Suppose that a is an idempotent of H y p ( 2 ) . By Lemma 6.4.6, a oh a cannot equal a, unless we are in one of the cases ( E l ) t o ( E I O ) . But all of these possibilities reduce to an element o of ExU E, U{ax, a,, aid). Clearly, each element from ExU E, U{ax, a,, aid}is idempotent. In any semigroup or monoid the set of idempotents can be ordered by the relationship e f iff e o f = f o e = e. It is easy t o see that for any a in Ex,ax a aid. Dually, for any a E E,, a, a aid. But none of the idempotents in ExU E, are comparable t o each other. We have seen that H y p ( 2 ) contains countably infinitely many idempotents, plus the special element of(,,,) of order two. All regular elements in IFtyp(2) were determined in [13].

<

< <

< <

Theorem 6.4.9 ( [ 1 3 ] )If a E H y p ( 2 ) is an regular element, then 0 is in E x UE, U{ax, a,, a i d 7 ~ ~ ( ~ , x ) ) . We now show that any other element of H y p ( 2 ) has infinite order.

Theorem 6.4.10 Let a be a hypersubstitution from H y p ( 2 ) . T h e n a has order 1, 2 o r infinite. Proof: We will show, if a is not an idempotent in ExU E, U{ax, a,, aid},then for any n 1 , an # an+',and a has infinite order. Except

>

6 Monoids of Hypersubstitutions

126

in the special cases listed in possibilities ( E l ) t o (E10) of Lemma 6.4.6, we always have an+' = an oh a corresponding t o a longer term than a n . As before, all the exceptions correspond t o elements of Ex U EyU{az, O v , a i d ) Or to Of (),. Now let G be the set of elements of Hyp(2) of infinite order. We can show that G is not a subsemigroup of Hyp(2), since it is easy to find two elements of G whose product is an idempotent. In fact, for any idempotent of(,,,), with u E W({z)), one can easily verify that for any term k # z, y, we have

with a-f (&u) and

6.5

af(k,,)

both in G.

Green's Relations on H y p ( 2 )

In this section we investigate Green's relations for the monoid IFtyp(2). A special role will be played by the terms t' and 2, for a term t, which were defined in Definition 6.4.5. For any set S of terms, we denote by 3 the set { S I s E S), and similarly for S'. We will also use analogous notation for sets of hypersubstitutions. We write for example and EL. By Lemma 6.4.6 for any term t , we have at oh of(,,,) = 0%and of(,,,) oh at = at,. This proves the following result.

Lemma 6.5.1 For a n y t e r m t, the hypersubstitution at is R related t o 0%and C-related t o at,. Therefore for a n y t, the hypersubstitutions a t , a%,at(, and a?! are all in the same 2)-class. We now determine the classes of these Green's relations for special elements of Hyp(2). Using the results from 6.1 it is not difficult to prove the following results:

Lemma 6.5.2 (i) T h e element a, is C-related only t o itself, and R-related t o only itself and a,. T h u s the D-class of a, is { a x , a?/). (ii) T h e element of(,,,) has R-, L-, and D-class all equal t o {aid,

of(y,x)).

6.5 Green's Relations on Hyp(2)

Lemma 6.5.3 For any terms t and s, at

127

R a,

ifl s = t or s = 2.

Proof: The proof follows from Corollary 6.1.2. We will skip the details here. Since the relation 2) is just the composition of the relations R and C, Lemma 6.5.3 allows us t o describe the 2)-class of any hypersubstitution at: it is the union of the 3-class of ot with the 3-class of OF. Thus we continue with the problem of describing the 3-classes. We need a preliminary lemma as well.

Lemma 6.5.4 If c and d are terms in W({x)) (or dually both in W({y))), then for any terms u and v the term corresponding to o f ( c , d ) O h o f ( u , v ) 2s in W({x}) U W({Y)). Proof: We have to consider the letters used in the term w = f (S2(c, 6[u],6[v]),S2(d,6[u],6[v])),where o = o f ( , , d ) . Since by assumption we have c and d both in W({x)), we need only examine what letters appear in the term 6[u]. We will prove by induction on the complexity of the term u that &[u]is a term in one letter only. If u is a variable, then 6[u] = u, which of course uses only one letter. Now suppose that u is a composite term f (a, b), with &[a]a term in one letter. Then 6[u] = f (S2(c,6[a],6 [ b ] ) , S2(d,6[a],6[b])). Since c and d are in W({x)), this term uses only the letters used in &[a],and so it is also a term in one letter only. Lemma 6.5.5 Let t = f (u, v) be an essentially binary term of length at least two which uses both letters x and y. Then the only elements which are C-related to at are at itself and at,. Proof: Suppose that o f ( a , b ) is C-related to of(,,,).Then there exist terms c, d, p and q such that ~ f ( ~ , oh d ) of(u,v) = o f ( a , b ) and of(,,,) O h o f ( a , b ) = o f ( u p ) . If f (c, d) is either f (x, Y) Or f (Y x), we just have f (a, b) = t or t'. So we now need only check what happens if f (c, d), and similarly f ( P , q), is not equal to either of these. Now if f (c, d) and f (p, q) use both letters x and y, Lemma 6.4.7 would mean that f (a, b) is longer than f (u, v) and vice versa. This is impossible, so one of the terms f (c, d) and f (p, q) uses only one letter. Now Lemma 6.5.4 gives us that f (a, b) and f (u, v) also use only one letter. Thus for an essentially binary term t, we see that

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6 Monoids of Hypersubstitutions

only atand at,can be C-related to at. This shows that when the term t is essentially binary, the hypersubstitution athas a two-element C-class. The next lemma shows that one-letter terms t can result in larger C-classes.

Lemma 6.5.6 The C-class of the element of(,,,) is precisely the set E, uE,. Proof: For any two idempotents e and f , it is known that eCf if and only if e o f = e and f o e = f . (See for example [59].) Since Ex is a left zero band, it follows that of(,,,) is C-related t o any element of E,.By Lemma 6.5.1, we also have each member of E,,and hence C-related t o its corresponding prime element, which is in O I .f> . ( Ek = E,. This shows that the C-class of of(,,,) contains at least ExUE,. For the opposite inclusion, suppose that some element Of is C-related t o of(,,,). Then there exist terms a , b, c, and d such that of(a,b) O h of(u,v) - of(x,x) and of(&)O h of(x,x)- of(u,v). From these two equations one may work out directly that of(,,,) must be in E, U 2,. 7-

Corollary 6.5.7 The D-class of the element of(,,,) is precisely the set E U E. We have now identified the C-class of all hypersubstitutions otfor which t is a term using both letters, or which is an idempotent. This still leaves the cases where t has the form f (a, b) for a and b terms in a single letter and both of length at least two. In what follows we will analyze C-classes for elements atwhere t E W({x)); 1 analogous arguments will hold for t E W({y)). For each n we inductively define the symmetric term s, in W({X)), by sl = f (x, x), and s,+~= f (s,, s,). We note that such terms satisfy s; = s,, justifying the name symmetric, and that s, has leftdepth and rightdepth both equal t o n. The idempotent element of(,,,) is precisely a,,,and in general atis in Ex when t E W({x}) and t has leftdepth one. Moreover, we see that the at in the D-class E UF = E U E' of the idempotent of(x,x) are precisely those where t uses only one letter and has either rightdepth or leftdepth equal to one. To generalize this result, we inductively define for any term d E W({x)) a sequence wt, n E N+, of terms, by wf = f (x, d)

>

6.5 Green's Relations on Hyp(2)

f ( x ,S2(d7x, x ) ) and w : ,

129

f (w:, S2(d7w:, w t ) ) .Note that the leftdepth of the term wt is equal t o n. =

Lemma 6.5.8 For any n as, .

=

> 1 and

any term d E W ( { x ) ) ,a,$ C

Proof: It is easily t o verify by induction on n that for any term u of leftdepth n , we have of(,,,) aha, = asn.In particular, we have of(,,,) oh a d = as,. Now let c = f ( x ,d). We will show by induction on n , , that a, oh as, = ow$.For the base case n = I , the term associated with a, oha,, is S 2 ( a c ( f )x,, x ) = f ( x ,S 2 ( d x, , x ) ) = w f ; and for any n 1, the term for a, ohasn+,is S 2 ( a c ( f i) ,,[ s n ]6,[sn]) , = f (&[sn],S 2 ( d ,&[sn],q % L ] ) ) = f S 2 ( d w:, , = w:+,. This shows that a,$ is C-related t o osn.

>

(4,

4))

Lemma 6.5.9 Let u and w be terms in W ( { x ,y } ) . Assume that a, C a,, so that there exist terms b and c such that ab oh a, = a,

and a, oh a,

=

ow. Then a, oh ab and ab oh a, are both idempotents.

Proof: Let o = a, oh ab. By repeated substitution we have a, = ak oh a,, for all k 1. It is clear that none of a , ab, a,, a, or a, can be among a,, a,, aid, or Suppose that a had infinite order as an element of the monoid 'Ftyp(2).Then there must exist a natural number n for which w is shorter than the term for on. But then the equation a, = an oh a, does not fit any of ( E l )t o (ElO),so by Lemma 6.4.6 we must have w longer than the term for an. This contradiction shows that a cannot have infinite order. By Theorem 6.4.10, it follows that a = a, oh ab must be an idempotent. A similar argument applies t o ab O h ac.

>

Lemma 6.5.10 Let w be a term in W ( { x } )with lefidepth (rightdepth) n, and let ab be an idempotent.

(i) If b = f ( x ,d ) E W ( { x ) ) ,then aboh a, related to a,,. (ii) If b = f ( a ,y ) E W ( { y ) ) ,then ab the term a' E W ( { x ) ) .

oh

a,

=

a,$, which is C-

=

a(,$),, where d is

Proof: (i) We use induction on n. For n = 1, we have leftdepth of w is 1 , so we can write w = f ( x ,e) for some e E W ( { x } ) .Then

130

6 Monoids of Hypersubstitutions

ow uses the term S2(ab(f), x, ib[e])= f ( x , S2(d,x, eb[e]))= f (x, S2(d, x, x)) = wf , since d E W ({x)) . Inductively, if w = f (t, e) for terms t and e in W({x)), where t has leftdepth n, then a b oh 0, Uses the term S 2 ( 0 b ( f ) ,eb[t], eb[e])= f (eb[t], S2(d,eb[t], eb[e])) = f(eb[t],s2(d,eb[t]:eb[t])) = f(w;, S2(d:w;,w$)) = w;+I. BY Lemma 6.5.8, the hypersubstitution o b oh ow is then C-related to a b oh

(ii) This can be proved similarly, by induction on n. With these lemmas we are now ready to complete our analysis of the C-classes of a, for terms w in W({x)). We have seen that when w is a symmetric term s,, ow is C-related to o, for certain terms u = w$, d E W ({x)). Our next theorem will show that these and their primes are the only a, which are related t o a,,, and that any non-idempotent a, which is not C-related t o one of the a,, is C-related only t o itself and a,,.

Theorem 6.5.11 Let u and w be terms in W({x)) for which a, and ow are not in M5 U E U E'. If o, C ow and w is not just u or u', then either w or w' has the form w; for some d and n equal to the leftdepth(or rightdepth, resp.) of w, and both o, and ow are C-related to a,,. Proof: We assume that a, C a,, where u and w are terms in W({x}). As in Lemma 6.5.9, there exist b and c such that ob oh ow = o, and o, oh o, = ow, and we have ow = o oh ow for the idempotent o = o, oh ob. By Lemma 6.5.10, this means that either w or w' has the form wt for some term d E W({x)) and n equal to the leftdepth (or rightdepth, resp.) of w. In either case we see that a, is C-related to aSn. We conclude this section with some remarks on other properties of the monoid IFtyp(2) which follow from knowledge of its C-, R-, D-, IFt- and 3-classes. By Theorem 6.1.1 every element of a D-class is regular iff the Dclass contains an idempotent element. Since we have described the D-classes of all idempotent elements of IFtyp(2), and these clearly do not exhaust the elements of Hyp(2), it follows that IFtyp(2) is

6.5 Green's Relations on Hyp(2)

131

not a regular semigroup. In fact we have precisely three regular {a,, a,) and E U E;all other elements are D-classes: {aid,af(y,x)}, non-regular. All the regular elements of 'Hyp(2) were already determined in Theorem 6.4.9. Our knowledge of the D-classes of all the idempotents of 'Flyp(2) now gives us the following result on local subsemigroups of the semigroup 'FlypP(-r) := (Hyp(-r);oh) (see 6.1):

Lemma 6.5.12 The monoid ' H y p ( 2 ) has three (up to isomorphism) local subsemigroups: S1 = a i d E y ~ ( 2 ) ~ a= i dE Y P ( ~ ) ~ , S2 = o,'Flyp(2)-a, = {a,), and S3 = af(z,x)'Fl~~(2)-af(x,x). It follows from Lemma 6.5.10 that the local subsemigroup S3 is in fact equal t o the set {asnI n I), of all the hypersubstitutions corresponding t o the symmetric terms. Let M, be the submonoid with the universe S3 U{aid). As we pointed out earlier, the collection of all type (2) varieties which are M,-solid form a complete sublattice of the lattice of all varieties of type (2). We can now characterize which varieties V of type (2) are M,-solid, that is, have the property that for any identity u u v of V, the equation eSn [u] u eSn [v] is also an identity of V for every n 1. It is straightforward t o prove by [u] induction that for any term u of leftdepth m, and any n 1, ism if the first variable in u is x, but s,,if the first variable in u - s, is y. We will call a variety V leftmost if for every identity u u u of V, the terms u and v start with the same first variable. We also will be interested in whether V satisfies any identities u u v in which u and v have different leftdepths. Let us set

>

>

>

D(V):= { u u u 1 u u u E I d V and u and u have different leftdept hs}. Then we have four cases t o consider. 1. If V is a leftmost variety and D(V) is the empty set, then V is M,-solid.

132

6 Monoids of Hypersubstitutions

2. If V is leftmost, but D(V) is non-empty, then let p be the smallest leftdepth occurring in an identity u FZ u in D ( V ) , and k be the greatest common divisor of the numbers I(leftdepth of u) - (leftdepth of v)I, for u = v E D(V). Then V is Ms-solid iff V satisfies the identity s, = s,+k. 3. Suppose that V is not leftmost, but D ( V ) is empty. Then there exists a smallest natural number n for which V satisfies a nonleftmost identity u FZ u where u and u both have leftdepth n. Then V is M,-solid iff V satisfies the identity s, FZ =:s,.

4. If V is not leftmost and D(V) is non-empty, then we can formulate a condition which combines cases 2. and 3.

Lemma 6.5.13 The largest congruence contained in 'Ft is the trivial one, (so that 'Ftyp(2) is a fundamental semigroup, see e.g. [59]). Proof: Let y be any congruence contained in 'Ft. Suppose a,yob. Then a, R o b also, so by Lemma 6.5.3, the hypersubstitution a, is either ab or 6. If a, = 6, we have 6 y ab. Then since y is a congruence, we also have a, oh b yoz oh ab. Since composition on the left with a, merely picks out the first variable letter used in the term, and and ab begin with different letters, this gives ox y a,. But then y C R C =: 'Ft contradicts Lemma 6.5.2(i).

n

Lemma 6.5.3 shows that each R-class of Hyp(2) has size two, and hence that each IFt-class can have size at most two. In order for an IFt-class to have size two, we must have the two elements t and 2 which are R-related to t also being C-related. One way in which this can happen is if t = t'. The following lemma shows that in fact this is the only way it can happen.

Lemma 6.5.14 Let t be a binary term of type (2). Then the following are equivalent: (i) at has an IFt-class of size two. (ii)

2 = t'.

(iii) t

=

f (u, v) with v

=

a'.

6.5 Green's Relations on Hyp(2)

133

Proof: (i) ==+ (ii) When at has a two-element E-class, it means that t and ? are C-related. If t uses both letters x and y, Lemma 6.5.5 tells us that we must have ? = t'. But if t E W ( { x } ) ,then since (o o h ot)( f ) is also in W ( { x ) )for any o, it is impossible for oz t o be C-related t o at. (ii) ==+ (i) If t' = 2,it is clear that at is both 2- and C-related t o at,,and the 'Ft-class has two elements. (iii) ==+ (ii) If t

=

f (u, a'),then t'

=

f (a,u')

=

f (u, a')= 2.

(ii) ==I+ (iii) Let t = f (u, v) with t' = 2. This means that f (v', u') f (a,a). Therefore v' = a, making v = v" = a'.

=

This lemma tells us how t o find hypersubstitutions with twoelement 'Ft-classes; for example, the element of(f(z,z),f(y,y)) has this property. Moreover, if t satisfies the conditions of Lemma 6.5.14, then so does f (t,t). This gives us a way t o produce countably infinite chains of such hypersubstitutions, and shows that the set of all such hypersubstitutions is a countably infinite one. This means that there are a countably infinite number of two-element E-classes in Hyp(2). It is also easy to see that the set of such hypersubstitutions forms a subsemigroup of 'Ftyp(2). We now consider the relation 3 on Hyp(2). Let I (a,) be the (principal) ideal generated by an element o,, the set of all elements of the form o, o h o, o h oq for some o, and a, in Hyp(2). Then we observe that two elements of Hyp(2) are 3-related iff they generate the same ideal of Hyp(2). Note also that by definition D C 3, so that the D-related elements described above are also 3-related. (From 6.1 we know that D = 3 in the transformation semigroup

' F ~ .)A Lemma 6.5.15 I(oid) = Hyp(2) = I ( O ~ ( ~ and, ~if )I(o) ), = Hyp(2) then a is one of aid or of(,,,). Thus the 3-class of of(,,,) is equal to its D-class, {aid,af(,,x)}.

Proof: Any hypersubstitution a from Hyp(2) can be written as a = g o h a i d o h a i d = ~ o h ~ f ( , ~, h~a)f ( , , ~ )SO , both a i d and of(,,,) generate all of Hyp(2) as an ideal. This means that they are 3-equivalent.

134

6 Monoids of Hypersubstitutions

Now suppose that a is an element which generates all of Hyp(2) as an ideal. Then in particular, we must be able to write a i d as S o a o p for some S and p in Hyp(2). But by Lemma 6.4.1 this forces a to be one of a i d or of(,,,). This means that no other elements can be 3-related to these two elements.

Lemma 6.5.16 The 3-class of a, is {a,, a,), which is also its 2)-class. Proof: As we pointed out at the beginning of Section 6.4, both a, and a, generate the same ideal, {a,, a,). This means that they are 3-related. Moreover, since for any 6 and p in Hyp(2) we have S oh axoh p E {a,, a,}, no other element can be 3-related to a,. Theorem 6.5.17 (i) The ideal I(af(,,,)) is I W ( b H UW({y>>>.

=

{at

I

t E

(ii) If a is in I but not in M5 U E U E, then I(o) is properly contained in I . (iii) The 3-class of of(,,,) is equal to its 2)-class, E U E.

Proof: (i) Let a and 6 be any elements of Hyp(2). Then by Lemma 6.5.4, of(,,,) oh S is in I, and hence so is a oh af(x,x) oh 6.This shows is contained in I . Conversely, for any terms u and that I(afi,%,,) v in ~ ( { iwe } have ) , ~ / ( u , v )oh a / ( x , r ) oh a f ( r , v ) - ~ ~ ( C ~ ~ I L I= ,C.~~I) of(,,,). Along with a similar argument for terms in W({y)), this shows that I is contained in I(of(,,,)). at be in I, so that t E W({x)) U W({y)), but not in M5 U E U E.In particular, this means that t has length at least four. We will show by contradiction that of(,,,) cannot be in I ( a t ) . Suppose there existed terms u, u, p and q such that of(,,,)oh at oh - of(,,,). First consider the term w which corresponds to the prod-

(ii) Let a

uct

=

at Oh Of ( p , q ) .

By Lemma 6.4.7, w will also have length at least four; the only possible exceptions would be (E9) and (EIO), but we know that t # f (x, y), f (y, x). Now we consider the term z corresponding to the product of(,,,) oh ow,which is supposed to produce of(,,,). By Lemma 6.4.7 this is impossible if the term f (u, u) uses both letters

6.5 Green's Relations on Hyp(2)

135

x and y. So we must have f (u, v) in either W ({x)) or W ({y}). But again the equation of(,,,) o h ow = of(,,,), with w of length at least four, does not fit any of ( E l ) t o (ElO), so by Lemma 6.4.6 we must have f (u, v) shorter than f (x, x). This is a contradiction. (iii) Since D C 3, we must have E U E contained in the 3-class of of(,,,). For the opposite inclusion, let a be any element of this 3-class. Then it must generate the same ideal as of(,,,), so I ( o ) = I . Then we must have o itself in I, but by (ii) o must also be in M5 U E U E. We can rule out elements o,, a,, aid and of(,,,) by Lemmas 6.4.1 and 6.4.2. Hence a must be in E uE.

Theorem 6.5.18 Let t be any essentially binary term (diflerent from f (x, y) or f (y, 2)). Then the 3-class of the hypersubstitution at is equal to its D-class, {at, OF, at! oF}. Proof: Suppose that a, is 3-related to at. Then there must exist terms u, v, p and q such that a u o h a t o h a v = a, and a p o h a , o h a q = at. Combining these two equations gives at = a, o h auo h at o h a, o h aq. Since the term t uses both letters x and y, it follows (using Lemma 6.5.4) that none of the terms u, v, p or q, or terms corresponding t o intermediate products, can use only one letter. Thus we can apply Lemma 6.4.7, to conclude that the term corresponding t o at = (apo h au)o h (at o h a, o h aq)is longer than the term corresponding t o the product at o h o, o h oq. By Lemma 6.4.6, this term in turn is longer than t - which would make t longer than itself - unless the product at o h (ovo h oq)fits one of ( E l ) t o (EIO). But since none of the terms involved use only one letter, we see that cases ( E l ) , (E2), (E5), (E6), (E7) and (E8) are impossible. We can also rule out (E9) and (ElO), since they would require that t equal one of f (x, y) or f (y, x ) , which we have excluded. This leaves only (E3) and (E4) as possibilities. For (E4) t o hold we must have the term corresponding t o the product a, o h aqequal t o f (y, x). Then we would have ot = op o h ou o h ot o h o, o h o q op O h ou O h ot O h of(,,,) = op O h QU O h OF. But by Lemma 6.4.7, the term corresponding t o this last product is longer than 2,which is the same length as t , and this is clearly impossible.

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6 Monoids of Hypersubstitutions

Finally, suppose it is (E3) which holds for at oh (av oh a,). Then we have a, oh a, = aid By Lemma 6.4.1, this can only happen in two ways: either a, = a, = aid, or a, = a, = of(,,,). In the first case, we see from our original two 3 equations that a, oh ot = os and o, oh os = ot. This shows that as is 3-related to ot, so by Lemma 6.5.5 s is one of t or t'. In the second case, our original two and at = 3 equations reduce t o a, = a, oh at oh af(?l,z) = a, oh a, oh a, oh of(,,,). Now multiply this second equation on the right by of(,,,). This gives OF = at ohof(,,,) = 0, oh 0, of(,,,) of(,,,) = opoh os. Altogether we have as = o, oh oz and oz = opoh os. This shows that o, is C-related to a?, and by Lemma 6.5.5, s must be one of t or (2)' = t'.

As a result, we see that the only way for o, t o be 3-related to ot is for s to be one of t , t,t', or t'.This completes the proof.

Theorem 6.5.19 Let w be a t e r m i n W ( { x ) ) for which ow is not i n {o,) U E uE. Suppose that ow is 3-related to o,, for some u other than w,w',E or .ru'. T h e n a, is C-related to one of a, or 0,'. Proof: Let a, 3 a,, so that there exist terms b, c, p and q such oh owoh ap = a, and a, oh a, oh a, = ow. By substitution we that obtain the equation a, = a oh ow oh 6,where a = a, oh ab and S = op oh 0,. As in the proof of Lemma 6.5.9, we argue that o must be an idempotent. We consider the equation ow = a oh ow oh 6,using the two cases of Lemma 6.5.10. First, suppose that the idempotent a = o f ( , + ) for , some term d in W ( { x ) ) . Then by Lemma 6.5.10 we have o oh ow = ow$,where n is the leftdepth of the term w. This means that owg oh 6 is equal t o 0,. We claim that this forces w = w;.To see this, suppose that 6 = for some terms r and t. For n = 1, we can write w = f ( x , e ) for some term e. Then the term for awg oh 6 is f (8,: ( r ), S2( d , ( r ), GWd( r ) ) ), and the only way this can equal w = f ( x , e ) is for r = x and e = d. This shows that w = f ( x , e ) = f ( x ,d ) = wf, in the case that n = 1.

eW:

Suppose now that w = f ( s , e ) for some terms s and e , with s of leftdepth n. Then oh 6 uses the term f (-, S 2 ( d ,,; z ) ) , where 2 =

6.5 Green's Relations on Hyp(2)

137

[r], &w;+l [r]). We need this term t o equal w = f ( s ,e ) . Now w: has leftdepth of n,and using any inputs into it other than x or y would result in an even greater leftdepth, so we must have r equal to x or y. But this means that x = S2(w;, r, T ) , and w therefore is f (2, S2(d,z, 2 ) ) = f (S2(wt,r, r),S2(d, S2(w;, r, r ) , S2(w;, r, T))). This shows that r = x and w has the form w;+,.

S2(wt ,

In the second case, we have a = of(,,,) for some term a E W({y}). In a similar way we can show that w' must have the required form. Further we have:

Corollary 6.5.20 In the monoid 'Flyp(2) of all hypersubstitutions of type (2), we have 2) = 3 . There is a natural partial order defined on the 3-classes of Hyp(2), as follows. Let us denote the 3-class of an element at of Hyp(2) by J,,, or sometimes for convenience by Jt only. We say that Jt J, iff I(ot) C I(a,). Our results let us build up a complete picture of the poset of all 3-classes of Hyp(2). We have a minimum element, namely Jx= J,, and a maximum element Jf(,,,)= Jf(,,,). We have = E U E.If t is any term in W({x)) an infinite 3-class Jf),,,( or W({y)), then we have I ( % ) C I(af(z,,)),so that Jt I Jf(,,,). If t is a term in both letters x and y but has the special form f ( s ,v) or f (v, s), where s is one of x or y, then it is easy t o see one of which is an that I(at)contains the elements acz[ t ~and acy[t~, idempotent. This means that I(at)contains the ideals such elements generate, so that I(at)contains I . But this containment is proper since elements at where t uses both letters are not in I . Thus we have Jf(,,,)< Jt in this case. In fact we can find an infinite chain of such 3-classes above Jf(,,,). Let t = f (x, f (y, y)), and consider - where a = at. It is easy t o verify by induction the sequence (an),>' that the term corresponding to an always has the form f (x, w) for some term w using only the letter y. In particular, a has infinite 1, we order, and each an has a four element 3-class. For any n can write an+' = a oh an oh a i d , showing that an+' E I ( a n ) . This means that J o n + l 5 Jon. Since the 3-classes are all distinct, we have an infinite descending chain of 3-classes above Jf(),.

<

>

138

6.6

6 Monoids of Hypersubstitutions

Idempotents in H y p ( 2 , 2 )

Notationally, any hypersubstitution a of type (2,2) is determined by the binary terms a and b t o which it maps the operation symbols f and g and we write a,&. Therefore the situation is much more complicated than it is for type (2). An easy observation is:

Lemma 6.6.1 Let Then

be a hypersubstitution of type T = (2,2). is idempotent if and only if Ga,b[a]= a and Ga,b[b] = b.

In 6.1 we introduced already the leftdepth of a term t as the number of occurrences of an operation symbol before the first occurrence of a variable letter. There are a number of natural submonoids of the monoid Hyp(2,2) we can define based on various properties of hypersubstitutions.

Definition 6.6.2 (i) The set of all projection hypersubstitutions of type (2,2) is denoted by

(ii) A hypersubstitution a is called a weak projection hypersubstitution if a (f ) or o(g) belong to {xl, x2). Let W P ( 2 , 2 ) denote the set of all weak projection hypersubstitutions of type 7 = (2,2). (iii) We call a hypersubstitution a of type (2,2) a pre{xl, x2) and a(g) {xl, x2). hypersubstitution if o ( f ) Let Pre(2,2) := Hyp(2, 2) \ W P ( 2 , 2 ) be the set of all prehypersubstitutions of type (2,2).

6

6

It is easy t o see that Pre(2,2), W P ( 2 , 2 )U {aid)and P ( 2 , 2 ) U {aid) are universes of submonoids of Eyp(2,2) and that P ( 2 , 2 ) U {aid} forms a submonoid of the monoid with the universe W P ( 2 , 2 ) U {aid). Further it can be checked that P ( 2 , 2 ) is an ideal in

6.6 Idempotents in H y p ( 2 , 2 )

139

E y p ( 2 , 2 ) and that W P ( 2 , 2 ) is a left ideal in E y p ( 2 , 2 ) . Clearly, every projection hypersubstitution is idempotent. Therefore we have t o consider weak projection hypersubstitutions which are not projection hypersubstitutions and moreover we have to consider pre-hypersubstitutions. For the hypersubstitution o,,b, a , b E W(2,2) ( X 2 ) ,aa,bE W P ( 2 , 2 ) \ P ( 2 , 2 ) we have exactly the following possible cases:

1. a E X 2 and op(b) = 1 , 2. b E X 2 and o p ( a ) = 1 , 3. a E X 2 and op(b) > 1 , 3.1. a

= x1

and op(b) > 1,

3.2. a

= x2

and op(b) > 1,

4. b E X 2 and o p ( a ) > 1 , 4.1. b = x l and o p ( a ) > 1, 4.2. b = 2 2 and o p ( a ) > 1. It is clear that case 1 and case 2 give similar results and that in the cases 3 and 4 we have also similar results. First of all, for the cases 1 and 2 we will give a necessary condition for a,$ to be idempotent.

Lemma 6.6.3 Let a,$ E W P ( 2 , 2 )\ P ( 2 , 2 ) be a n idempotent element. If a E X 2 and op(b) = l , t h e n f i r s t o p s ( b ) = g . If b E X 2 and o p ( a ) = 1 , then f i r s t o p s ( a ) = f . Proof: Assume that a E X 2 and op(b) = 1. By Lemma 6.6.1 we have ~ , , ~ [ = b ]b. Since op(b) = 1 , the term b begins with an operation symbol. Assume that b = f ( b l , b2), bl, b2 € X2. Since o,,b maps the operation symbol f t o a variable we have ~ , , ~ [ # b ]b, which is a contradiction. Hence f i r s t o p s ( b ) = g. For b E X 2 and o p ( a ) = 1 we conclude in a similar way. By using of Lemma 6.6.3 we can answer t o the first and the second cases: (i) I f a E X2 and op(b) = 1, t h e n a a , b is Proposition 6.6.4 idempotent i f and only if b E { g ( x l ,x 2 ) , g ( x 1 , X I ) , g ( x 2 , ~ 2 ) } .

6 Monoids of Hypersubstitutions

140

(ii) If b E X2 and op(a) = 1, then a a , b is idempotent if and only , f (x2,x2)1. if a E {f (x1,22)7f ( ~ 1x1),

Proof: (i) Since op(b) = I , the element b belongs to g ( ~22)). , We exclude the the set XI, X I ) , g ( ~ l ~7 2 ) g7 ( n , case b = g(x2,XI). Assume that a a , b is idempotent and b = g(x2,x1). By Lemma 6.6.1 we have g(x2,xl) = 6 a , b [ g (x~) 2] , = g(xl, x2), a contradiction. It is easy to see that 0 ~ 1 , g ( ~ 1 , x,20)~ 2 , . 9 ( ~ 1 , ~7 2O) x ~ , g ( x ~ 7, x0 x~ 2) , g ( x ~ , x 7~0x~,g(x2,x2) ) and O ~ a , ~ ( ~ a ,are ~ a )idempotent. The proof of (ii) is similar. Now we consider the case 3.1 and assume that a = XI and op(b) > 1. If a a , b is idempotent, then g E ops(Lp(b)) since otherwise 6a,b[b] E X2 and this contradicts op(b) > 1 because of 6a,b[b] = b. Therefore, if Lp(b) = F 1 F 2 . .. F,, Fi E {f,g), there must be a least element i E { I , . . . , n) such that Fi = g and a subterm 6 of b, which has the form 6 = F;(t, t"). Because of Fl, . . . , FG1E { f ), for an idempotent hypersubstitution a a , b we have:

b

= Ga,b[b] = ~ a , b [ 6 = ]

6a,b[g(t7t)]= S2(aa,b(g)7 6a,b[tI7G a , b [ f l ) s2(b, Ga,b[t],Ga,b[fl).

=

have t o be variables. There follows that both, 6a,b[t]and 6a,b[t"] Then we obtain the following propositions: (C1) if x1 E var(b) and t E X2, then t (C2) if x2 E var (b) and

= 21,

t" E X2, then t" = 2 2 ,

(C3) if x1 E uar(b) and t $ X2, then ops(Lp(t)) leftmost(t) = x l , (C4) if x2 E uar(b) and l ef tmost(t") = 22.

t" $

=

{ f ) and

X2, then ops(~p(t"))= { f ) and

Indeed, (C1) and (C2) follow directly from the equation b = S 2 ( b , 6a,b[t], 6a,b[t"]).w e show (C3). Since t $ x2,the term t has the form t = F(tI,ti), F E { f , g), tl, ti E W(2,2) (X2). By induction on the length of Lp(t) (i. e. induction on op(Lp(t)) where t occurs in b = f (t, t") and where op(Lp(t)) is the number of operation symbols occurring in Lp(t)), we show that ops(Lp(t)) = {f). If the lenght of Lp(t) is 1, then t = F ( x i , t i ) , xi E X2. If F = g, then

6.6 Idempotents in Hyp(2,2)

141

which contradicts op(b) > I. Therefore F = f . Inductively, assume that ops(Lp(t*))= { f ) if the length of Lp(t*) is n - I. Consider t = F ( t * ,t i ) , then the length of Lp(t) is n and in a similar way as before the assumption F = g gives a contradiction. Hence F = f . By induction hypothesis, ops(Lp(t))= { f } . If le f tmost(t) = 2 2 , then because of ops ( L p ( t ) )= { f ) we have

This is a contradiction. Thus le ftrnost(t)

= xl

(C4) can be proved in the same manner. Altogether we have the following proposition: Proposition 6.6.5 If a = x l , op(b) > 1 and if Lp(b) = Fl . . . FiplFiFi+l. . . F, with Fl, . . . , F, E { f , g } , then g a , b is idempotent if and only if there exists a least i E { I , . . . , n } such that Fi = g with the subtenn b = Fi ( t ,t"), t , t" E W(,,,)( X 2 ) and the fol-

lowing conditions are satisfied: (i) if var(b) = { x l ) , then t

= xl

or ops(Lp(t))= { f ) ,

t" = x2 or o p s ( ~ p ( t "=) ){ f ) , { x l ,x 2 ) then t = x1 and t" =

(ii) if var(b) = { x 2 ) , then (iii) if var(b)

= 2 2 or ops(Lp(t)) = { f } and le f t m o s t ( t ) = 2 1 , o p s ( ~ p ( i )= ) { f } and le ftmost(t") = x2, or ops(Lp(t)) = - { f } and le f trnost(t) = x l , o p s ( ~ p ( E = ) ) { f ) and l e f tmost(t) = x2.

Proof: Assume that aa,b is idempotent. Then by the previous observation in Lp(b) = Fl . . . Fi-lFiFi+l . . . F, there must occur at least one g. Let i be the least index with Fi = g and consider the subterm b = g(t,t"), t , t" E W ( 2 , 2 ) ( X 2 of) b. From 6a,b[b] = b = S 2(b, 6a,b[t], e a , b [ f l ) , there follows that both, e a , b [ t ] and e a , b [ 4 are variables and we have the following cases:

6 Monoids of Hypers~bstitut~ions

142

For each of these cases we consider the four subcases:

We obtain: (lA),(lD) t E X2, var(b) = {xl}: then by (CI) we have t

(2A),(2c) i

6 X 2 , unr(b) = {z}then : by

{f} and leftmost(l)

= XI.

((%),we have ops(Lp(i))

=

=zr

(3A) t E X2, i $ X2, var(b) = {zl,22): then by (C4), we get o p s ( ~ p ( i )= ) {f) and leftmost(i) = 22. ( 3 ~ t) $ X2, t X2, var(b) = {xI,x2}: then by (C3) we have ops(Lp(t)) = {f) and leftmost(t) = 21. (3c) t $ X2, i $ XZ, var(b) = (51,2%): (C3) and (C4) give ops(Lp(t))-= {f), o p s ( ~ ~ ( l=) ){f}, leftrnost(t) = sl and l ef tmost (t) = 22.

From this we obtain conditions (i)-(iii). Conversely, it is left t o prove that in each of the conditions (i)-(iii) the hypersubstitjution a,,b is idempotent. But this is routine work. By exchanging of a = x1 with a = x2, of Lp(b) with Rp(b), of le f tmost(t) with rightmost(t) and of lef tmost(i) with rightmost(t") we get a similar proposition.

6.6 Idempotents in Hyp(2,2)

143

The next case is t o consider op(a) > 1, b = xl and Lp(a). Then we have to exchange g by f a,nd f by g and get a similar characterization and finally we consider op(a) > 1, b = x2, Rp(a) and instead of le f tmost (t) and i e f tmost(i) we get rightmost(t), rightmost (l) and have a similar characterization. Now we consider the case that in o a , b the terms a and b are not variables. If op(a) = 1 and op(b) = 1, then we have

Proposition 6.6.6 If op(a) = 1 and op(b) = 1, then potent if and only if o a , b € El U E2U E3 for

ga,b

is idem-

Proof: Assume that g a , b is idempotent. Then 6 a , b [ ~ =] a and 3a,b[b] = b by Lemma 6.6.1. If a = f (x2,x l ) , then we must have

which is a contradiction. Thus we exclude the case a = f (x2,XI). Similarly, we exclude the case that b = g(x2,xl). If a = g(xi,xj), b = f (xk, ~ 1 xi, ) xj, ~ xk, 2 1 E X27 then

a contradiction. Therefore, for an idempotent hypersubstitution we exclude the case firstops(a) = g and firstops(b) = f . If a = f ( x i , x i ) and b = f ( x i 7 x j ) , i # j7 then we have ea,b[b] = S2(f (xi, xi), xi, xj ) # b and in the same way we exclude the case a = g(xi,xj), i # j7b = g(xi7xi). Hence precisely the hypersubstitutions from El U E2U E3 are left. Conversely, it is easy t o check that all these hypersubstitutions are idempotent.

144

6 Monoids of Hypersubstitutions

Lemma 6.6.7 (i) If op(a) = 1 and op(b) > 1, then 6,,b[a]= a zf and only if a belongs to the set { f ( X I , x2), f ( X I , X I ) , f (x2,x2)}. (ii) If op(b) = 1 and op(a) > 1, then 6,,b[b]= b zf and only if b belongs to the set { g ( ~ lQ, ) , g(x1,X I ) , g ( x 2 , ~ 2 ) )

Proof: (i) Assume that 6,,b[a] = a. If a = g(xi,x j ) , xi, xj E X 2 , then 6,,b[a] = S 2 ( ~ a , b (xi, g )x7j ) = S2(b,xi, x j ) # a since op(b) > 1, a contradiction. If a = f ( x 2 x, l ) , then 6 ~ , ~ [=a ] S 2 ( o a , b ( fx2, ) , x l ) = f ( x l ,x 2 ) # a, which is also a contradiction. Hence a E { f ( x l ,x 2 ) ,f ( x l ,x l ) ,f ( x 2 x, 2 ) ) .Conversely, it is easy to see that in these cases we have 6,,b[a]= a. In the case (ii) we conclude in a similar way. Lemma 6.6.7 shows that we have t o consider the following cases if op(a) = 1 or op(b) = 1:

It is clear that case 1.1 and case 2.2 as well as case 1.2 and case 2.1 are similar. We consider at first the case 1.2 and obtain:

Proposition 6.6.8 If op(a) = 1, op(b) > 1 and b = g(t,i), t,d t W ( 2 , 2 ) ( X 2then ) 7 o,,b is idernpotent i f and onlg if a t { f ( x l ,x 2 ) ,f ( x l ,x l ) ,f ( x 2 x, 2 ) ) and the following conditions hold: (i) var(b) = { x l ) and b = g(x1,d ) , or (ii) var ( b ) = { x 2 ) and b = g ( t , 22).

Proof: Assume that oa,b is idempotent. Since 6a,b[a]= a, by Lemma 6.6.7 we have a E { f ( x 1 , x 2 f) (, x l , x l ) ,f ( x 2 , x 2 ) ) .ASsume that var(b) = { x l ,22). Since b = 6,,b[b] = 6,,b[g(t7i)]= s2(b,6a,b[t]G a , b [ i ] ) ,then

6.6 Idempotents in Hyp(2,2)

145

This is a contradiction. Thus var(b) = { X I ) or var(b) = { x 2 } . If var(b) = { X I ) , then from b = S2(b,6a,b[t], 6,,b[q) there follows b = g ( xl , t). Similarly, for var ( b ) = { x 2 )we have b = g ( t ,x 2 ) . It is easy to check that all these hypersubstitutions are idempotent. From Proposition 6.6.8 we obtain a similar result which solves the case 2.1 if we exchange in Proposition 6.6.8 the term b = g(t,i) by a = f ( t , t ) ;var(b) = { x l ) by var(a) = { X I ) ;b = g(x1,E) by a = f ( x l ,t"); var(b) = (22) by var(a) = { x 2 ) and b = g ( t , x 2 ) by

a = f ( t ,22). If op(a) = 1, op(b) > 1 and f irstops(b)= f , then b has the form b = f ( t ,i ) . It is easy t o see that in this case the following implications hold:

( C 5 ) if x1 E var(a),then f irstops(t) = f or t ( C 6 ) if x2 E var(a),then firstops(E) = f ot

=

xl,

t = x2.

Indeed, if xl E var(a),t # xl and assume that t = g ( t l , t i ) ,t l ,ti E W ( 2 , 2 )( X 2 ) ,then 6a,b[t]= S 2( b , 6a,b[tl] , e a , b [ti]) and we have = 6a,b[f

(t7

i)]= S 2 ( a 7S2(b7ea,b[tl],&a,b[ti]), &a,b[i]),

but O P ( ~ )= op(S2(a,S2(b,&a,b[tl], ia,b[t',]), i a , b [ i ] ) ) > op(b), a contradiction. If we carry on this procedure we get that ( C 5 )holds. The implication ( C 6 )can be proved in a similar way. To attack the case 1.1 we introduce the following notation: Let t E W ( 2 , 2 ) ( X be 2 ) a term containing only one of both variables, i.e. v a r ( t ) = { x l } or var(t) = (22). Then we define (i) t 1 := t , (ii) tn := S 2( t ,t n p l ,t n p l ) if n

> 1,

(iii) t:%= S 2( t n ,xi, xi) if xi E X 2 , n E N+. Let length(Lp(b))(length(Rp(b)))for a term b be the number of symbols occurring in the semigroup word Lp(b) ( R p ( b ) ) . Then we have:

146

6 Monoids of Hypersubstitutions

Proposition 6.6.9 If op(a) = 1, op(b) > 1 and b = f(t,t") with t , t" E W ( 2 , 2 ) ( X 2then ) , a,,b is idempotent if and only if a E { f ( x l ,x 2 ) ,f ( x l ,x I ) ,f ( x 2 ,2 2 ) ) and the following conditions hold:

(ii) If a

x l ) , then b

=

a&ngth(L~(b)), xi = le f tmost ( b ) ,

(iii) If a = f ( x 2 ,x 2 ) , then b

=

aktngth(R~(b))7 xi = rightmost ( b ).

=

f

(XI,

Proof: Assume that a,$ is idempotent. From 6,,b[a]= a (Lemma 6.6.1) we obtain by Lemma 6.6.7

Since 6a,b[b] = b we have

(i) Assume that a

=

f

(XI,

22). By (*) we get

We consider the following three cases:

( I ) From ( w ) there follows b = f ( e a , b [ t ] ,z k ) . Then ( C 5 )implies firstops(t) = f . Since f is mapped to f ( x l , x 2 ) , we have also f z ~ s t o p s ( i , , ~ [= t ]f). We will show by induction on the complexity of t occurring in b = f ( t ,i),that O P S ( ~ ~ , ~=[ {~ f] )) . If t = f ( X i 7 xj), xi, x j E X 2 , we are done. Let t = f ( t l , t 2 )t,l , t 2 E W(2,q(X2)and assume that O P S ( ~ , , ~ [ = ~ ~ {]f ) ) and ops(ea,b[t2]) = { f ). Then

Therefore ops(b) = { f ). In the second case we obtain the result in a similar way. ( 3 ) By (Ci) and ( C 6 )we have f z ~ s t o p s ( t = ) f and firstops(t") = f . Then using (x*) by induction on the complexities of t and of t",

6.6 Idempotents in Hyp(2,2)

147

respectively, we can show that ops(b) = { f ) since f is mapped t o f ( x l ,x 2 ) and g is mapped to a term starting with the operation symbol f .

(ii) Assume that a = f ( X I , x l ) . By (*) we have b = f ( e a , b [t], e a , b [t] ). From (C5) we get firstops(t) = f or t = X I . The last case is impossible since otherwise b = f ( x l ,x l ) , which contradicts op(b)> 1. With t = f ( t l ,t i ) ,tl, ti E W(2,2) ( X 2 ) we get

(Notice that ops(b) = { f ) . ) Now we set tl W(2,2) ( X 2 )and obtain b = f ( f ( f ( e a , b [ t 2 ] ,e a , b [ t 2 ] ) ,f ( e a , b [ t 2 ] ,G a , b [ t 2 ] ) ) ,f f ( e a , b [ t 2 ] ,ea,b[t2]))).

=

f (t2,t',),t 2 ,t', E

( f ( e a , b [ t 2 ] ,G a , b [ t 2 ] ) ,

This procedure stops with a variable and then we have b a y h ( L ~ ( bfor ) ) xi , = le f tmost(b).

=

(iii) can be proved in a similar way. The converse direction is straightforward.

The case 2.2 can be solved in a similar way and we get: if op(b) = 1, op(a) > 1 and firstops(a) = g , then aa,bis idempotent if and only if b E { g ( x l ,x 2 ) ,g ( x l ,x l ) ,g ( x 2 ,x 2 ) ) and the following conditions are satisfied: (i) if b = g ( x l ,x 2 ) ,then ops(a) = { g } ,

(ii) if b = g ( x l ,x l ) , then a

zength(Lp(a)), = bZz

xi

=

le f tmost ( a ) ,

Now we assume that op(a) > 1 and op(b) > 1. Assume that a = Ga,b[a], firstops(a) = g and firstops(b) = f . Then a and b have the form a = g ( t , i),b = f ( s ,s'), t , i,s, s' E W(2,2) ( X 2 ) .Since

it follows f irstops(a) = f . This is a contradiction and proves that if aa,bis idempotent, then the case firstops(a) = g and firstops(b) = f is impossible. Then we will consider the following cases:

6 Monoids of Hypersubstitutions

148

1. f irstops(a) = f and f irstops(b) = f , 2. firstops(a)

=g

3. firstops(a)

=

and firstops(b) = g,

f and firstops(b) = g.

We obtain the following necessary condition for the idempotency of ga,b:

Lemma 6.6.10 Let op(a) > 1, op(b) > 1. If then Iuar(a)l = 1 and I uar(b)l = 1.

ga,b

is idempotent,

Proof: Assume that au,b is idempotent. Then 6 ~ , ~ [= a ]a and 6a,b[b]= b. We consider the cases 1,2 and 3: 1. In this case we have a = f (t,t"),b = f (tl,t',),t l ,t',,t,t" E W ( 2 , 2 ) ( X 2If) .u a r ( a ) = { x l , x 2 ) from op(a) > 1, op(b) > 1, then we obtain

This is a contradiction. Thus u a r ( a ) = { x l } or u a r ( a ) = { x 2 ) and Ivar(a)l = 1. In the case u a r ( a ) = { x l ) we have b = S 2 ( a ,6 a , b [ t l6] ,a , b [ t i ] ) . Clearly, f i r s t o p s ( t l ) = f and tl can be written in the form tl = f (t2,t',) E W(2,2) ( X 2 ) .This gives

length(L~(b)). Therefore and continuing in this way, we get b = alejtmost(b) uar(b) = {le f t m o s t ( b ) ) and Iuar(b)1 = 1. For u a r ( a ) = { x 2 } , we have var(b) = {rightrnost(b)) and Ivar(b)1 = 1. 2. can be proved in a similar way.

3. In this case a and b have the form a then

=

f ( t , i ) , b = g ( t l , t i ) and

6.6 Idempotents in H y p ( 2 , 2 ) if v a r ( a ) = { x l , 22). Therefore Ivar(a)1 show Ivar(b)I = 1.

149 =

1. In the same way we

For the three possible cases for the first operation symbol in a and in b we have the following results:

Theorem 6.6.11 If o p ( a ) > 1, op(b) > 1, f i r s t o p s ( a ) = f and f i r s t o p s ( b ) = f , then o a , b is idempotent if and only if lvar(a)l = Ivar(b)I = 1 and the following conditions hold:

( i ) zf v a r ( a ) = { x l ) , v a r ( b ) = { x l } , t h e n a a'' E W(2,2) ( { x l } ) and b = alength(Lp(b)),

=

f ( x l ,a") where

(ii) if v a r ( a ) = { x 2 ) , v a r ( b ) = { x 2 ) , then a ( { x 2 ) ) and b = azength(Rp(b)), a' t W(2,2)

=

f (a', x 2 ) where

(iii) zf v a r ( a ) = { x l ) , v a r ( b ) = { x 2 } , t h e n a a" t W(2,2) ( { x l ) ) and b = aF2gth(L~(b)),

=

f ( x l ,a") where

(iv) if v a r ( a )

=

f (a', x 2 ) where

a' t

=

{ x 2 ) , v a r ( b ) = { x l ) , then a b = aF:gth(R~(b)).

W ( 2 . 2 )( { x 2 } ) and

Proof: ( i ) Assume that is idempotent. Then by Lemma 6.6.10, Ivar(a) 1 = Ivar(b)1 = 1. From 6a,b[a]= a , 6a,b[b]= b, a = f ( a l ,a 2 ) , b = f ( b l ,b 2 ) ,with a l , a 2 , bl, b2 E W(2,2) ( X 2 )together with v a r ( a ) = v a r ( b ) = { x l ) we obtain the equations

Since o p ( a ) > 1 and op(b) > 1, the term 6a,b[al]must be a variable and thus al must be a variable and then a l = x l , so

Since b from bl obtain

S2( a ,Ga,b[bl],ea,b[bl]),we get that f i r s t o p s ( b l ) = f and = f (b',, by), with b', , by E W ( 2 , 2 )( X 2 ) and v a r ( a ) = { x l ) we

=

150

6 Monoids o f Hypersubstitutions

This procedure stops after finitely many steps with the variable xl and then b = azength(Lp(b)). The cases (ii), (iii) and (iv) can be proved in the same manner.

For t h e opposite direction we assume that Ivar(a)1 = Ivar(b)1 = 1 and t h e conditions ( i ), (ii), (iii), ( i v ) hold. In t h e first and t h e third cases we have

since u a r ( a ) = { x l } . In the second and the fourth cases we have

6a,b[a]= e a , b [ f (a', Q ) ]

=

S 2 ( a ,6a,b[a'], x 2 )= a,

since u a r ( a ) = { x 2 }

T o show that ea,b[b]= b, we prove b y induction o n k that 6u,b[ak] = ak for every k E w+. Indeed, 6u,b[a1]= 6u,b[a]= a. Assume that = a k p l . Then 6a,b[akp1]

In the other cases we use a similar calculation t o show that ea,b[b]= b.

I f firstops(a)

=

firstops(b)

=g

we have a similar result:

Theorem 6.6.12 If op(a) > 1, op(b) > 1, firstops(a) = g and firstops(b) = g, then aa,b is idempotent 2f and only if I uar(a)l = I uar(b)l = 1 and the following conditions hold: ( i ) 2f v a r ( a ) = { x l ) , var(b) = { x l ) , then b ( { z l ) ) and a = b'ength(L~(a)), bt' E W(2,2)

=

g ( x l , b") where

(ii) 2f u a r ( a ) = {x2}, uar(b) = { x 2 ) , then b ( { x l } ) and a = bzength(R~(a)), bt E W(2,2)

=

g(bl,x 2 ) where

(iii) 2f v a r ( a ) = { x l ) , var(b) = { x 2 ) , then b b" E W(2,2) ( { x 2 ) ) and a = bk:gth(Rp(")),

=

g ( x 2 ,b") where

(iv) 2f u a r ( a ) = {x2}, uar(b) = { x l ) , then b ( { x 2 ) ) and a = bF2gth(L~(a)). b' t W(2,2)

=

g(bt,x l ) where

6.7 The Order of Hypersubstitutions of Type ( 2 , 2 )

151

In the last case we have:

Theorem 6.6.13 If o p ( a ) > 1, op(b) > 1, f i r s t o p s ( a ) = f and f i r s t o p s ( b ) = g, t h e n aa,b is idempotent zf and only if 1 v a r ( a ) l = I var(b)l = 1 and the following conditions hold: (i) zf v a r ( a )

=

{ x l ) , v a r ( b ) = { x l } , t h e n a = f ( x 1 7 a 2 ) ,b dx17 b2) where a2 E W ( 2 , 2 )( { X I ) ) b2 , E W ( 2 , 2 )( { x l ) ) ,

(iii) i f v a r ( a )

=

g(b17 2 2 )

{ x l ) , v a r ( b ) = { x 2 ) , t h e n a = f ( x 1 7a 2 ) , b where a2 E W ( 2 , 2 )( { ~ l ) )bl, E W ( 2 , 2 )( { x 2 ) ) ,

(iv) zf v a r ( a )

{ x 2 ) , v a r ( b ) = { x l } , t h e n a = f ( a l 7x 2 ) , b g(x17 b2) where a1 E W ( 2 , 2 ) ( { ~ 2 b2 } ) ,E W ( 2 , 2 ) ( { ~ 1 } ) . =

=

=

=

Proof: Assume that aa,b is idempotent. Then Ivar(a) 1 = Ivar(b)1 = 1 by Lemma 6.6.10. In the first case from a = f ( a l 7a 2 ) , b = g ( b ~b ,2 ) , with a17 a27 b17 b2 E W ( 2 , 2 ) ( X 2 ) we get

Since in a and b only x l occurs, this implies 6a,b[al]= x1 and thus = x1 and 6a,b[bl]= X I , thus bl = X I . In the other cases we conclude similarly. The opposite direction can be proved straightforward.

a1

6.7

The Order of Hypersubstitutions of Type ( 4 2 )

In this section we study the order of any hypersubstitution of type T = ( 2 , 2 ) , that means, the cardinality of the cyclic subsemigroup of H y p ( 2 , 2 ) generated by o. In the previous section we found all idempotent hypersubstitutions of type ( 2 , 2 ) .The hypersubstitution o is idempotent if and only if the order of o is 1. Now we want to determine the order of each hypersubstitution of type T = ( 2 , 2 ) . The results presented here can be found in [15].As we did before we

152

6 Monoids of Hypersubstitutions

denote by f and g , respectively the both binary operation symbols. We will consider the following cases for a and b of a,,b:

(11) o p ( a ) > 1 and o p ( b ) > 1,

In the first case a and b are elements of the set X 2 U { f ( X I , x z ) , f ( ~ 2 > ~ 1f )( x7 ~ 7 x 1 ) 7 f ( x 2 , x 2 ) > g ( x ~ > x 2g )( >x 2 > x 1 ) > g ( x i , x i ) > g ( z 2 , x 2 ) ) . For o a , b there are 100 cases. The order in each of these cases can be listed in a 10 by 10 table, shown below. The results can be obtained by simple calculation.

6.7 The Order of Hypersubstitutions of Type ( 2 , 2 )

153

We denote by H,"p the set of all hypersubstitutions a,,b with op(a),op(b) > 1. First we determine the order of any hypersubstitution from Hip. We may assume that a := F ( a l ,a 2 ) ,b := G(bl,b 2 ) , F, G E { f , g ) , al , a2,bl , b2 E W(2,21 ( X 2 )and consider the following subcases of (11):

(11-1) var ( a ) = { x l ,x 2 ) ,var (b) = { x l ,x 2 ) ,

, (b) = { x 2 ) , (11-7) var(a) = { x 2 } war

Note that cases (11-3), (11-4) and (11-5) can be proved in the same way as case (11-2),while cases (11-7),(11-8)and (11-9)can be proved in the same way as case (11-6).Therefore, we will consider the cases ( 1 1 - I ) , (11-2) and (11-6). The following formula for the operation symbol count of the composite term S n ( s ,t l , . . . , t,) E W , ( X ) was proved in [38]:

here wbj(s) is the number of occurrences of the variable xj in the term s.

>

Using the fact that o p ( e [ t ] ) op(t) for all t E W, ( X )if a E Hyp(-r) is regular (that is, w a r ( a ( f i ) ) = Xni for every i E I ) , and the formula above, we obtain the following result:

Theorem 6.7.1 Let a, b E W ( 2 , 2 ) ( X 2If) . op(a) > 1, op(b) > 1, war(a) = { x l ,22) and var(b) = { x l ,x 2 } , then the order of a,,b is infinite.

154

6 Monoids of Hypersubstitutions

Proof: Let t = F ( t t ,t") for some F E { f , g } and some t', t" E W(2,2) ( X 2 ) . We have

This shows that 0 ~ ( 6 , , ~ [>t ]op(t) ) for a11 t E W(2,2)(X2) \ X2. Conk+l sequently, o p ( ~ , ,[~a ] )= 0 ~ ( 6 ~ , b [ 6 : , ~>[ oa p] ]( )~ : , ~ [for a ] all ) ic E w+. Hence the order of aa,b is infinite.

-

Let t E W ( 2 , 2 ) ( X be 2 ) a term with uar(t) = { x l } or uar(t) = (22). In the sequel we will use the notation tn introduced in 6.6. The following results can be proved straightforward.

Lemma 6.7.2 Let a, b , t E W ( 2 , 2 ) ( X 2be) such that op(a) > 1, op(b) 1. If t $ X 2 , then 6;,b[t] is not a variable for all n E W+.

>

Lemma 6.7.3 Let a, b, t E W(2,2) ( X 2 ) .If var ( t ) = { x i ) for some i E { 1 , 2 ) , then var(6:,,[t])= { x i ) for all n E N+.

>

Lemma 6.7.4 Let a, b , t E W ( 2 , 2 ) ( X 2be) such that op(a) 1, op(b) 1, and uar(a) = X 2 . If uar(t) = X 2 and ops(t) = { f } , then ~ a r ( 6 ; , ~ [=t ]X) 2 for all n E w+.

>

Lemma 6.7.5 Let a, b E W(2,2) (X2)\X2 be such that uar(b) = { x l } and ~ a r ( i , , ~ [ = a ]X) 2 . Then for every t E W ( 2 , 2 ) ( X 2 ) ,

a ]X2 ) for all n E N+. In particular, ~ a r ( i , , ~ [ = a ]X) 2 + ~ a r ( 6 ; , ~ [=

6.7 The Order of Hypersubstitutions of Type ( 2 , 2 )

155

Proof: Assume that ~ a r ( 6 , , ~ [ a=] )X 2 . Then var(a) = X 2 . Let t E W(2,2) ( X 2 ) .W e will prove by induction on the complexity o f the term t that

This is clear if t is a variable. Let t = F ( t t ,t") for some t',t" E ( X 2 )and some F E { f , g ) and assume that for s E {t',t"),

We assume now that ~ a r ( e , , ~ [= t]X ) 2 . The following conditions

(*) and (**) are satisfied:

We consider two cases: F = g and F = f . If F = g , then Q,,b(F) = b. Since var (b) = { x l ) , by (*) we have ~ a r ( 6 , , ~ [ t=' ]X) 2 . By induc-

tion hypothesis, ~ a r ( 6 ~ , ~=[ X t /2]. )Since , : a ~ ( 6 , [b]) _ ~ = { x l ) , using (**) we have

Assume that F

=

f . Then a,,b(F)

=

a. Since var(a) = X 2 , using

(*) we have X 2 = var (6,,b[tt]) U~ar(6,,~[t"]). We consider three subcases: (a) ~ a r ( 6 , , ~ [ t=' ]{)x i ) and var(6,,b[ttt]) = { x j ) with i

# j,

( c ) var(6a,b[t"]) = X2. Using ~ a r ( 6 , , ~ [ = a ]X) 2 and by ( w ) ,we get that i f ( a ) or ( b ) or ( c ) holds, then ~ a ~ ( 6 2 , ~ =[Xt 2].)

Lemma 6.7.6 Let c, d E W(2,2)(X2)be such that op(c) > 1, op(d) > 1, and c = f (el,C Z ) , d = g ( d 1 , d 2 ) for some cl, ~ 2 d ,l , d2 E W(2,2) ( X 2 ) .Then the following hold:

156

6 Monoids of Hypersubstitutions

(i) if var(c) = {xl} and cl

# XI, then the order of

is infinite;

(ii) if var(c) = {x2} and c2

# 22,

then the order of

is infinite;

# xl,

then the order of oc,di s infinite;

(it) if var(d)

=

{xl) and dl

(ii') if var(d)

=

{x2) and d2 # x2, then the order of

a,,d

is infinite.

Proof: (i) Assume that var(c) = {xl) and cl # XI. Then cl $ X2. Let n E N+. By Lemma 6.7.2, &td[cl]$ X2. Since var(c) = {xl), by Lemma 6.7.3, var(6:d[c]) = (2,). Now

for all n € N+. Hence we have Therefore, op(ir,nil[c]) > 0&)(62~[c]) the claim. A similar argument works for the remaining three cases: (ii), (it), and (ii') , using Lemma 6.7.2. Now we return t o case (11-2), where var (a) = {xl, x2) and var(b) = {xl). We consider the following 4 subcases:

Case (2.1) can be divided into 3 subcases:

Note that for case (2.1.3) we can proceed in the same way as in case (2.1.2). Case (2.1.2) can be divided into 7 subcases:

157

6.7 The Order of Hypersubstitutions of Type ( 2 , 2 )

Note that for the cases (2.1.2.5)-(2.1.2.7) we can proceed in the same way as for the cases (2.1.2.2)-(2.1.2.4), respectively.

Theorem 6.7.7 Let a , b E W ( 2 , 2 ) ( X 2 be ) such that o p ( a ) > 1 , op(b) > 1, v a r ( a ) = X 2 , v a r ( b ) = { X I ) and a = f ( a l , a2), b = f ( b l ,b2) for some a l , a2, bl, b2 E W(2,2) ( X 2 ) . T h e n the following hold: (i) if a and b satisfy (2.1.1), (2.1.2.1) or (2.1.2.4), t h e n

aa,b has

infinite order; (ii) if a and b satisfy (2.1.2.2) or (2.1.2.3), t h e n the order of

aa,b

is less than or equal t o 3 . Proof: (i) Assume that a and b satisfy (2.1.1). Let k E N+. Using Lemma 6.7.5, var(8,h,,[a])= X 2 . Since o p ( a ) > 1 , we have a1 Sf X 2 or a2 Sf X 2 . By Lemma 6.7.2, we have 6:,, [all Sf X 2 or 6:,b [a21 Sf X 2 . Now

This shows that op(6:,i1 [ a ] )> op(6;,,[a]) for all n E N+. Hence has infinite order.

aa,b

Assume that a and b satisfy (2.1.2.1). By Lemma 6.7.2, 6:,b[al] Sf X 2 for all n E N+. Since ~ a r ( 6 , , ~ [ a= ] ) { X I ) , by Lemma 6.7.3 v ~ r ( d ; , ~ [ a=] {) X I ) for all n E N+.Therefore, (* * *) for any k E N+ is satisfied. It follows that the order of aa,b is infinite.

158

6 Monoids of Hypersubstitutions

Assume that a and b satisfy (2.1.2.4). Since ~ a r ( i , , ~ [ a=] {) x l ) , 6:,b[a] = 6a,b[a]for all n E N+. Since g E o p s ( L p ( b l ) ) ,so 6a,b[bl]= S2(a17 S2 . . . S 2( a m , S2(b7 C a , b [ 7~ea,b[v]) ] 7 &a,b[vm]) 7 . . . 6 a , b [ ~ 2 7] ) 6 a , b [ ~for l ] some ) rn E w+, some a1 = a2 = . . . = a , = a and some U , v , v l , . . . , v , E W ( 2 , 2 ) ( X 2 Let ) . k E w+. Since ~ a r ( 6 , , ~ [ a=] ){ x ~ ) , it follows that o P ( 6 k i 1[ b l ] )= o p ( S 2((6:,,[a])'",S 2(6:,b[b](6:;' [u],

62;' [vl,S 2( e , b [bl, 62t1[ ~7 62i1 l [ v l ) ) )> 0 M , b PI). Thus oP(62;'[b]) = o p ( S 2 ( a 762:' [ b l ] 6;;' , [$I)) > 0 p ( 6 ! , ~ [ b ]which ) b ]op(A;,,[b]) ) for all n E N+. Hence the orimplies that o P ( 6 ~ , ~ ' [ > der of

0a.b

is infinite.

(ii) Assume that a and b satisfy (2.1.2.2). Since v ~ r ( i , , ~ [ a=] {) x l ) , a l = X I and f i r s t o p s ( a ) = f , we obtain 6:,b[a]= 6a,b[a].This gives

Since bl

~ a r ( 6 , , ~ [ a=] ){ x l } , and f i r s t o p s ( b ) = f , we have 3 6:,b[b] = 6a,b[a].s o o:,b(g) = o a , b [ q = 6a,b[z,b[b]] = 6a,b[6anb[a]] = 6 i , b [ a ]= 6u,b[a]= 6i,b[b]= c T : , ~ ( ~Hence ). ' T : , ~ = 's:,~. This shows that the order of o a , b is less than or equal t o 3. = XI,

-

Assume that a and b satisfy (2.1.2.3). As for (2.1.2.2), we have 6z,b[a] = &,b[a], which implies o$,,(f ) = o;,,(f ) . Since o P s ( L ~ ( b 1 )=) { f I-, we get ea,b[b] = S 2 ( a l ,S 2 ( a 2 ,. . . 7 S2(am-1, S 2 ( a m ,2 1 , X I ) , ea,b[vm-I]), . . , 6 a , b [v2])7 ea,b[vl]) for some v l , . . . , urn-1 E W(2,2) ( X 2 ) with m = o p ( L p ( b ) ) and a1 = a2 = . . . - a , = a . Since v ~ r ( i , , ~ [ a=] ) { x l ) , it follows that 2 'Ta,b[b] = ( 6 a , b [ a ] ) mSince . 6z,b[a]= 6 a , b [ a ]we , get

-

Consequently,

Hence

= ( T : , ~ , SO

the order of

ou,b

is less than or equal to 3.

In case ( 2 . 2 ) , where f i r s t o p s ( a ) = g and f i r s t o p s ( b ) sider 5 subcases:

=

g we con-

6.7 T h e Order o f Hypersubstitutions o f T y p e ( 2 , 2 )

159

Theorem 6.7.8 Let a , b E W ( 2 , 2 ) ( X 2 be ) such that o p ( a ) > 1 , op(b) > 1, u a r ( a ) = X 2 , u a r ( b ) = { X I ) , a = g(a1, a2) and b = g ( b l , b2) for some a l , a', bl, b2 E W(2,2) ( X 2 ) . T h e n the following hold:

( i ) if a and b satisfy (2.2.1) or (2.2.3), t h e n aa,bhas infinite order; ( i i ) if a and b satisfy (2.2.2), (2.2.4) or (2.2.5), t h e n the order of aa,b is less than or equal t o 2. Proof: ( i ) A s s u m e t h a t a and b satisfy (2.2.1). Since f i r s t o p s ( b ) = g , v a r ( b ) = { x l ) and bl $ X 2 b y L e m m a 6.7.6 ( i f )t h e order o f o a , b is infinite.

A s s u m e t h a t a and b satisfy (2.2.3). Since f i r s t o p s ( b ) = g , v a r ( b ) = { x l ) and b1 = 2 1 , w e get 6:,,[b] = b for all n E N+. W e set L p ( a ) = Fl . . . 4 F L + l . .. F , w i t h F I , . . . , FZ E { g ) , 4 + 1 = f and 4+',.. . , Fm E { f , g } , and t := FL+1(tf,t") for some tf,t" E W(2,2) ( X 2 ) .For k E N+,w e have ' + l [a] = Oa,b S2(bz, 62;' [t], & 2 i 1[ t ] ) 1 f = S2(b1>S2(6k,b[a]>o [t]>'2i1[t"])>S2('k,b[a]>'k$1[tf]> a,b '+'[t"])). Oa.b k+l [ a ] )> 0 ~ ( 6 ~ , ~ for [ a ]all) R E w+. T h u s t h e Consequently, op(oa,, order o f o a , b is infinite. A

-

-

(ii) Assume t h a t a , b satisfy (2.2.2). Since u a r ( b ) = { x l ) , w e have 6a,b[b]= I) and 6a,b[a]= s2( b ~ p ( ~ p l(e~f t)m) ,o s t ( a ) ,l e f t m o s t ( a ) ) ,w e have

-

2

aa,b[a]

=

[s'( ~ o P ( ~ P ( le~ )f )t,m o s t ( a ), lef t m o s t ( a ) ) ]

=

s ' ( ( ~ ~ , ~ [ ~ ] ) o P ( ~ Pl e( f" t) m ) ,o s t ( a ) ,l e f t m o s t ( a ) )

=

~ ~ ( b ~ ~le f (t m~o s~t ((a )~,be)f t)m o, s t ( a ) )

=

&a,b[a].

160

6 Monoids of Hypersubstitutions

This gives

(T:,~

2 = (T,,~.

If a and b satisfy (2.2.4), then ea,b[a]= b and 6,,b[b] = b; and if a and b satisfy (2.2.5),then we have 6,,b[a] = b ( x 2 ,x 2 ) and 6a,b[b] = b. These give o : , ~= ( T2 , , ~ . In cases (2.3),where f irstops(a) = g, f irstops(b) = f , we consider 5 subcases:

Theorem 6.7.9 Let a , b E W ( 2 , 2 1 ( X 2be) such that op(a) > 1, op(b) > 1, v a r ( a ) = X 2 , v a r ( b ) = { z l ) , a = g(a1, a2) and ( X 2 ) . Then the following b = f ( b l ,b2) for some a l , a2, bl, b2 E W(2,2) hold: (i) if a and b satisfy (2.3.1), (2.3.2) or (2.3.4), then finite order;

o,,b

(ii) if a and b satisfy (2.3.3) or (2.3.5), then the order of

has in-

(Ta,b

is

less than or equal to 3 . = Proof: (i) Assume that a and b satisfy (2.3.1). We consider a6a,b[a],6a,b[b]. Since ea,b[b] = e a , b [ f (bl, b2)] = s2(a7 ea,b[bl], 6a,b[b2]) = ~ ( ~ ~ea,b[bl]: ( ~ 6a,b[b2]), 1 , S2(a2,&a,b[bl], 3a,b[b2])), so f i r s t o p ~ ( 6 , , ~ [ b=] )g. Since v a r ( b ) = { x l } , ~ a r ( i , , ~ [ b=] {) x l ) . Using our assumption, we get S 2 ( a l ,6,,b[bl],6,,b[b2]) 6 X 2 B y Lemma 6.7.6 (i'), the order of ~ 6 , , b [ a ~ ~ ,is, , infinite. [b~ Since (a:,,) is a subsemigroup of we conclude that the order of a,$ is infinite. A similar argument works if a and b satisfy (2.3.2) or (2.3.4).

(ii) If a and b satisfy (2.3.3), then 6,,b[a] = 6i,b[a]and b = 62,b[b]. This gives ( T : , ~ = o:,~.A similar argument works for the case that

6.7 T h e Order o f Hypersubstitutions o f T y p e ( 2 , 2 )

161

a and b satisfy (2.3.5). In cases (2.4), where f irstops(a) = f , firstops(b) 4 subcases:

= g,

we consider

Note that case (2.4.4) can be proved in the same way as case (2.4.3).

In case (2.4.3) we separate t o 2 cases:

Theorem 6.7.10 Let a , b E 1/17(2,2) ( X 2 ) be such that op(a) > 1, op(b) > 1 , v a r ( a ) = X 2 , uar(b) = { x l } , a = f ( a l , a 2 ) and b = g(bl, b2)for some a l , a2, bl, b2 E W(2,2) ( X 2 ) .Then the following hold: ( i ) if a and b satisfy (2.4.l ) , (2.4.2) or (2.4.3.I ) , then a , , b has infinite order; (ii) if a and b satisfy (2.4.3.2), then the order of a,$ is less than or equal to 3. Proof: ( i ) I f a and b satisfy (2.4.I ) , t h e n b y Lemma 6.7.6 ( i ' ) we get that the order of aa,b is infinite. If a and b satisfy (2.4.2),then using Lemma 6.7.6 (i) we get that the order of a,$ is infinite. Assume that a and b satisfy (2.4.3.1). Consider = (T&a,b[a],6a,b[b]. Since al $ X 2 , we obtain 6,,b[a] $ X 2 . Clearly, f ~ r s t o p s ( 6 , , ~ [ a= ]) f . Since ~ a r ( 6 , , ~ [ a=] {) x l ) , b y Lemma 6.7.6 ( i ) we have that t h e order of aea,b[al,ea,b[b] is infinite. Hence the order of a,$ is infinite. (ii) Assume that a and b satisfy (2.4.3.2). T h e n 6,,b[b] = b. I f ~ 1 1= x1, t h e n b y ~ a r ( 6 , , ~ [ a= ] ) { x l } and have 6:,b[a] = Ga,b[a].T h u s (aa,b) = {aa,b,a & ) since o ~ ( e a , b [ a ]>) O P ( ~ )I f. a1 = x2, then %,b[a]= S2(ir,,b[a], 2 2 , x 2 ) . It follows that 6&[a] = S 2 ( 6 a , b [ a5]2, , x 2 ) = 6z,b[a].This gives =

162

6 Monoids of Hypers~bstitut~ions

W e conclude our investigation o f case (11) w i t h subcase (11-6), where u a r ( a ) = { x l } , u a r ( b ) = { x l } . Hence w e have t h e following subcases:

Theorem 6.7.11 Let a , b E W(2,2j ( x 2 ) be such that o p ( a ) > 1, op(b) > 1 , v a r ( a ) = { X I ) , and v a r ( b ) = {.xl). T h e n the following hold. ( i ) Let f i r s t o p s ( a ) = f and f Zrstops(b) = f . If a , b satisfy (6.1.1) is infinite. or (6.1.2), t h e n the order of (ii) Let f i r s t o p s l a ) = g and f i r s t o p s ( b ) = g . If a , b satisfy (6.2.1) or (6.2.2), t h e n the order of a,,b is infinite.

6.7 The Order of Hypersubstitutions of Type (2,2)

163

(iii) Let firstops(a) = f and firstops(b) = g. If a , b satisfy (6.3.1) or (6.3.2), then the order of a,$ is infinite. (iv) Let f irstops(a) = g and f irstops(b) = f . If a , b satisfy (6.4.1) or (6.4.2), then the order of a,$ is infinite.

Proof: (i) If a and b satisfy (6.1.1), then by Lemma 6.7.6 (i) we have that the order of a,$ is infinite. Assume that a and b satisfy (6.1.2). We set Lp(b) = Fl . . . F&+l.. . F, with F l , . . . , Fl E {f}, Fl+l= g, and F1+2,.. . , F, E {f,g); and t := Fl+l(t', t") for some t', t" E W(2,2)(X2).For k E w+, we have k+l 0 a . b [bl = S2@, 6:,l [tl,6 y [tl) k+l I = S2(a1,~2(6,h,b[b],6,,b [t],62$1[t"]), s2(6;,, [b],622 [t'],622 [t"]) ) . Consequently, op(6;$'[b]) > 0p(6:,~[b]) for all k E w+. Thus the order of a,.b is infinite. A

(ii) This can be proved similarly as (i). (iii) If a , b satisfy (6.3.1), then a l 6 X2. By Lemma 6.7.6 (i), we get that the order of is infinite. A dual argument works for the case that a, b satisfy (6.3.2), using bl X 2 and Lemma 6.7.6 (if).

6

= (iv) Assume that a and b satisfy (6.4.1). We consider X2,6a,b[a] X2. Clearly, f i r ~ t o p s ( 6 , , ~ [ a=] ) [bl . S'ince a 1 Oea,b[,I >ea,b f . By Lemma 6.7.6 (i), the order of aea,b[a],ea,b[b] is infinite. We conclude that the order of a,,b is infinite. A dual argument works for the case that a and b satisfy (6.4.2), using the following conditions: = g and Lemma 6.7.6 (if). Ga,b[a] X2, f~rstops(6,,~[b])

6

6

6

Theorem 6.7.12 Let a , b E W(2,2) (X2) be such that op(a) > 1, op(b) > 1, var(a) = {zl), var(b) = {xl). Then the following hold. (i) If firstops(a) = f and firstops(b) = f and if a, b satisfy (6.1.3) or (6.1.4), then the order of o,,b is less than or equal to 2. (ii) If firstops(a) = g and firstops(b) (6.2.3) or (6.2.4), then the order of to 2.

=

g and if a, b satisfy is less than or equal

164

6 Monoids of Hypersubstitutions

(iii) If firstops(a) = f and firstops(b) = g and if a1 bl = XI, then the order of a,$ is 1.

= xl

and

(iv) If firstops(a) = g and firstops(b) = f and if a1 = xl and bl = XI, then the order of a,,b is less than or equal to 2.

Proof: (i) If a , b satisfy (6.1.3), then 6,,b[a] = a and 6i,b[b] = 8n,b[b]. Thus oi,b = o:,~. If a, b satisfy (6.1.4). then 6a,b[a] = n and ea,b[b] = b. So is idemyotent. (ii) This can be proved using a dual argument as we used in the proof of (i). (iii) We have 6,,b[a] idempotent.

=

a and ea,b[b]

(iv) w e have 6,,b[a] = b and Ga,b[b]

=

=

b which implies that o,,b is

a which implies o:,b

=ga,b

Now we will deal with the case 111, i.e. op(a) = l,op(b) > 1 or op(a) > 1, op(b) = 1. We consider 9 subcases, denoted by (111-1)(111-9) which correspond to the subcases (TI-1)-(11-9) of case (11), respectively. Also, we will consider the cases (111-I), (111-2) and (111-6), where

In case (111-I), we consider the following subcases:

6.7 The Order of Hypersubstitutions of Type (2,2)

165

Note that cases (3), (4), (7) and (8) can be considered as cases ( I ) , (2), (5) and (6), respectively. We have the following results

Proposition 6.7.13 Let b = f (bl, b2) for some bl, b2 E W(2,2)(X2) be such that uar(b) = X2 and op(b) > 1. T h e n the following hold: (i) if a

=

f (xl, 2 2 ) t h e n the order of

(ii) if a = f ( 2 2 , xl), t h e n the order of 3 o r infinite.

Proof: (i) Assume that a two cases:

=

(a) This gives 6a,b[b] = b, so

aa,b aa,b

is 1 o r infinite; is less t h a n or equal t o

f (21, 22). Then 6a,b[a]= a. We consider

aa,b

is idempotent.

(b) We may assume that g E ops(bl). Let k E N+. Then o~(e,h$'[bl]) O P ( ~ : , ~ [ ~ ] ) .

>

A similar argument works for g E ops(b2). This shows that the order of a u , b is infinite. (ii) We also consider the two cases (a) and (b) from the proof of

166

6 Monoids of Hypersubstitutions

2 (a) This gives 6:,,[a] = 6:,,[a] and 6,,b[b] = 6:,,[b]. Thus a:,, = a,,,. Then the order of a,,b is less than or equal t o 3. (b) This can be proved similarly as (i) (b).

Proposition 6.7.14 Let b = f ( b l , b2) for some b ~ b2, E W(2,2)(X2) be such that var(b) = X2 and op(b) > 1. If a = g(xl,x2) or a = g(x2,xl), the order of o a , b is infinite. Proof: Assume that a = g(xl, x 2 ) Then 6,,b[a] = b and 6,,b[b] = g(6a,b[bl], 6a,b[b2]).Since var(a) = X2 and var(b) = X2, we have u ~ T ( ~ , , ~ [= u ]!a~(6,,~[b]) ) = X2. By Theorem 6.7.1, o:,, has infinite order. Hence the order of a,,, is infinite. A similar argument works also for a = g(x2,xl). In Case (111-2), we consider the 8 subcases of case (111-1). Also, we will consider the following cases:

The following proposition is proved similarly as Proposition 6.7.13. Proposition 6.7.15 Let b = f (bl, b2) for some bl, b2 E W(2,4(X2) be such that var(b) = {xI) and op(b) > 1. T h e n the following hold: (i) if a

=

f

( X I , x2),

then the order of a,$ is 1 or infinite;

(ii) if a = f (x2,xl), then the order of oa,b is less than or equal t o 3 or infinite. Proposition 6.7.16 Let b = f (bl, b2) for some b ~ b2, E W(2,2)(X2) be such that var(b) = {xI) and op(b) > 1. T h e n the following hold: (i) if a = g(xl, x2), then the order of 3 or infinite; (ii) if a = g(x2,XI), then the order of 4 or infinite.

oa,b

is less than or equal to is less than or equal to

167

6.7 The Order of Hypersubstitutions of Type (2,2)

Proof: (i) Assume that a = g(xl, x2). If bl = x l , then b = 6z,b[b] m d 6A,b[a]= b = 6:,b[a]. This gives CT;,~ = CT;,~. Hence the order of a a , b is less than or equal to 3. Assume that bl $ X 2 . We consider oa,,b~a~,a,,b~ai. Since ii,,b[a] = h, using Lemma 6.7.6 (i) we get that the order of is infinite. Therefore the order of a a , b is infinite. A dual argument gives (ii). Finally we will consider the case (111-6), where uar(a) {xl). We consider the following subcases:

=

uar(b)

=

We consider the following sets of hypersubstitutions.

The mapping 9 : SI + S2 defined by a f ( z l , z l ) , b ag(zl,zl),b is a bijection. Similarly, the mapping 9 : TI + T2 defined by og(xl,xl),bH of(zl,zl),bis a bijection. Hence we will investigate the order of hypersubstitutions satisfying (6.1) and (6.2), respectively.

Proposition 6.7.17 Let a = f ( x l , x l ) , firstops(b) 1, and uar(b) = {xl). Then the following hold:

=

f , op(b)

>

168

6 Monoids of Hypersubstitutions is infinite;

(i) if b satisfies (6.1.1), then the order of

(ii) if b satisfies (6.1.2) or (6.1.3), then the order of than or equal to 2.

o,,b

is less

Proof: (i) Since a = f ( X I , X I ) , firstops(b) = f and g E ops(Lp(b~)),it follows that 6,,b[b] = S2(f (21,XI)', S2(b, U, v), S2(b, u', vt)) for Some U, v, u', vt E W(2,2)(X2) and some k E w+. Hence the order of is infinite since for n E w+,we have

(ii) Assume that a , b satisfy (6.1.2). Since a = f (xl, x l ) and g !$ 6a,b[b]7 W have 62,b[b] = 6,,b[b]. (11~arly,62,b[(7]= (I. Hence o3 a,b = o:,~. If a , b satisfy (6.1.3), then o,,b is idempotent. Thus the order of o a , b is less than or equal to 2.

Proposition 6.7.18 Let a = f ( x l , x l ) , firstops(b) 1, and uar(b) = {xl}. Then the following hold:

=

g, op(b)

>

(i) if b satisfies (6.2.1), then the order of o,,~is infinite; (ii) if b satisfies (6.2.2), then the order of

o,,b

is 2.

Proof: (i) Using Lemma 6.7.6 (ii) we have the claim. (ii) Since a = g(xI, x l ) and var(b) = {xl), we have 6,,b[a] = b # a. Since bl = xl and firstops(b) = g, we get 6a,b[b] = b. It follows that 6&[b] = b for all n E @. Therefore ( o g ( x l , x l ) , b ) = { o g ( x ~ , x l ) _ bob,b). ;

In case (IV) we will only consider the case when op(a) = 0 and op(b) > 1, the case op(a) > 1,op(b) = 0 can be proved similarly. We consider the following subcases:

6.7 The Order of Hypersubstitutions of Type (2,2)

169

Note that cases (IV-4), (IV-5) and (IV-6) can be proved in the same way as cases (IV-1) , (IV-2) and (IV-3), respectively. Moreover, case (IV-2) can be proved in the same way as case (IV-I). For case (IVI ) , we have the following result.

Proposition 6.7.19 Let a = xl, var(b) = {xl), and op(b) > 1. Then the order of g a , b is less than or equal to 2 or infinite. Proof: When op,(Lp(b)) is the total number of occurrences of the operation symbol g in Lp(b), we consider three cases:

(b) In this case

oa,b

is idempotent.

(c) We have 6a,b[b] = S 2 ( b , S 2 ( b , u, v), t) for some u, v, t w ( 2 , 2 ) ( x 2 )Let . k t w+. Then 0p(6;,~[b]) 1. Now

>

E

since var(6k,b[b])= {x1}. Therefore, the order of oa,b is infinite. rn To prove (IV-3), we need the following result which can be proved similarly as Lemma 6.7.5.

170

6 Monoids of Hypersub~titut~ions

Lemma 6.7.20 Let a, b E W ( 2 , 2 ) ( X 2be) such that a = xl and ( X 2 ) ,zf ~ a r ( d , , ~ [= t ]X ) 2, uar(b) = X 2 . Then for every t E W(2,2j then ~ a r ( i i , 2 , ~= [ tX ] )2 . Proposition 6.7.21 Let a = X I , uar(b) = X2, b = F(b1,b2) for some b l , b2 E W(2,21 ( X 2 ) ,F E { f,g ) and op(b) > 1. Then the following hold: ( i ) if ops(Lp(b))= {f),then the order of (ii) if op,(Lp(b)) = 1, then the order of to 4 or infinite; (iii) if op,(Lp(b)) > I , then the order of to 4 or infinite.

is 2;

o,,b

is less than or equal

o,,b

is less than or equal

3 Proof: ( i ) This gives ( a a $ )= {0,,,b7 azl,leftmost(b)}since 02,b = Oa,b. (ii) Assume that ~ , ( L p ( b ) ) = 1. T h e n we have e a , b [ b ] = S 2( b , lef tmost ( b ), 6a,b[b2]). W e consider three cases:

( a ) This gives

=

2 o,,~.

( b ) W e consider three subcases: (b-1) b2 E X2, (b-2) ops(Lp(b2))= { f ) , (b-3) g E ops(Lp(b2)). Both, (b-1)and (b-2) give that the order of Oa,b is less than or equal t o 4. (b-3) Let k E w+. Then op(6,*~'[b2]) > 1. B y Lemma 6.7.20, we have x2 E 1 1 0 ~ ( 6 k , ~ [Therefore b]). we have

6.8 Green's Relations in Hyp(n, n ) Hence the order of

aa,b

is infinite.

(c) This can be proved similarly as (b). (iii) Assume that op,(Lp(b)) > 1. Then e a , b [b] = S2(b, S 2 ( b , u, v), ea,b[b2]) for some u, v E W(2,2)(X2). Let k E w+. Then we have the equation: = 6;$[6a,b[b]] 6;,b [S2 (b7S2(b, u72)). 6a,a [bn])] = [b]). s2 (';,b[b] S2(6k,b[b] &;,b [u]I &;,b [u])7 Using this equation, we have t o consider the cases (a), (b) and (c) in the proof of Proposition 6.7.21 (ii).

[4

(a) This gives 0 p ( 6 [q) ~ ~> ~0 ~ ( 6 ; , ~ [ b ] which ) implies that the order of a a , b is infinite. (b) By Lemma 6.7.20, we have vor(6;,,[b]) = X2. Then op(&:$' [b]) > op(6;,,[q) which implies that the order of a,,b is infinite. (c) We consider three cases (c-l), (c-2), and (c-3) corresponding t o (b-1)-(b-3) in the proof of Proposition 6.7.21 (ii),respectively. (c-1), (c-2) give that the order of o a , b is less than or equal to 4.

4

(c-3) Let k E w+, then e have 6:$'[b2] X2. Then oP(6:$'[b]) = op(S2(A:,, [q,&:$I [b2], 6 : ; ' [b2])) > o P ( q b[b]) . Therefore, the order of o a , b is infinite.

6.8

Green's Relations in H y p ( n , n)

>

Now we consider the type T = (n, n ) for n 1, with two n-ary operation symbols f and g. We study Green's relations for this type. As usual we denote by the hypersubstitution which maps the operation symbols f and g, respectively to s and t. By Corollary 6.1.2 we have (a,,t,a,,,,) E R iff the clones ({s,t}) and ({sf,t'}) generated by s, t and by sf,t', respectively are equal.

6 Monoids of Hypersubstitutions

172

For any term t E W(,,,)(X,) we define the following set of terms St := {Sn(t,x ~ ( .~. .) ,,xp(,)) 1 y : { I , . . . , n } + { I , . . . , n } is a mapping}. The mapping which produces the term 1 E St from the term t is denoted by ~ l ,Clearly ~ . St is a subset of the universe of the clone generated by the term t. Therefore we have:

Therefore we need not to consider hypersubstitutions which map both operation symbols to variables and we have to check the following pairs of hypersubstitutions: 1. o,,, and

where x E X,, s , t E W(,,,)(X,) \X, and s # t,

In the first case we have:

Theorem 6.8.1 Let x E X, and let s, t E W(,,,) (X,) Then ~,,,Ro,,~ zf and only if s E St and t E S,. Proof: If t E S,, then

=

Oh

\ X,, s # t.

ox,g(xYt,s( ~ ) > . . . > x ~ ~ ,and ~(n))

Ox,s = Ox,t O h ax,g(x,s,t(~), . . . , x , ~ , ~ ( " ) )and

therefore ax,sRax,t.The next step is t o show that from ax,sRax,t there follows op(s) = op(t). Assume that op(s) # op(t). Without loss of generality, we may assume that op(t) < op(s). We show by contradiction that there is no q E W(,,,) (X,) such that t = ex,,[q]. Indeed, if q E X,, then t E X,, a contradiction. If q $ X,, then q = F(ql, . . . , q,) where F E { f ,g} and inductively we assume iX,,[qk]# t, for k E (1, . . . , n}. Then

If F

=

f , then

from a,,, ( f ) = x we get

e,,, [q] = e,,, [qk]for some

k E {1,2, . . . , n}. But by the induction hypothesis each ix,s[qk] is equal to t, and we are done. If F = g, then by the formula before Theorem 6.7.1 and our assumption,

6.8 Green's Relations in Hyp(n,n )

173

which implies t # 6,,,[q]. This proves 6,,t[W(n,nl (X,)] # 6,,,[W(,,,)(X,)] and by Corollary 6.1.2 we get (a,,t, a,,,) gf R, a contradiction. Now we show that ax,,Rax,t implies t E S, and s E St. First of all, we realize that if t is not in S,, then there is no q E W(,,,)(X,) with t = 6,,,[q]. Indeed, if q E X , and t = 6,,,[q], then t E X,, a contradiction. Assume that q = F(ql, . . . , q,), F E { f , g) and assume inductively that 6,,,[qk] # t for k = 1 , . . . , n . Then

If F = f , then we have 6,,, [q]= 6,,, [qk]for some index k , but again by the induction hypothesis no 6,,,[qk] can be equal to t . If F

= g,

then

We set t k := 6,,,[qk]for k = 1 , . . . , n . If there is a ko E ( 1 , . . . , n ) such that tkogf X , and if xko E v a r ( s ) ,then

which implies t

# i,,,[q]

If for all k E { I , . . . , n) the term tk is a variable or xk $ var ( s ) , then 6,,,[q] E S,, so 6,,,[q] # t , by our assumption. Altogether, we (X,)] # 6,,, [W(,,,)( X , ) ] , a contradiction t o Corolhave 6x,t[W(n,nl lary 6.1.2. This shows that t E S,. In the same way one shows s E

st. We mention the following result from [14]:

Corollary 6.8.2 Let x E X , and let s , t E W(,,,) (X,) \ X , with s # t . Then o,,~R~,,, if and only if there is a permutation cp* : {17 . . . n } (17 . . . n } such that ax,, = ax,t oh ~ X , ~ ( X , * ( , ) ,...,x(DX;(n)). 7

+

1

Now we consider the second and the third case and prove at first several necessary conditions for ax,tRa,,, and as,tRa,,,.

174

6 Monoids of Hypersubstitutions

We need the following sets of terms:

Ct =: { S n ( 46s,t[al17. . . , es,t[an])I a17. . . , a, E W(,,,)(X,)),

Lemma 6.8.3 Let x E X , and let s , t , u , v E W(,,,) (X,) s # v and t # u. (i) If as,tRa,,,, then u , v E C,

\ X,

with

u Ct.

(ii) If ax,tRa,,v, then u , v E Ct.

Proof: (i) Since o,,~R~,,,, there exists a hypersubstitution o,,b E Hyp(n, n ) such that o,,, = oh o a , b and then v = 6s,t[b]and u = 6s,t[a].Since u , v $ X,, also a, b $ X , and the term a has the form F ( a l , .. . ,a,), F E { f , g } and a l , . . . , a , E W(,,,)(X,). For F = f we have u = S n ( s ,6,,t[al],. . . , 6,,t[an]).For F = g , then u = S n ( t ,6 s , t [ a l .] ., . , 6s,t[a,])and for v we obtain a term of the same kind. (ii) In this case there exists a hypersubstitution aa,b E Hyp(n, n ) such that a,,, = ax,t oh aa,b. Thus u = 6x,t[a]and from u $ X , we obtain a Sf X,. Let a = F ( a l , . . . , a,), al, . . . , a, E W(,,,) (X,), where F E { f , g ) . If F = g , then

u

=

S n ( t ,GX,t[all,. . . ,6,,t [a,]),

as required. If F

u Sf F'E

If F'

f , there is a k E (1,.. . , n ) such that u = 6x,t[ak]. Since (1) X,, we have ak $ X , and then ak = F1(a1 , . . . , a (,I ) ) , where =

{ f , g l , a1( 1 ), . . . , a , = g,

E W(n,n) (Xm) \ Xm.

then

as required. If F' = f , we conclude as before. For v we obtain a term of the same kind. Therefore u , v E Ct.

6.8 Green's Relations in Hyp(n, n )

175

A consequence of Lemma 6.8.3 and the formula before Theorem 6.7.1 is: Lemma 6.8.4 Let s, t ,u,v E W(,,,) (X,)

\ X,.

Then we have:

Proof: (i) From u , v E Ct and the formula before Theorem 6.7.1 one obtains op(u) op(t) and op(v) op(t). (ii) can be proved in a similar way.

>

>

Lemma 6.8.4 and o,,~Ro~,, show that for O,,~RO,,, the following

mark that the third case cannot happen since in this case there is no term q with t = 6,,,[q]. Indeed, q cannot be a variable since t is not a variable and from q = F ( q l , .. . , q,), ql, . . . , qn E W(,,,)(X,) \ X,, F E { f , g } for F = f we obtain op(t) = 0p(Sn( u ,&,v [ R ] ., . . , &,v [qn])) op(u) and for F = g we have op(t) op(v) which contradicts the third case.

>

>

>

In the fourth case we must have op(t) = op(v) since op(t) op(u) cannot happen and since op(t) > op(v) gives also a contradiction. Altogether, exactly the following cases are left:

Because of o,,,Ro,,,

we have not to consider the case (111).

For any w ,z E W(,,,)(X,), we denote by S,,, the union S,,, Sw U S,. Then we have

:=

Lemma 6.8.5 Let x E X,, t ,u , v E W(,,,) (X,) \X, . If (a,,,, a,,,) E R, then t E S,,, .

176

6 Monoids of Hypersubstitutions

Proof: Assume that (a,,t,a,,,) E R . Then there exists a hypersubstitution a,,, E Hyp(n, n) such that ax,t= a,,, oh ap,qand thus t = i,,,[q]. Since t $ X, we have q $ X, and then there exist elements ql, . . . , q, E W(,,,)(X,) such that q = F(ql, . . . , q,) where F E {f,g ) . We consider at first the case F = f . Then

We set tk := i,,,[qk], for k = 1 , . . . , n. If there is a ko E ( 1 , . . . , n} such that tko$ X, and xko E var(u), then

which gives a contradiction in both cases op(t) = op(u) and op(t) < op(u). Therefore for every ko E { I , . . . , n ) we get tkoE X, or xko $ var(u). In both cases we have t E S, C S,,,. Now assume that F = g . Then because of a,,,Ro,,,, we can conclude as we did in the first case. This completes the proof.

Lemma 6.8.6 Let z E X,, t , u, v ( a x $ , a,,,) E 2 . Then

E W(,,,) (X,)

(i)

If op(t) = op(u) = op(v), then u , v E St.

(ii)

If op(t) = op(u) < op(v), then u

(iii)

If op(t) = op(v) < op(u), then v E St.

E

\ X,

be such that

St

Proof: (i), (ii) By Lemma 6.8.3 we have u = S n ( t , [al],. . . , [a,]) for terms al, . . . , a, E W(,,,) (X,). With tk := [ak],for k = 1,. . . , n we get u = Sn(t, tl, . . . , t,). Then the case that there is a number ko E ( 1 , . . . , n) such that tko$ X, and xko E var(t) is impossible because of

Therefore for all k E { I , . . . , n ) we have tk E X, or zk $ var(t) and then we have u E St and in the first case also v E St. (iii) follows from (ii) considering a,,,Ra,,,.

6.8 Green's Relations in Hyp(n, n)

Lemma 6.8.7 Let t , u, u E W(,,,) (X,) have:

177

\ X,

and x E X,. T h e n we

Theorem 6.8.8 Let x E X,, t, u, v E W(,,,) (X,) \ X,. Then o,,~Ro,,, if and only if one of the following conditions is satisfied: (i) t E S,,,, and u, v E

St,

(ii) t E S,,,, u E St and v E Ct, (iii) t E S,,,, u E St and u E Ct.

Proof: One direction follows from Lemma 6.8.3, Lemma 6.8.5 and Lemma 6.8.6. (i) Using the mappings pt,, and pt,, which produce u,v E St from

(iii) is similar to (ii). Now we come to the third case.

Lemma 6.8.9 Let s , t , u , v E W(,,,)(X,)\X, w i t h u # t a n d u If op(t) < op(u) and op(t) < op(v), then ( ~ s , t~, u p Sf) R .

# s.

Proof: Suppose that op(t) < op(u) and op(t) < op(v). To show that t # i,,,[q]for all q E W(,,,) (X,) , let q E W(,,,) (X,) . If q E X,, then 6,,,[q] E X,. Since t $ X,, we have t # 6,,,[q]. Suppose that q = F ( q l , . . . , q,), where F E {f,g ) and 6,,,[qk] # t for k = 1 , . . . , n. Then

If F = f , then b y assumption,

178

6 Monoids of Hypersubstitutions

If F = g, then by assumption,

which implies t f &,,

[q].

We suppose that s, t , u, u E TW),( (X,) \ X,. Without loss of generality we may assume that op(s) op(t) and op(u) 5 op(u). This gives the following possibilities:

<

From Lemma 6.8.9 and Lemma 6.8.4 (ii), the equation op(s) = op(u) can be derived. Therefore in the first case, we have op(s) = op(t) = op(u) = op(u). In the second case, we get op(s) = op(t) = op(u) < op(u). In the third case, we get op(s) = op(u) = op(u) < op(t). In the fourth case, we obtain 3 subcases:

Hence, if a,,tRa,,, for any s , t, u,v E W(,,,)(X,) enough t o consider the following cases:

\ X,,

then it is

6.8 Green's Relations in Hyp(n, n )

Clearly, (11) and (111) are equivalent and ( V ) and (VI) are also equivalent. Then we have

Lemma 6.8.10 Let s, t , u , v as,tRa,,, . Then

E

W(,,,) (X,)

\

X , be such that

( i ) If op(s) = op(t) = op(u) = op(v), then s, t E S,,, and u , v E SSJ. (ii) If op(s) = op(t) = op(u) < op(v), then s , t E S, a n d u E Ss,t. (iii) If op(u) = op(v) = op(s) < op(t), then s E S,,, and u , v E S,. ( i v ) If op(u) = op(s) < op(t) = op(v), then s, t E S,,, and u , v E Ss.t.

Proof: ( i ) There exists a hypersubstitution a,,, E Hyp(n, n ) such that = a,,, oh Thus s = 6,,,[p] and t = 6,,,[q]. Since s , t Sf X,, so p, q Sf X,. Then we set p = F l ( p l , .. . ,p,) and q = F2(ql,...,qn) where Fl,F2 E { f 7 g ) ,p17...,pn,q1,...7qm E W(,,,) (X,) . We set tk := 6,,, [pk]and sk := 6,,, [qk],for k = 1, . . . , n . Therefore

s

=

Sn(au,,(F1),tl, . . . , t,) and t

=

Sn(a,,,(F2),sl , . . . , s,).

I f tk E X , or xk @ (war( u ) U war ( v ) ) for all k E ( 1 , . . . , n ) , then s , t E S,,,. I f tk @ X , and z k E v a r ( u ) or x k E v a r ( v ) for some k E { I , . . . , n),then op(s) > op(u)or op(s) > op(v)or op(t) > op(u) or op(t) > op(v),which contradicts our assumption. Because of the symmetry of R in the same way we get u , v E Ss3t.

180

6 Monoids of Hypersubstitutions

(ii) In this case we conclude in the same way as in case (i). (iii) Here we use the symmetry of R .

(v) In this case we conclude similarly. (vi) This can be obtained by using the symmetry of R . Further we have

Lemma 6.8.11 Let s, t , u, v E W(,,,) (X,) \ Xn with u # t and v # s . If o,,~R~,,,, then u and v belong to Ct U C, from Lemma 6.8.3. From the symmetry of R we conclude that s and t belong to C,UCv. Altogether, we obtain the following result:

Theorem 6.8.12 Let s, t, u, v E W(,,,) (X,) \ X, with u # t and v # s . Then o,,~Ro,,,if and only if one of the following conditions is satisfied:

(ii) s, t E S,, u E S,,t and v belongs to Ct U C,, (iii) s E S,,,, u E S,, v E S, and t belongs to C, U Cv, (iv) s E S,, u E S, and t E S,, v E St, (v) s E S,, u E S,, v belongs to C, U Ct, t belongs to C, U Cv.

Proof: If o,,~R~,,,, then we get these possibilities by Lemma 6.8.10 and Lemma 6.8.11. If one of the cases (i)-(v) is satisfied we obtain o,,~Ro,,, by use of one of the hypersubstitutions

Of (a,, ...,an),f ( b l , ...,b n ) 7

a

s,t

E

n

g

11

, h

S,,,, and u, v

e E

Of ( a i ,...,an),g(bl,...,bn)7

1

,

7

1

7

n

Og(bl, ...,bn),f (al,...,an) 7

E

W(n,n)(Xn). If

S,,t, then the terms a l , . . . ,a,, bl,. . . , b,

are variables arising from the variables occurring in u, v (in s, t ) by using the permutation introduced at the begin of this section.

6.9 The Monoid of Hypersubstitutions of Type (n)

6.9

181

The Monoid of Hypersubstitutions of Type

(4

In this section we generalize some of the results from sections 6.3, 6.4, 6.5 to the type (n) for n 2. As usual, any hypersubstitution a of type (n) is determined by the n-ary term t to which it maps the fundamental operation symbol f , and we will denote by at the hypersubstitution mapping f t o t . The results presented here can be found in [105].

>

The set P ( n ) = {axiI 1 5 i 5 n) of all projection hypersubstitutions has the following properties

Corollary 6.9.1 (i) P ( n ) forms together with a i d a submonoid and a two-sided ideal of 'Flyp(n). I n fact P ( n ) U {aid)is the smallest possible two-sided ideal of 'Flyp(n), called the kernel of 'Hyp(n). Thus ' H y p ( n ) is not a simple semigroup. (ii) P ( n ) is the set of all right zero elements of 'Hyp(n), so that ( P ( n ) U {aid);o h , aid)itself is a right zero band. (iii) Hyp(n) contains n o left zero elements.

Proof: All propositions follow directly from oZio h a E P ( n ) and a o h ax, = azi for a11 a E Hyp(n). Let ;.r : (1,. . . , n} + (1,. . . , n} be a permutation. A hypersubstitution of type (n) which maps the n-ary operation symbol f to the term f (x,(ll, . . . , x,(,)) is called a dual hypersubstitution. Let D ( n ) be the set of all dual hypersubstitutions.

Corollary 6.9.2 The set D ( n ) of all dual hypersubstitutions of type (n) forms a submonoid of 'Hyp(n) which is even a group. The set D ( n ) is the largest set of hypersubstitutions i n Hyp(n) which have inverses. Thus D ( n ) is the largest subgroup of 'Flyp(n). The proof of this corollary is straightforward (see [105]). Therefore D ( n ) is equal to B i j ( n ) , the set of all bijective hypersubstitutions of type (n). In 6.3 we considered a finite submonoid M 5 of hypersubstitutions of type (2) which consisted of six elements at,

182

6 Monoids of Hypersubstitutions

where t ranged over ~ 1 ~ xf 2( X,I , x2), f ( 2 2 , X I ) , f ( X I , X I ) , f ( 5 2 , ~ ) . The first two of these form the set P ( 2 ) and the third and fourth are the dual hypersubstitutions. The remaining two are special idempotent elements, and we can generalize them to type (n) also. Let F (n) be the set of hypersubstitutions of the form ~ f ( ~ , , . ,. .for ,~,) 1 i n. It is not difficult t o see that M := P ( n ) U D ( n ) U F ( n ) is a submonoid of IFtyp(n) of size 2n n! (see [105]).

< <

+

Next we describe the idempotent elements of Hyp(n). Clearly, all i n we projection hypersubstitutions are idempotent. For 1 let we put Ex, = {at I t = f ( u l , . . . ,u,),ui = xi and u j E W({xi}) for all j = 1, . . . , n} and E = Ex, U . . . U Ex, For n = 2, the projection hypersubstitutions and the hypersubstitutions in E = Ex, U Ex, are the only kind of idempotent elements in Hyp(2). But for n > 2, other types of idempotent elements are possible as the following example shows.

< <

Let r be any ternary term in the variables xl and x3. Then for t = f (xl, r, x ~ )the , hypersubstitution at is idempotent in Hyp(3).

Theorem 6.9.3 ([105]) A hypersubstitution at of type (n) where E W,(X,) \ X, is idempotent iff t satisfies the following condition X , + 'v'k E (ID) t = f (tl, . . . , t,), war (t) = {xi,, . . . , xi,) { I , . . . , m}(tik = xik).

t

c

Proof: First we prove that any hypersubstitution satisfying (ID) is idempotent. Indeed, a2(f ) = Gt [t] = G [f( t . . . , t ) ] = Sn(t,et[tl],. . . et[t,]) = t = at( f ) since tik = xik.

>

1 If conversely at is idempotent, then because of depth(t) the term t has the form t = f (tl, . . . , t,). We may assume that uar(t) = {xil,. . . ,xi,) C X, and then o:(f) = a t ( f ) = t +Sn(t,et[tI], . . . , et[tn]) = t +- xi, = t i l , . . . ,xi, = ti, and this finishes the proof. Now we consider Green's relation R on IFtyp(n). It is not difficult to see that any hypersubstitution from P ( n ) is R - related to all elements of P ( n ) , and not to any other hypersubstitution. Moreover, any dual hypersubstitution from D ( n ) has R-class equal t o D(n). Further we have

6.9 The Monoid of Hypersubstitutions of Type ( n )

183

Theorem 6.9.4 ([105])Let a,, at any hypersubstitutions of type ( n ) not in P ( n ) U D ( n ) . Then a,Rat ifl there is a bijection ;.r : ( 1 , . . . , n } + ( 1 , . . . , n } such that s = S n ( t ,x , ( ~ ).,. . , x,(,)). From this result and Theorem 6.9.3 using Theorem 6.1.1 we are able t o determine all regular elements of H y p ( n ) . By Theorem 6.9.3 the hypersubstitution at for t E W ( , ) ( X n ) is idempotent iff the condition (ID) is satisfied. But for any bijection ;.r : ( 1 , . . . , n ) + { I , . . . , n ) we have: the term t E W(,)\ X , satisfies (ID) iff the term S n ( t ,x,(l), . . . , x,(,)) satisfies (ID). This gives the following result:

Corollary 6.9.5 A hypersubstitution at E W(,)(X,) ular element of H y p ( n ) ifl ot is idempotent.

\ X,

is a reg-

A direct proof of this corollary can be found in [13]. We notice that we have two different concepts of regularity. A hypersubstitution o of arbitrary type 7 is called regular if v a r ( o ( f i ) )= { x l , . . . , xni} for every i E I. In 6.2 we proved that a hypersubstitution of type ( n ) for n 2 is injective iff it is regular. Now we characterize Green's relation R on subsemigroups of the semigroup R e g ( n ) of all regular hypersubstitutions of type ( n ) .

>

Proposition 6.9.6 Let Rs be Green's relation on a subsemigroup S with D ( n ) c S of R e g ( n ) . Then otRsotl ifl there is a hypersubstitution o, E D ( n ) such that ot = ot(oh oT Proof: If at = at(oh a, is satisfied, then a,-I E D ( n ) gives atRsat,. If conversely o t R s o t / ,then there exist hypersubstitutions o,, ob E S such that ot = at! oh a, and ott = at oh ob. Since a 6 X , we may assume that a = f ( a l ,. . . , a,). From the formula before Theorem 6.7.1 we obtain

>

and op(tf) op(t). Altogether we obtain o p ( t f )= op(t). Consequently we have op(a) = op(b) = 1. Now it is not difficult t o see that a has the form a = f ( x , ( ~ ). ,. . , x,(,)) with a permutation ;.r on { I , . . . , n } . rn In [13]the author proves also that the order of arbitrary hypersubstitutions of type ( 3 ) is 1 , 2 , 3 or infinite. In 6.4 for T = ( 2 )

6 Monoids of Hypersubstitutions

184

we obtained 1,2 and infinite as possible orders. For arbitrary type (n),n 4, this is a open problem. For more results on 'Hyp(n) we refer t o [105].

>

6.10

Left-Seminearrings of Hypersubst it utions

In this section we will define a second binary operation on Hyp(7) such that Hyp(7) forms a left-seminearring. By

we define a hypersubstitution which maps for each i E I the ni-ary operation symbol fi t o the ni-ary term Snt ( a 2(f i ) , al ( f i ) , . . . , ol (f i ) ) . Therefore ol 0 2 is a hypersubstitution of Hyp(7). We

+

0 O h (01

+0 2 ) =

( 0O h 0 1 )

+

(0O h 0 2 )

is satisfied. Indeed, we have (0

(01

+ 0 2 ) )( f i )

= = = =

6[(01+ 0 2 ) ( f i ) ] 6 [ S n t ( 0 2 ( f i01 ) , ( f i ) ,. . . , 0 1 ( f i ) ) ] S n z ( 6 [ 0 2 ( f i ) ] , 6 [ o l( f i ) ] ,. . . , 6 [ o i( f i ) ] ) S n z ( ( o 0 2 ) ( f i ) , ( 0 0 1 ) ( f i ) ,. . . , (0O h ~ l ) ( f i ) )

=

((0

01)

+

( 0 oh 0 2 ) ) ( f i ) .

Here we used that for each hypersubstitution o its extension 6 is a clone endomorphism and thus hypersubstitution and superposition are permutable. This shows the left distributivity. The following counterexample shows that the right distributive identity is

6.10 Left-Seminearrings of Hypersubstitutions

185

not satisfied. Assume that r = (2), with a binary operation symbol f , and that a l , 0 2 , a3 are defined by al ( f ) = f ( x ,y ) , a2( f ) = f(y,x),4 f= ) f ( ~ , f ( ~ , Then d). = ( 0 1 +0 2 )( f f ( f ( x ,Y ) , f ( x ,Y ) ) , = ((01 0 2 ) Oh 0 3 ) ( f ) (01 + 0 2 ) [ f ( ~ f 7 ( Y ,Y ) ) ] , = f ( f ( x ,f ( f ( Y , y)7 f ( Y , Y ) ) ) , f ( x ,f ( f ( Y , Y ) , f ( Y , Y ) ) ) ) , (01 Oh 0 3 ) ( f ) = f (x7 f ( ~ Y7 ) ) , ( 0 2 Oh 0 3 ) ( f ) = f ( f ( ~ Y7 ) , 4 7 = S2((02Oh 0 3 ) ( f ) 7 (01 Oh 0 3 ) ( f ) , ( ( 0 1 Oh 0 3 ) ( 0 2 Oh Q ) ) ( f )

+

A

+

(01 Oh 0 3 ) ( f

))

(.,

=

S 2 ( f( f ( Y , Y ) , 4 ,f f ( x ,f ( Y , Y ) ) )

=

f(f(f(x,f(y,y))7f(x,f(y,y)))7

f

( Y ,Y ) ) ,

f

(x7 f ( Y , Y ) ) ) . On the set H y p ( 7 ) not only operations, but also relations can be defined. Let o l ,a2 E H y p ( 7 ) .Then we define a1 dR a2 if and only if there is a hypersubstitution a such that a1 = a2 oh a . Since H y p ( r ) is a monoid, dR is reflexive and transitive, i.e. a quasiorder. Similarly, we define a1 5 o2 if and only if there is a hypersubstitution a such that ol = a oh a2. The relation dL is also a quasiorder. Then it is easy to see (and well-known) that R =dR n 5;' and L =dL n 3,' are Green's relations R and L which we considered before. The relations dc and dR induce partial order relations on the quotient sets H y p ( r ) / R and H y p ( r ) / L , respectively. In [18]we considered special submonoids of 'Flyp(7) consisting of permutationally full and of strongly full hypersubstitutions, respectively. The images of the hypersubstitutions from these submonoids are called permutationally full terms and strongly full terms.

Definition 6.10.1 A hypersubstitution a is called permutation) all i E I and strongly full if for all ally full if a ( f i ) E W F F ( X n i for i E I the images of the operation symbols fi belong to W,SF( X n a ) . By H y p P F ( 7 )and H y p S F ( 7 )respectively, , we denote the set of all permutationally full and the set of all strongly full hypersubstitutions of type 7 . Then it is easily checked that

186

6 Monoids of Hypersubstitutions

Lemma 6.10.2 ([18]) The sets HypPF(r) and ~ y p ' ~ ( r form ) submonoids of the monoid E y p ( r ) . In 1.3 we defined already the well-known term complexity measures op, depth (mindepth). Using the depth we can define another set of terms.

Definition 6.10.3 Let t E W,(X,) be an n-ary term of type r . Then t is called path-regular (for short a pr-term) if mindepth(t) = depth(t). Let Wy(X,) be the set of all n-ary pr-terms of type r. A hypersubstitution is o called path-regular, if the terms o(fi) are path-regular for every i E I. Let H y f ( r ) be the set of all pathregular hypersubstitutions. Then it is easy to see that we have one more submonoid of E y p ( r ) (see [Is]).

Proposition 6.10.4 E y f l ( r ) forms a submonoid of E y p ( r ) . Several complexity measures are particular cases of the following valuation of terms:

Definition 6.10.5 Let & ( X ) = (W,(X); with f, : (tl, . . . , t n , ) H f i ( t l , . . . , t n i ) be the absolutely free term algebra of type r on a countable set X, and let N, = (N; (f?)iE1) be an algebra of type r defined on the set of all natural numbers. Then a mapping v : X + N is called a valuation of terms of type r into N, if the following conditions are satisfied: (i) There is an element a E N such that v(z) = a for all z E X , (ii) v(t)

> v(z) for every variable z and every term t (see

[39])

From the freeness of & ( X ) we obtain a uniquely determined homomorphism 6 : F,(X) + N, which extends v. For short, we denote this homomorphism also by v and will call it valuation of terms. In the case of depth(t) the operations f? are defined by 1 and for rnindepth(t) we have f?(al, . . . , a,%)= rnaz{al, . . . , a,,) fT(al, . . . , a n t ) = rnin{al, . . . , a,%) 1. Both kinds of operations are monotone with respect t o the usual order 5 on N. So, in many case the mapping v satisfies the following condition

+ +

6.10 Left-Seminearrings of Hypersubstitutions

< <

187

< <

If aj bj for 1 j ni and fi is an ni-ary operation symbol of type r then for the corresponding operations f: we have f r ( a 1 , . . . , ani) f:(bl, . . . , bni). Here we denote by the usual order on the set of natural numbers.

(OC)

<

For more background on valuations of terms see [39]. This can be applied t o hypersubstitutions as follows:

Definition 6.10.6 Let a be a hypersubstitution of type r. Then d e p t h ( a ) := m i n { d e p t h ( a ( f i ) ) I i E I } := m i n { v ( a (f i ) ) I i E I ) for any valuation v of ~ ( 0 ) terms of type r. By definition of S n , it is easy t o check that d e p t h ( S n ( t ,t l ,. . . , t,)) = d e p t h ( t ) m a z { d e p t h ( t i ) I 1 i n ) for t ,t l ,. . . , t , E W(,)( X , ) . For m i n d e p t h we have a similar formula. For type r = ( n )and for permutationally full terms in [31]was proved:

+

< <

Proposition 6.10.7 Let r = ( n ) n , permutationally full t e r m . T h e n

> 1 and let t E W F F ( X ) be a

As a consequence we obtain:

Proposition 6.10.8 A s s u m e that r = ( n ) . T h e mapping d e p t h : H y p P F ( n ) i N+ defined by o H d e p t h ( o ) i s a (surjective) homom o r p h i s m of ( H y p P F ( n ) o; h , aid) onto the m o n o i d (@; ., 1 ) of all positive integers. Proof: The mapping d e p t h is well-defined and for every natural number n 1 there is a permutationally full hypersubstitution a with d e p t h ( o ) = n . Therefore d e p t h is surjective. Further we have d e ~ t h ( a O1 h 0 2 ) = d e ~ t h ( ( a O1h 0 2 ) ( f ) ) = d e p t h (61[a2( f ) I ) = depth(,, ( f ))depth(,, ( f )) = depth(ol)depth(02) by Proposition 6.10.7 and d e p t h ( a i d ) = d e p t h ( a i d ( f ) ) = d e p t h ( f ( x l , . . . , x,)) = 1.

>

188

6 Monoids of Hypersubstitutions

By the homomorphism theorem, ( N + ; - , I ) is isomorphic to the quotient monoid H y p P F ( n ) / k e r d e p t h with k e r d e p t h = { ( a l ,0 2 ) I al, 0 2 E H y p P F ( n ) ,d e p t h ( a 1 ) = d e p t h ( a 2 ) ) . Proposition 6.10.7 has some consequences for Green's relations. First of all, if ol = a2 oh 03, then by Proposition 6.10.7 the integer d e p t h ( a 2 ) divides the integer d e p t h ( a l ) ( d e p t h ( a 2 )1 d e p t h ( a l ) ) and d e p t h ( a 3 ) divides d e p t h ( a l ) . One more consequence is that the monoid ( H y p P F ( n ) o; h , aid) is not finitely generated. The homomorphism d e p t h maps any generating system of H y p P F ( n ) onto a generating system of (N+; I ) . Every generating system of (N+; 1 ) contains the infinite set of all prime numbers. This shows that there is no finite generating system of ( H y p P F( n ) o; h , a i d ) . a ,

The homomorphism d e p t h preserves also the quasiorders d L on H y p P F ( 7 ) since

a ,

dR and

(Here 5 denotes the usual order on N . )

Corollary 6.10.9 Let al,a2 E H y p P F ( n ) and let K E { R , L, E , D , J ) , t h e n a l K a 2 implies d e p t h ( a l ) = d e p t h ( a 2 ) . Proof: For K = R we have olRo2==+ 3 p , p' E H y p P F ( n ) (01 = 0 2 oh p A 0 2 = 01 oh p') ==+ depth(02) < depth(o1) A depth(o1) < depth(o2) ==+ depth(o1) = depth(02). For the relation C we conclude in a similar way. For 'Ft we have: alEa2 ==+ a l ( R n L)a2 ==+ alRa2 A alLa2 ==+ d e p t h ( a 1 ) d e p t h( a 2 ) . Considering D we obtain: a l D a 2 ==+ al ( Ro L)a2 ==+ 30 E H y p P F ( r )(alLa A 0 2 0 2 ) ==+ depth(a1) =depth(o) A depth(o) = depth(02). Finally, for J we get r )= a oh a2 oh a' a 1 2 ==+ 30, a', y , y' E ~ y f ~ ( (01 A02 =

==+ ==+

Y O h 0 1 O h 7')

depth(ol)ldepth(02)Adepth(02)ldepth(ol) depth(o1) = depth(02).

=

6.10 Left-Seminearrings of Hypersubstitutions

189

For the depth of the hypersubstitutions p, p', y, y' which are needed in the definitions of R,C and 3 we have depth(p) = depth(pf)= depth(?) = depth($ = 1. Further we have

Corollary 6.10.10 a E H y p P F ( n )is invertible if and only if depth(a) = 1 and idempotent if and only if a = aid. Proof: If a is invertible, then there exists a hypersubstitution a' such that a oh a' = a' oh a = aid. NOWfrom Proposition 6.10.8 we obtain depth(a) . depth(af) = 1 and then depth(a) = 1. If conversely depth(o) = 1, then there is a permutation s on the set { I , . . . , n ) such that a(f ) = f ( x , ( ~.).,. , x,(,)), but then the hypersubstitution a' with a'( f ) = f ( X , - I ( ~ ) , . . . , xs-I(,)) satisfies ( 0 O h 0')( f ) = f ( x ( s o s - ~ ) ( I ). . . 7 x ( s o s - ~ ) ( ~ = ) ) f ( X I , . . . 7 2,) = aid(f) = f ( ~ ( , - l , , ) ( ~ ) ., . . , x(,-I,,)(,)) = (a' oh a ) ( f ) .Therefore a is invertible. For the second proposition we have

and therefore there exists a permutation s on ( 1 , . . . , n ) with ~ ( f =) f (xs(1). . . 7 xs(7)) 0 = aid

*

Clearly, the set of all invertible elements of ( H y p P F ( n o) h; , aid) is the maximal subgroup of ( H y p P F ( no)h;, aid)and is isomorphic t o the full symmetric group S, of all permutations on { I , . . . , n ) . Assume that al, o2 are permutationally full hypersubstitutions. In [31]was proved that the superposition of permutationally full terms is permutationally full. Therefore a1 a2 is a permutationally full hypersubstitution and we have:

+

Theorem 6.10.11 ( H y p P F ( no)h;, +,aid) is a left-seminearring with identity and the function depth is a homomorphism onto the semiring (N+; +, 1) of natural numbers with identity. 0

,

Proof: We proved already that all defining identities of a leftseminearring are satisfied. We have

190

6 Monoids of Hypersubstitutions

d e p t h ( ( a l + a 2 ) ( f)) d e ~ t h ( ~ " ( ~ 2~( lf )()>>f . . 7 ~ l ()))f = m a x { d e p t h ( o l ( f )) , . . . , depth(01( f ))) +depth(02( f )) (by the remark following Definition 6.10.6) = depth(a,) depth(a2). The rest follows from Proposition 6.10.8 depth(al+

02)

= =

+

Seminearrings were considered in [82]and [103].Further we have

Proposition 6.10.12 T h e both algebras ( H y p S F ( n ) oh, ; +,aid) and ( H y T ( n ) o; h , +, aid)are left-seminearrings and ( ~ y p '( n~ ) ; oh, aid) is a sub-left-seminearring of the left-seminearring ( H y p P F( n ); oh + 7 aid).

+,

Proof: ( H y p S F ( n )o; h , aid)is a submonoid of ( H y p P F( n ) o; h , aid). Assume that ol,o2 E H y p S F ( n ) . If we substitute in 0 2 ( f ) for x l the strongly full term o l ( f ) , etc., and finally for x , the strongly full term a l ( f ) , then by the inductive definition of strongly full terms the resulting term Sn (a2(f ) , al ( f ) ,. . . , al ( f )) is strongly full and a1 a2 E H y p S F ( n ) . Assume now that al,a2 E H y f l ( n ) , i.e. r n i n d e p t h ( o j ( f ) )= d e p t h ( o j ( f ) ) j, = 1 , 2 . Then we have also

+

rnindepth((ol

+

0 2 () f

))

= = = = =

and then

ol

+

0 2

E

m i n d e p t h ( S n ( 0 2 f( ), ol(f ), . . . , a d f 1)) m i n d e p t h(al( f ) ) + m i n d e p t h ( 0 2( f )) d e p t h ( m (f )) + depth(a2( f )) de~th(Sn(02(f), ~ l ( f ) ., .., ~ l ( f ) ) d e p t h ( ( a l + a 2 )( f ) ) by the remark following Definition 6.10.6

Hyy(n).

We remark that the set of all hypersubstitutions of arbitrary type r is also closed under our addition and is called a left-seminearring since the proof of the associativity and left distributivity did not use the type ( n ) . Further we have

Proposition 6.10.13 ( H y p S F ( n ) n H y f l ( n ) ;o h , +) is the leftseminearring generated by aid.

)

6.10 Left-Seminearrings of Hypersubstitutions

191

Proof: The one-element set {aid)is closed under the multiplication. Therefore, since the addition of hypersubstitutions is associative, every element of the left-seminearring generated by aid can be written as maid for some natural number rn. We show by induction on m that every hypersubstitution of the form maid belongs t o ~ y ~ ' ~ and ( nto) H y f ( n )and therefore t o the intersection. This is clear for aid. Assume that maid E H~~~~ ( n )n H y y ( n ) ,then ( m l)~id(= f ) (maid ~ i d ) ( f )= Sn(aid(f),maid(f),...,maid(f)) = f (maid( f ) , . . . , maid( f ) ) and by the definition of strongly full and of path-regular hypersubstitutions we have (m+l)aidE H y p S F ( nn) HYT

+

+

(4.

Conversely, assume that o E ~ y p ' ~ ( nn) H y y ( n ) . Since a is strongly full, there are terms t l , . . . , t , such that a ( f ) = f ( t l ,. . . , t,). We give a proof by induction on depth(a). If depth(a) = 1, i.e. depth(a(f))= 1, then a ( f ) = f ( x l , .. . , x,) = aid(f ) and thus a E (aid).Assume that every a with depth(a)= m belongs t o (aid)and let a' be a hypersubstitution with depth(ot)= m 1. Thus a'(f ) = f ( t l ,. . . , t,) where t l , . . . , t , are strongly full and path-regular terms. Consider the hypersubstitutions 0 1 , . . . , a, with ai(f ) = ti. Since depth(ai)= m , we have ai E (aid).But only for the hypersubstitution maid E (aid)there holds depth(maid)= m . Therefore for every i = 1,. . . , m we have ai = maid and therefore a' = ( m l)aidE (aid). This shows that (aid)= H y p S F ( nn ) H y f l (n).

+

+

A consequence of Proposition 6.10.13 is that the left-seminearring ( H y p P F ( noh, ) ; aid)has no finite sub-left-seminearring and that every left-seminearring of hypersubstitutions contains the infinite left-seminearring (aid)= HypSF( n ).

+,

Remark further that the mapping given by maid H m defines an isomorphism between (aid)and ( N + ; ., 1). This shows that (aid) is a semiring with cancellation rules for both operations, and with commutative addition. Furthermore (aid)has no zero divisors. Now we want t o generalize Corollary 6.10.9 t o the valuation of terms of type r into N+ introduced in Definition 6.10.5. We mention the following fact which was proved in [37]:

+,

6 Monoids of Hypersubstitutions

192

Fact: Let v be a valuation of terms into N+ which satisfies the condition (OC). Then for any n-ary terms t l , . . . , t, and for an arbitrary m-ary term t we have

Now we prove: P r o p o s i t i o n 6.10.14 Let R be Green's relation on EypPF(n) and let 01, 0 2 E HypPF(n). If olRo2, then for every valuation which satisfies (OC) we have v(ol) = v(02). Proof: If olRo2, then there exist hypersubstitutions o, of E HypPF(n) such that ol = o2 oh o and o2 = ol oh of and therefore al(f ) = i2 [a(f ) ] and a2(f ) = il[af( f )]. Since T = (n) and since a E ~ y p ~ ~ (the n )term , a ( f ) has the form f ( t l , . . . , t,) and then i 2 [ o ( f ) ] has the form S n ( 0 2 ( f ) , 62[tl],. . . ,i2[tn]). Applying the Fact we see that v((02 oh o)( f ) ) v(02(f ) ) and then v(ol) v(02). Using 0 2 = ol oh of we get v(o2) v(o1). Altogether, we have v(al) = v(a2).

>

>

>

Let E be Green's relation on EypPF(n). Clearly, if alEa2and if v satisfies (OC) we have also v(ol) = v(02).Because of (ol+02)( f ) = Sn(02(f), o l ( f ) ,. . . , o l ( f ) )from the Fact follows that v(ol 02) v(02), while v(ol oh 02) v(ol).

<

+

<

Now we consider again Green's relation R on EypPF(n). The condition depth(al) = depth(a2) turns out t o be necessary, but not sufficient for olRo2. Indeed, if 01, 0 2 E HypPF(2) and ol # o2 then olRo2 implies ol = 0 2 oh ozlz2or 0 2 = 01 oh oz2zl. For instance, for type r = (2) the hypersubstitutions ol with m ( f ) = f (f ( X I , 2 2 1 7 f (f ( X I , x2), f ( ~ 2 X7 I ) ) ) and 0 2 with a2(f) =

f (f (f ( X I , ~

2

f () ~

2~ X, I ) ) ,

f (XI, ~ 2 ) )

d e ~ t h ( a l )= d e ~ t h ( a 2 ) ,

but ol and o2 are not R-related. Therefore we need some more conditions. For any m-ary term t we denote by C,[t] the term arising from t if we substitute for each variable the variable x (see also Definition 6.4.5). Then we get:

6.10 Left-Seminearrings of Hypersubstitutions

193

Proposition 6.10.15 Let R be Green's relation on 'FLypPF(n). Assume that al,a2 E HypPF(n). If alRa2, then Cx[al(f)] = Cx[a2(f)l. Proof: If olRa2 then there are hypersubstitutions a and a' such that a1 = a 2 oh a and a 2 = al oh a' and then al = al oh (a' oh a ) . By Proposition 6.10.8 we get depth(al) = depth(al)depth(a)depth(at). From this we obtain depth(o) = depth(at) = 1. Since o and a' are permutationally full hypersubstitutions, we have o ( f ) = f ( z , ( ~.).,. , x,(,)), a t ( f )= f (z,,(~), . . . , x,I(,)) for permutations s,s' on (1, . . . , n). From this there follows ~

-

Every permutationally full term t E WzF(Xn) contains subterms of the form

for permutations sj on { I , . . . , n). Considering the tree of the permutationally full term t; by P(t) we denote the sequence of all permutations on { I , . . . , n) which are needed in t, written from the left to the right.

Example: Consider for r

t

=

f (f ( ~ 2 Z,

=

f (f ( X I , ~

~~ , 3 ) ,

(3) the term

f

f

2 > ~ 3 () ~> 2 X ,1 7 ~ 3 ) 7 ( ~ 3 > ~X 2 I ) )>,

f (23,217 22)). Then P ( t ) = ((12),( I ) , (12), (13), (132)) if we write the permutations which are needed as cycles. Clearly, two terms tl, t 2 E Wgf'(X,) are equal if and only if Cx[tl]= Cx[t2]and P ( t l ) = P ( t 2 ) . Now we have

Proposition 6.10.16 Let al, a2 E HypPF(n) and assume that P(al(f))= (ul, . . . , urn) and P(a2(f ) ) = (ul, . . . , ul). If alRa2then 1 rn = 1 and ul o u,' = . . . U, 0 urn .

6 Monoids of Hypersubstitutions

194

Proof: Assume that a l R a 2 . There are hypersubstitutions a, a' such that a1 = a 2 oh a and a 2 = a1 oh a' and therefore depth(a) = depth(at) = 1. It follows that there are permutations s, st : { I , . . . , n) { I , . . . , n) such that Q(f) = 6 2 [ f (xs(l),. . . , xs(n))] and 0 2 ( f ) = 61[f(xsf(1)7 . . . , xSf(n))].Then p(al (f )) = p ( 6 2 [f(xs(1)7 . . . , xs(n))]) = p ( S n ( a 2 (f),Xs(1) 7 . . . 7 ~ s ( n ) ) ) = (sov1, . . . , s o u 1 ) . Similarly, we get P ( o 2 ( f ) ) = P(61[f( & ( I ) , . . . 7 G ( n ) ) I ) = P(Sn(ol (f ) 7 xsf(1)7 . . . , xs/(n))) = (st0 u1, . . . , st 0 urn). By Proposition 6.10.15 from alRa2 there follows Cx[al(f)]= Cx[a2( f )]. Since the trees of Cx[al(f )] and al ( f ) differ only in the labeling of the leaves, the structures of the trees of a l ( f ) and of 0 2 ( f )are equal and therefore the number of permutations s occurring in o l (. f .) and 0 2 ( f )is equal, i.e. we have m = l and then (S o ul, . . . , s o urn) = (sto ul, . . . , S' o urn) implies s o uj = st o u j for every 1 j m. From this equation we obtain u j o u;' = s o s" 1 for every 1 j m and this means ul o u,' = . . - - urn 0 urn . f

.

< < < <

It turns out that both conditions, Proposition 6.10.15 and Proposition 6.10.16 together, characterize Green's relation R on 'HypPF (n).

Theorem 6.10.17 Let al,a2 E HypPF(n). Then the following conditions are equivalent: (i) a l R a 2 . and ul o u,' = . . . - urn o u i l where (ii) Cx[al(f)] = Cx[a2(f)] p ( o l ( f ) ) = (,I, . . . , urn) and P(a2(f))= ( ~ 1. ,. . , urn).

Proof: (i)+(ii) was already proved. (ii)+(i) We form s = ul o u,', spl = ul o u,' and consider o, o' E HypPF(n) defined by o(f ) = f (xs(l),. . . , xS(,)) and a'(f ) = NOWwe prove that 01 = f (2,-1(~),. . . , xs-linl). Clearly, 0' = o a2oh a using the remark before Proposition 6.10.16 and show that Cx[m(f)] = Cx[(a2 Oh a)(f and P ( ~ l (1)f = P((a2Oh 4 ( f )). Indeed, we have

)I

6.10 Left-Seminearrings of Hypersubstitutions

7 M-Solid Varieties of Semigroups In Chapter 3 we developed the general theory of M-solid varieties. As often happens, the development of this subject has involved an interplay between the general theory and specific examples. In this chapter we present results about M-solid varieties of semigroups for various choices of M. Semigroups seem to be simple enough to be accessible, yet rich enough to provide an interesting structure. A lot is known about the lattice of all varieties of semigroups, where the associative law plays an important role. Section 7.1 will give a general introduction t o the study of M solid varieties of semigroups. In Section 7.2 we discuss regular-solid varieties of semigroups. After the description of all regular-solid varieties of commutative semigroups by identities we characterize the complete lattice of all regular-solid varieties of semigroups. Sections 7.3 and 7.4 deal with solid and pre-solid varieties of semigroups respectively.

7.1

Basic Concepts on M-Solid Varieties of Semigroups

A semigroup is an algebra of type 7 = (2) with a binary operation which satisfies the associative law. As usual W ( 2 ) ( X 2is) the set of all binary terms of type r = (2). Throughout we will write also x, y, x, . . . instead of X I , ~ 2 ~ x 3. .,, .sometimes in the same context, t o distinguish between variables occurring in an identity and variables in terms which have to be substituted for the operation symbol, and use f as binary operation symbol. So the associative law has the form f (f (z,y), x) = f (z,f (y, 2)). We shall follow the custom of omitting the operation symbol and the brackets in semigroup identities (in the associative law only the operation symbol f )

7 M-Solid Varieties of Semigroups

198

indicating the operation by juxtaposition instead, so that we write for instance (xy)z = x(yz) and xyzt = xzyt for the associative law and the medial law, respectively. That means that a semigroup identity is an equation u = v where u,v are non-empty sequences, which are also called words, in X. A variety of semigroups (or semigroup variety) V = ModC is a class of semigroups consisting of all semigroups which satisfy some defining set C of semigroup identities. The class Vl is a subvariety of V2, written as Vl C V2, if every semigroup in Vl is in V2. Throughout this chapter we will use the following notation for a word t: the power tP ( I p E N) of t is defined inductively as follows: t1 := t, t p + l := t p t for 1 p E N.

<

<

As defined in 6.1 uar(t) denotes the set of variables occurring in t. By the length o f t we mean the total number vb(t) of all occurrences of variables in t. As already defined in 6.1, le f tmost(t) denotes the leftmost variable in t ; and rightmost(t) is the rightmost variable in t. Sometimes we indicate the operation symbol explicitly when considering M-hyperidentities. Throughout this chapter we shall deal with the problem of testing whether a particular identity u = u is an M-hyperidentity of a given variety V. In doing so we shall make frequent use of the following observation. Our testing involves examining equations of the form 6[u] = 6[v] for o E M . Informally, we can think of such an equation as obtained by substituting the binary term o ( f ) for the binary operation symbol f , in u and u. But how the letters in u and u are grouped makes a difference to what happens when we substitute a ( f ) in them. For example, consider substituting o ( f ) = x2 in the medial law xyxt = xzyt. If this identity is written as f ( f ( x , y ) , f ( z , t ) ) = f ( f ( x , z ) , f ( y , t ) ) , using x2 for f gives the trivial identity x4 = x4. But if the identity is written as f (x, f (y, f (z, t ) ) ) = f (f (x, z), f (y, t ) ) we get instead the non-trivial identity x2 = x4. This example illustrates that we cannot speak of substituting a binary term into an identity u = v without specifying precisely how the variables in u and u are grouped. This means that we have to be careful in our informal approach to testing Msolidity.

7.1 Basic Concepts on M-Solid Varieties of Semigroups

199

If C an identity basis of a variety V of semigroups, i.e. V = ModC, then to show that V is M-solid we only have t o check that all u = v E C are M-hyperidentities. Moreover, it is clear that for a variety V of semigroups to be M-solid, the associative law must be an M-hyperidentity in V. This fact is crucial in any discussion of M -hyperidentities for semigroup varieties. In this context, the associative law being an M-hyperidentity in a variety V means precisely that it makes no difference how we group our identity, since all such groupings are equivalent. Thus when in V the associative law is an M-hyperidentity we may use our informal approach, and speak of substituting a binary term into an identity, without any confusion. The collection S E M of all semigroups forms a variety of type r = (2), so S E M = Mod{(xy)z FZ x(yz)}. We will use C s g for the collection of all subvarieties of S E M , i.e. the collection of all varieties of semigroups. There are an unaccountably infinite number of such varieties, forming a lattice under inclusion, with S E M as largest element and the trivial variety T(2) consisting of all oneelement algebras of type r = (2) as the smallest. C s g forms a complete lattice (see section 1.4). The meet of any collection of semigroup varieties is simply their intersection and their join is the smallest variety which includes all of them. As usual, we denote the meet of Vl, V2 by Vl A V2 and their join by Vl V V2. More generally, C(V) will denote the sublattice of C s g consisting of all subvarieties of the variety V. If Vl is defined by C1 and V2 by C2 then Vl A V2 may be defined by C1 U C2. To find defining identities for Vl V V2 for given defining identities for Vl and V2 may be quite difficult, if not impossible. The structure of this lattice C s g is complicated, and not entirely known, C s g is the disjoint union of two large parts: the ideal Per of all periodic varieties (that is, varieties consisting of periodic semigroups) and the coideal (filter) OC of all overcommutative varieties (that is, varieties containing the variety C of all commutative semigroups). Without much exaggeration, one can say that, up to the mid-1990's, practically all studies concerning the lattice C s g deal

200

7 M-Solid Varieties of Semigroups

with the lattice Per only. As a result, there is considerable and diverse information about the latter lattice ([47], [61], [74], [75], [79], [87]). However, almost nothing was known about the structure and the properties of OC up to the mid-1990's. One can consider two large ideals in Per that intersect in the trivial variety only: the ideal C R of completely regular semigroups (that is, varieties all of whose members are unions of groups) and the ideal Nil of all nilsemigroups (that is, varieties satisfying xny = y for some n E @). The two ideals mentioned play a key role in the investigation of the whole lattice Per. Many different questions about periodic semigroup varieties are reduced, in some sense, to completely regular varieties and to nil-semigroups. The lattice C R is well studied. A considerable information about its structure had been obtained in [79]. On the contrary, informations about the lattice Nil obtained before the 1990's may be characterized as poor and fragmentary. We see that there are two large and important parts of the lattice of semigroup varieties, the lattice OC of all overcommutative semigroup varieties and the lattice Nil of all nilsemigroup varieties that were rarely investigated. The investigation of the lattice C R of completely regular varieties is based on the structure theory of semigroups. But this theory is not applicable, in fact, to overcommutative and nilsemigroup varieties. Significant progress in the investigation of lattices OC and Nil has been obtained by M. Volkov and co-workers during the last 15 years (see [99], [loo], [loll). For a survey we refer the reader also to a paper of Evans ([43]). Throughout this chapter we shall refer to the following varieties of semigroups: C := Mod{(xy)z = x(yz), xy = yx} - the variety of commutative semigroups LZ := Mod{xy = x) - the variety of left zero bands N B := Mod{(xy)z = x(yz), x2 = x, xyzt = xzyt} - the variety of normal bands RB := Mod{(xy)z = x(yz), x2 = x, xyz = xz} - the variety of rectangular bands RegB := Mod{(xy)z = x(yz), x2 = x, xyzx = xyxzx) - the variety of regular bands RZ := Mod{xy = y} - the variety of right zero bands S L : Mod{(xy)z = x(yz), x2 = x, xy = yx} - the variety of

7.1 Basic Concepts on M-Solid Varieties of Semigroups semilattices Z := Mod{xy semigroups) .

= zt)

-

201

the variety of zero-semigroups (or of constant

An identity s = t is called normal, if both s and t are the same variable or neither s nor t are variables. It is a well known fact in semigroup theory that the normal identities are precisely the identities in the variety Z of zero-semigroups. An identity u = v is called regular if both terms u and v use the same variables. The set of all regular identities agrees with the set of all identities satisfied by the variety SL. An identity u = u is said to be outermost if le ftmost(u) = rightmost(u) and le f tmost(u) = rightmost(u). The set of all outermost identities is equal t o the set of all identities satisfied by RB. In Section 3.1 we have defined for any type r the monoid E y p ( r ) := (Hyp(r);oh,aid)of a11 hypersubstitutions of type r and in Section 3.3 we introduced the concept of an M-solid variety of type r for any submonoid M of E y p ( r ) . The set of all M-solid varieties forms a complete sublattice of the lattice of all varieties of type r . For each variety V we may study the lattice of all M-solid subvarieties of V. Now we will examine M-solid varieties of semigroups. Throughout this section we use the notation E y p for the monoid of all hypersubstitutions of type r = (2). To check whether an identity of a given variety of semigroups is an M-hyperidentity of the variety, we need to apply all possible a E M t o the identity. Since we have only one operation symbol involved, we can represent any a E Hyp as at, where t is the binary term t o which a maps the operation symbol f . In particular, the identity hypersubstitution a i d which maps f t o the fundamental term zlx2can be represented as a,,,,. Suppose that s and t are binary terms such that s = t is an identity of a variety V.Then for any identity u = u of V it follows that 6, [u] = 6,[u] E I d V iff Gt [u] = et[u] E I d V since s = t implies a, --v at (see Proposition 3.1.14). This means that we can partition the set of all binary terms of type r = (2) into classes according t o the partition of Hyp by the equivalence relation -v, and t o test whether an identity u = u of V is an M-hyperidentity of V we need only check the hypersubstitutions at E M for one t

202

7 M-Solid Varieties of Semigroups

from each class. When one representative term is chosen from each class we speak often of "the binary terms of V". But which term of any class we choose makes a difference to what happens when we check the M-solidity. For example, consider the variety S L of semilattices. It is easy t o check that we can choose xl , x2, and xlx2 as well as x:, xi, and 21x2 as the binary terms of SL. If M contains a,, and a,, but neither a,, nor a,, , then we have t o choose x:, x;, and xlx2 as the binary terms of SL. This example illustrates that one cannot speak of the binary terms of a variety V irrespective the monoid M. We have t o choose from each class such a term t with ot E M except for such classes with o, $ M for all terms s of this class. That means that we must be careful in our choice of the binary terms of the variety V. Example: To examine the M-solidity of R B we need to consider all binary terms of RB. Using the identities x2 = x and xyz = xz we see that we need t o consider only four binary terms in R B , for example: xl, x2, xlx2, and x2xl. Let M = Hyp. Then we have t o check whether et[XI = et[x2],et[ ( x y ) ~=] et[ ~ ( y x ) ]and , et[xyx] = et[XX] for t E {xl, 2 2 , 2 1 2 2 , x2x1) are identities in RB. If t = xl or t = x2 then we get the trivial identity x = x (x = 2). If t = 21x2 then at is the identity hypersubstitution. If t = 22x1 SO we get x = x 2 , (xy)x = x(yx), and xyx = xx which hold in RB. This shows that the basis identities of R B , i.e. x = x2, (xy)x = x(yx), and xyx = xx, are M-hyperidentities in RB. Consequently, R B is M-solid. In our example we tested that each identity is an M-hyperidentity. However it is not always the case that an identity holds as an Mhyperidentity in that variety. In fact, something very basic goes wrong right away. Example: To examine the M-solidity of Z we need to consider three binary terms of Z, for example: X I , 2 2 , and 21x2. Let M = Hyp. Then we have t o check whether 8,,[xy] = 8,,[xt], eZ2 [xy] = eZ2 [xt], and eid[xy] = eid[2t]are identities in 2.These equations provide x = x, y = t, and xy = xt, but both x = x and y = t are not satisfied in Z. Thus xy = xt is not an M-hyperidentity in Z and Z is not M-solid.

7.1 Basic Concepts on M-Solid Varieties of Semigroups

203

This example illustrates the crucial role played by the two projection hypersubstitutions a,, and a,,. Suppose that u = u E I d V is an M-hyperidentity in a non-trivial variety V, where a,, , a,, E M . Then, choosing binary terms of V we always have the option of using the term xl. The identity which results by applying a,, to u = v will have the form le f tmost(u) = le f tmost(v). For this to be satisfied in a non-trivial variety V, we must have leftmost(u) = leftmost(u). A dual argument shows that the last variable on either side of u = u must be the same too; rightmost(u) = rightmost(v). Graczyriska ([55]) has called an identity which has the same first and the same last variable on each side an outermost identity. It is a well known fact in semigroup theory that I d R B , the identities of the variety of rectangular bands, are precisely the outermost identities. We have thus proved the following result which also follows from Theorem 5.1.10 since R B is the variety RA2 of rectangular algebras of type (2): Proposition 7.1.1 Let M be a submonoid of 'Ftyp containing a projection hypersubstitution and let V be a non-trivial M-solid variety of semigroups. Then R B C V. We have already mentioned a related example. We saw that the variety Z is not M-solid for a monoid M containing a projection hypersubstitution. The best we can do is to omit the projection hypersubstitutions from our consideration; the Pre-hypersubstitutions consist of all hypersubstitutions except the projection hypersubstitutions. We will use the notation P r e for the set of all Prehypersubstitutions, so

Proposition 7.1.2 P r e forms a submonoid of 'Ftyp. Proof: The identity hypersubstitution a i d = a,,,, belongs to P r e . Let a,, at E P r e . Then (a, oh at)( f ) = 6,[at(f )] = 6,[t]. Since s , t E W(2)(X2)and s , t # xl,x2 there are sl, s2,tl,t2 such that s = f ( s l ,s2) and t = f (tl,t2). Thus 6,[t] = S2(s,6s[tl],6s[t2]) = f (S2(s1,6,[tl], 6s[t2]),S2(sZ7 6S[tl],Gs[t21)) # X l , X 2 This shows that a, oh at E P r e .

204

7 M-Solid Varieties of Semigroups

In the sequel we will write pre-solid instead of Pre-solid. Example: In order to examine the pre-solidity of Z we need only to consider the binary term ~ 1 x in 2 Z. Since aid = o,1,2 it is clear that Z is pre-solid. However it is not always the case that an identity holds as P r e -hyperidentity in a variety. Example: To examine the pre-solidity of S L we need only to consider the following three binary terms in SL: x:, x;, and 21x2. We have to check whether At[x] = At[x2],At[(xy)z] = At[x(yz)], and At [xy] = At [yx] for t E {x:, x;, xlx2) are identities in SL. In particular, A,: [xy] = A,: [yx] provides x2 = y2 which does not hold in SL. This shows that the commutative law is not a Pre-hyperidentity in S L and S L is not pre-solid. A hypersubstitution ot using both variables xl and x2 is called regular. Let Reg be the set of all regular hypersubstitutions. A binary term which uses both variables xl and x2 is said to be essentially binary. We will use the notation WfT(X2) for the set of all essentially binary terms. Then Reg := {at I t E Wf$(X2)). L e m m a 7.1.3 If o E Reg and t E W(2)(X), then var(t) var(A[t]).

=

Proof: Assume that t E W(2)(X).We proceed by induction on the complexity of t. If t E X then A[t] = t and var(t) = var(6[t]). Let t be a compound term of the form t = f (tl, t2) and suppose that var(ti) = var(A[ti]) for i = 1 , 2 . Since a is regular, o ( f ) can be written as o ( f ) = f ( s l , s 2 ) . Then A[t] = S2(o(f), A[tl], A [ t 2 ] ) = f (S2(s1,A[tl], A[t21), S2(s2,dtlI,dt21)). Since o ( f ) uses both variables xl and x2, the term A[tl] ( A [ t 2 ] ) is a subterm of S2(sl,6[tl],A[t2]) or S2(s2,A[tl],6[t2]). This shows that var(A[t])= var(6[tl])uvar(A[t2])= var(tl) uvar(t2) = var(t).

This lemma shows that the application of a regular hypersubstitution t o a term t E WfT(X2) gives an essentially binary term. Thus Reg forms a submonoid of Nyp.

7.1 Basic Concepts on M-Solid Varieties of Semigroups

205

P r o p o s i t i o n 7.1.4 Reg forms a submonoid of ' H y p . Proof: The identity hypersubstitution a i d = a,,,, belongs t o Reg. Let o,, ot E Reg. Then (o, oh o t ) ( f ) = 6,[t], where by Lemma 7.1.3 6,[t] uses both variables xl and x2 since t does. Thus o, oh ot belongs t o Reg. Example: To examine the Reg-solidity of S L we need only to consider the binary term xlx2 in S L (the other classes contain no essentially binary terms). Since a,,,, is the identity hypersubstitution, S L is Reg-solid. (Instead of a Reg-solid variety we will also speak of a regular-solid variety.) The application of the hypersubstitution a,,,, to any term t gives the dual term t' which is inductively defined by t' = t for t E X and t' = f (t',, ti) if t = f (tl,t2). L e m m a 7.1.5 For t E W(2)( X ) there holds 6,,,, [t] = t'. Proof: If t E X then 6,,,,[t] = t = t'. If t = f (t1, t2) and suppose that 6,,,,[ti] = ti for i = 1 , 2 , then 6,,,,[t] = S2(o,,,, (f),6 Z , Z l [tl],&,z, [t21) = f (S2(x2,ti, t',), S2(x1,ti, t',)) = f (t',, ti) = t'. The hypersubstitutions aid and a,,,, are called dual hypersubstitutions, with D being the set of dual hypersubstitutions. (Clearly, D forms a submonoid of 'Flyp). For arbitrary type 7 = (n),n 2, a hypersubstitution a which maps the n-ary operation symbol f t o the term f ( x , ( ~ ). ,. . , x,(,)) for a permutation s on (1,. . . , n} is called a symmetric hypersubstitution.

>

We have the following relationship between the submonoids D , Reg, and P r e of 'Hyp. Clearly, P r e C H y p by definition of P r e . Since xl, x2 WfT(X2) we have Reg C P r e , but not every Prehypersubstitution belongs to Reg (for example o,,,,). Finally, every symmetric hypersubstitution belongs t o Reg, but not every regular hypersubstitution is symmetric (for example a,,,,,,). Altogether we have the following relationship:

6

D c Reg c P r e c H y p . In Chapter 3 we showed that the set of all M-solid varieties of a given type r forms a complete sublattice of the lattice of all

7 M-Solid Varieties of Semigroups

206

varieties of this type. For any submonoid M we denote by C z the collection of all M-solid varieties of semigroups. This lattice is the intersection of the complete lattice Csg of all varieties of semigroups and the complete lattice of all M-solid varieties of type 7 = (2). Thus C E is a complete sublattice of Csg. As mentioned before, the structure of this lattice is complicated and not entirely known. We can get more insight into its structure by describing its complete sublattices. In this chapter we will characterize the lattices C g , Ggeg, CFre, and Cgu,. We have the following relationship between these lattices. From D Reg C P r e Hyp there follows

c

c

We will see later that these inclusions are proper. The varieties from C g are called dual-solid or self-dual.

Proposition 7.1.6 A variety V of semigroups is dual-solid ifls' E I d V for all s = t E IdV.

=

t'

Proof: V is dual-solid iff 6,,,,[s]= e,,,,[t] E I d V and eid[s]= eid[t] E I d V for all s = t E IdV. The assertion will be clear by Lemma 7.1.5 and the fact that eid[w] = w for w E W(2)(X). The elements of Lgeg are the regular-solid varieties. The varieties belonging t o CFTeare the pre-solid ones. Ggu, is the lattice of all solid varieties. These three lattices will be characterized in the following sections.

7.2

Regular-solid groups

Varieties of Semi-

In this section we will examine the complete lattice of all regularsolid varieties of semigroups. As a first step we characterize all regular-solid varieties of commutative semigroups. The greatest one is HRegMod{(xy)z = x(yz), xy = yx), the Reg-hyperequational class defined by the associative law and the commutative law. Let us denote the variety Mod{(xy)z = x(yz), xy = yx, x2y = xy2) = x(yz), xy = by VRC. We will prove that VRC = HRegM0d{(xy)~ yx}. Clearly, HRegMod{(xy)z= x(yz), xy = yx) C VRC. First of all we prove that VRC satisfies the identities x3y = x4y and x4 = x 5 .

7.2 Regular-solid Varieties of Semigroups Lemma 7.2.1 Let V

C VRC. Then

(i) x3y = x4y E I d V (and thus x4 = x5 E IdV) (ii) x2yz = x y 2 z E IdV(and thus x2yz = x"z

E IdV).

Proof: (i) Using x2y = xy2 E I d V we get x4y = ( x 2 ) 2 y= x2y2 = x x y 2 = xx2y, i.e. x3y = x4y E IdV. Finally by substitution we get x4 = x5 E IdV. (ii) In x2y M x y 2 we substitute yz for y. Using the commutative law and x2y M xY2 we get x2yz = xy2z2 = x3yz E IdV. Let t be an essentially binary term. Then because of the commu1. Using tative law, t = x t x i for some natural numbers k , 1 x2y = xY2 we get t = x f + z p 1 x 2and finally t = x y x 2 for some m E { 1 , 2 , 3 ) (because of x3y M x 4 y ) . This shows that the essentially binary terms of the variety VRC are of the form xTx2 for m E { I , 2,3).

>

In general, not every subvariety of a greatest M-solid variety is M-solid. But luckily in our case we have:

Theorem 7.2.2 Each subvariety of VRC is regular-solid. Proof: Let V E C(VRc). We have only t o consider the regular hypersubstitutions a,,,,, a,:,, and a,:,,. Let u = v E IdV, u # v be an identity in V.Clearly, 6,,,,[u] = 6,,,,[u] E IdV. There are 1 and u l , . . . , u,, ul, . . . , us E X such that natural numbers r, s u = ~ 1 ~ 2. u7 . .and u = 2.1212...us. If T = s = I then 6,;,,[u] = u = v = 6,;,, [v]E I d V for 2 = 2,3. If one but not both of r, s = 1, then from u = v E I d V we get x = xk E I d V for some natural number k 2 by substitution. Then x4 = x5 E I d V gives us x = x2 E IdV. Therefore both hypersubstitutions a,:,, , and a,:,, are V-equivalent t o a,,,,. If r = s = 2 and { u l , u 2 } = { u l , u 2 ) then u = u is the 2 u2u: = uiul = 6,;,,[v] E commutative law and 6,.,,[u]= u ~ u = IdV, i.e. 6 x ~ x 2= [ ~6xa152[v] ] E I d V for 2 = 2, 3. If r = s = 2 and { u 1 , u 2 }# {u1,u2)or r = 2 and s 3, or r 3 and s = 2, then from u = u we get x2 M xk for some natural number k 3 by substitution. Together with x4 = x5 we have x2 = x3. Further,

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7 M-Solid Varieties of Semigroups

208

using the identities x2 % x3, x2y M xY2 and xy obtain

[

-

]

6i;Z2[v]

=

2 ulrr2.. . u, 2 v1v2 . . . u s

= =

%

yx for i

u 21 22 . .. u 2 2 2 v1v2 . . . us.

=

2,3 we

and

From u l u 2 . . . u , % ulu2.. .us E IdV it follows u&;. . . u,2 % ufu;. . . u: E IdV by substitution. Hence [u];;:6iii2 [u]E IdV for i = 2,3. If r, s 3 then using the identities x32 M x42, x2y % xy 2 and xy % yx for i = 2,3 we get

>

[ u ] 6iii2[u]

% %

u:u2.. . u , u;u2.. . u s

% %

u 31 u3 2 . .. u: and u;u;. . . u;.

From u l u 2 . . . u , % ulu2.. .us E IdV there follows u;u:. . . u," % u;ug. . . u: E IdV by substitution. Hence [u];;:6iii2 [u]E IdV for i = 2.3.

Theorem 7.2.3 VRC is the greatest regular-solid variety of commutative semigroups, i.e. VRC = H R e g M o d { ( x y )%~ x ( y z ) , xy % YXI. Proof: VRC is regular-solid by Theorem 7.2.2, so VRC C HRegMod{(xy)2% x ( y z ) , xy FZ yx). The opposite inclusion is clear. Theorem 7.2.2 and Theorem 7.2.3 show that the lattice of all regular-solid varieties of commutative semigroups is equal to the subvariety lattice of VRC,i.e. the subvarieties of VRC are precisely all regular-solid varieties. Now we describe the elements of C(VRc) by identities. If w is a term with var ( w ) = { x l ,x 2 ,. . . , x,), then we will also write w ( x l ,. . . , 2,). We will use the convention of replacing the identity w ( x l ,. . . , x,) % w ( y l ,. . . , y,) by the equation w = 0.

Lemma 7.2.4 Let V c VRC. If there are natural numbers a , b, m , n 1 and u l , . . . ,urn,ul, . . . ,u, E X with U ~ .L.urn ~ . % u!u2...u, E IdV \ I d S L then 2 3 2 . . . x , = 0 E IdV and x ; x 2 . .. X , = 0 E I d V .

>

7.2 Regular-solid Varieties of Semigroups

209

b Proof: Since u:u2.. .urn = u1u2.. . u, $ IdSL without loss of generality there is a letter r E X with r E { u l , . .. ,urn} and 7- $ ( ~ 1 ,. . , u,}. b If m = 1 then we have u;" = v1v2.. .v, E IdV with u1 $ b { v l ,. . . , v,}. This implies xa M v t v 2 .. . v, E IdV, ya M vlv2.. . v, E b IdV, u: M x 1 x 2 . .. x, E IdV and u? = y?y2... y, E IdV. Thus xa = ya E I d V and x ! x 2 . . . x, = y;y2.. . y, E IdV.

>

If m 2 then without loss of generality we assume that urn = r . So we have u;"u2. . . urn-1x = ufu2 . . . u, E I d V and uTu2 . . . urn-1y = b v1v2 . . . v, E I d V , i.e. u;"u2. . . u,-lx u;"u2. . . u,-ly E IdV. This implies u:u2 . . . u,-lu, u?u2. . . u,-1 (x?x2. . . x,) = $ x 2 . . . x,-1 (u:u2. . . u,-lx,) M x ? x 2 .. . x,. In the same way we get ~ $ 2 . . u,-~u, = yTy2.. . ym E IdV. Thus x ; C X .~..X , = yTy2.. . y, E IdV. Let w l , . . . , w, E X with { w l , . .. , w,} n { u 1v } = 0. Then ~ $ 2 . . x , = y;"y2... y, E IdV and u ? u 2 . .. u , = v bl v 2 . .. v, E IdV implies w:w2.. . w, = b v b1 v 2 . ..v, E IdV. This gives wyw2... w, M x1x2. . . x, E IdV b b and w;Cw2... W , = yly2.. . y, E IdV, i.e. x!x2.. . x, = y1y2.. . y, E IdV. Altogether we have 2 3 2 . . . x , = 0 E I d V and x!x2 . . . x, = 0 E IdV. N

Lemma 7.2.5 Let V c VRCsuch that S L V . If there are natural numbers r , s, p > 1, r < s such that xix2 . . . xp = xyx2. . . xp E IdV, then x7x2.. . x, = 0 E I d V and x S x 2 . .. xp = 0 E IdV.

V it follows that there are natural numbers Proof: From S L a, b, m, n 1 and u l , . . . ,urn,ul, . . . , u, E X with u;"u2... urn = v b1 v 2 . .. v, E IdV \ IdSL. Then b y Lemma 7.2.4 we have x : x 2 . . . x, = 0 E IdV. From x i x 2 . . . xp M x s x 2 . . . xp E I d V , x32 = x42 E I d V and r < s it follows that x i x 2 . . . x, = x ; + ~ x. ~. xp . E I d V for all natural numbers k . This implies 2 2 . . . x p E IdV. Using x;"x2... x , = 0 E xTx2 . . . xp = xs+a+m-l 1 IdV we get xTx2.. . xp = yTy2.. . y,xsx2.. . xp E IdV and by Lemma 7.2.4 we have x;x2 . . . x p = 0 E IdV. In particular we have yyy2.. . yp = x y x 2 . . . xp E IdV. This gives x y x 2 . . . x p = 0 E I d V by Lemma 7.2.4.

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Corollary 7.2.6 Let V be a variety of semigroups with S L

V.

210

T h e n x2yz

7 M-Solid Varieties of Semigroups =

0.

Proof: By Lemma 7.2.l.(ii) we have x2yx Lemma 7.2.5 we get x2yx = 0.

= x3yx E IdV.

Using

We denote the subvariety of a commutative variety V of semigroups which is defined by the additional set C of equations by V(C). It turns out that C(VRc) consists of eight infinite chains. This becomes clear by the following theorem.

Theorem 7.2.7 For a variety V of commutative semigroups the following are equivalent: (a) V is Reg-solid,

(c) V either coincides with one of the varieties VRC, V({x2y = xy2,x2 = x3}), V({x2y = xy2,x3 = x4}), V({x2y = xy2 = x3y)), V({x2y = xy2,x2 = O)), V({x2y = xy2,x3 = O)), V({x2y = xy2 = o}), V({x2y = xy2,x2yz = o}), or is given i n VRC by one of the following sets of identities (where p is a positive integer): 2 2 I): y = xy2,x2 = x3, XlX2. . . Xp M XIX2. . . Xp ; 2 x2y = xy2, x2y = x3y, 21x2 . . . xp M xlx2 . . . xp where p 3 ; 2 x2y = xy2, x3 = x 4 , 21x2.. . xp M x1x2.. . xp where p 4 ; 2 x2y = xy2, ~ 1 x 2. . xp = xIx2. . . xp where p 4 ; 2 2 x y = xy , x2 = 0, ~ 1 x 2 .. .xp = 0 where p 3 ; x 2 y = xy2, x2y = 0, x1x2.. . xp = 0 where p 4 ; x2y = xy2, x3 = 0, 21x2.. . xp = 0 where p 4 ;

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Proof: By Theorem 7.2.2 and Theorem 7.2.3 we have the equivalence of (a) and (b). Further, (b) follows straightforward from (c). Thus, we have to show that (b) yields (c). Assume that V C VRC. We consider the following cases:

7.2 Regular-solid Varieties of Semigroups 1 1.1 1.2 1.2.1 1.2.1.1

SL

cV

zgv

ZCV x2 = x3 E IdV 2 21x2.. . x, = x,x2.. . x , E IdV for some natural number p 1 1.2.1.2 21x2 . . . x, = x:x2 . . . x, Sf IdV for all natural numbers p > 1 1.2.2 x2 = x3 Sf IdV 1.2.2.1 x3 = x4 E IdV 2 1.2.2.1.1 x1x2. . . x, = x,x2 . . . x, E IdV for some natural number p > 1 1.2.2.1.1.1 x2zy = x3zy E IdV 1.2.2.1.1.2 x2zy = x3zy Sf IdV 1.2.2.1.2 21x2 . . . x, = x:x2 . . . x, Sf IdV for all natural numbers p > 1 1.2.2.2 x% x4 Sf IdV 2 SL g V 2.1 x2 = 0 2.1.1 21x2. . . xp = 0 for some natural number p 1 2.1.2 ~ 1 x .2. . x, # 0 for all natural numbers p > 1 2.2 x2 # 0 2.2.1 "x 0 2.2.1.1 21x2.. . xp = 0 for some natural number p 1 2.2.1.1.1 x2zy = 0 2.2.1.1.2 x2y # 0 2.2.1.2 ~ 1 x .2. . x, # 0 for all natural numbers p > 1 2.2.2 "x 0.

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1.1 Then x = xz"E IdV for some natural number k E {2,3,4} (because of x4 = x 5 ) . Using x4 = x5 E IdV this implies x = x2 E IdV and thus V = S L = V ( { x 2 y= x y 2 , x2 = x" xxl= x f } ) .

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1.2.1.1 We assume that p is the least natural number r 2 with 21x2.. . x, FZ x : x 2 . . . x, E IdV. Then V C V({x2zy= xzy2,x2 = 2 x 3 ,2122.. . xp = x,x2.. . x p ) )=: W .

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Conversely, let u = v E IdV. Then there are 1 a, b 2 and 1 q E N such that u = xZ;C2.. . x q E IdW and v = xb,x2.. . xq E IdW. Without loss of generality let a b. If a = b,

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7 M-Solid Varieties of Semigroups

212

then we have u = u E I d W . Let now a = 1 and b = 2. The case q = 1 is impossible since Z C V . Suppose now that q 2. Then 21x2.. . x, M x 2l x 2 . . . x, E IdV and we have q p because of the minimality of p. So, from x l z 2 .. . z, = z 12 x 2 . . . z, E I d W it follows that z 1 x 2 . .. x, = x 21 z 2 . .. x, E I d W , i.e. u = v E IdW.

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1.2.1.2 Then V C V ( { x 2 = ~ x Y 2 , x 2 = x 3 ) ) =: W . Conversely, let u = u E IdV. Then there are natural numbers a , b, q with a, b 2 and q 1 such that u = ~ 3. . x , ~E I d. W and 1 v = zb,z2.. . z q E IdW. Without loss of generality let a b. If a = b, then we have u = v E IdW. The case a = 1 and b = 2 is impossible since 21x2. . . x, = x : x 2 . . . xp $ IdV for all natural numbers p 2 and Z C V .

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1.2.2.1.1.1 We assume that p is the least natural number r 3 with 21x2.. . x , = x : x 2 . . . x, E IdV. Then V C V ( { x 2 = ~ 3 z y 2 ,z2y = z y , ~ 1 x 2 .. .z p = z : z 2 . . . x,)) =: W. Conversely, let u = v E IdV. Then there are natural numbers a , b, r, s with 1 a, b 3 and q 1 such that u = x y x 2 . . . x , E IdW and u = xzb,x2... x q E I d W . If a = b, then we have u = u E I d W . Without loss of generality let a < b. The case q = 1 is impossible since z 2 = z 3 $ I d V and Z V. Suppose now that q 2. If a = 1, then q 3. Otherwise u = v and z 2 y = x3y E I d V imply z 2 = x3 E I d V , a contradiction. So we have q p because of the minimality of p. Then by 21x2 . . . x, = x:x2 . . . x, E IdW we get 21x2.. . x, = x : x 2 . . . x, E IdW and by x 2 = ~ x 3 E ~ IdW this b implies x 1 z 2 . .. x, = x 1 z 2 . .. z , E I d W , i.e. u = v E IdW. If a 2, then we get u = v E I d W immediately from x2y = z3y E IdW.

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1.2.2.1.1.2 We assume that p is the least natural number r 4 with ~ 1 x 2 . .z , = x : z 2 . . . z , E IdV. Then V V ( { z 2 y= z y 2 ,z3 = 4 2 x ,2122.. . xp = x1x2.. . x p } )=: W . Conversely, let u = u E IdV. Then there are natural numbers a , b, r, s with 1 a , b 3 and q 1 such that u = x y x 2 . . . x, E I d W and v = x!z 2 . . . x , E IdW. If a = b , then we have u = v E IdW. Without loss of generality let a < b. The case q 2 is impossible since x 2 = ~ x 3 $~ IdV and Z C V . Suppose now that q 3. If a = 1, then q 4. Otherwise u = u and x3 = x4 E IdV imply x 2 = ~ x 3 E~ I d V , a contradiction.

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7.2 Regular-solid Varieties of Semigroups

213

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So we have q p because of the minimality of p. Then from x3 M x4 E I d W and 21x2.. . xp M x:x2.. . xp E I d W it follows b that 21x2.. . x q M xlx2.. . x, E I d W , i.e. u M u E IdW. If a 2, then we get u M v E I d W using x2yx M x3yx E I d W (see Lemma 7.2.1(ii)).

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1.2.2.1.2 Then V C V({x2y M xy2,x3 M x4)) =: Wl if x2y M x3y 6 I d V and V V({x2y = xy2 = x3y)) =: W2 if x2y = x3y E IdV. Conversely, let u M v E IdV. Then there are natural numbers a , b, r, s with 1 a, b 3 and q 1 such that u M x2";x2...x,, u M x!x2.. . x q E IdWl n IdW2. If a = b, then we have u M u E IdWl n IdW2. Without loss of generality let a < b. Then from x4 M x5 and xTx2.. .x, M xb,x2... x, it follows that xyx2. . . x, ,-x;"+lx2... x,. This shows that a 2. If x2y M x3y E IdV, then we get u M u E IdW2 in the same manner as in case 1.2.2.1.1.1. If x2y M x3y 6 IdV, then we get u M u E IdWl in the same manner as in case 1.2.2.1.1.2.

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4 1.2.2.2 We assume that p is the least natural number r with 21x2.. . x, M x:x2.. .x, E IdV. Then V C V({x2y M 2 xy2, XlX2.. . xp M x1x2.. . xp)) =: W. Conversely, let u M v E IdV. Then there are natural numbers a , b, q , with 1 5 a , b 5 4 and q 1 such that u M x2";x2... x, E I d W and v M x!x2.. . x, E IdW. The case q = 3 and a = 1 is impossible. Otherwise u M u and x4 M x5 E I d V imply x3 M x4 E IdV, a contradiction. Now we can argue that u M u E I d W as in case 1.2.2.1.1.2.

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2.1 Clearly, x2 = 0 and x4 M x5 imply x2 M x 3 .

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2.1.1 We assume that p is the least natural number r 1 with 21x2.. .x, = 0. Then V C V({x2y M xy2,x2 = O,xlx2.. . x p = 0)) =: W. Conversely, let u M u E IdV. Then there are natural numbers a, b, r, with s, 1 a, b 2 and r, s 1 and ~ 1 ,. .. , u,, vl, . . . , us E X such that u M uTu2. . . u, E I d W and v M v?v2...us E IdW. If a = b and {ul,. . . , u,) = {vl,. . . ,us), then u M u E IdW. Let a # b or { u l , . . . ,u,) # {ul,. . . ,us}. Without loss of generality let a b. If a = 1 , then by Lemma 7.2.4 and Lemma 7.2.5, respectively, we get 21x2 . . . x,

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214

7 M-Solid Varieties of Semigroups

>

0 and we have r p because of the minimality of p. If b = 1, then s p by the same arguments. So from 21x2. . . xp = 0 follows ~ 1 ~ 2. U., .= ulu2.. .us E I d W , i.e. u = u E I d W . If b = 2, then u1u2 . . . u , = ( u ~ ) ~ - ~. .v. us f u (~x 1 x 2. . . xp = 0 ) bv l v 2 . ..us $c2 = 0 ) . i.e. u = v E IdW If a = 2 , then b = and uyu2 . . . U , = u($&2 . . . u , ( x 2= 0 ) = vtv:v2 . . . v, ( x 2 y z= 0 by Corollary 7.2.6) =

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=

ufu2 . . . u S ( x 2 = O ) ,i.e. u = u E I d W . 2.1.2 Then V C V ( { x 2 y = xy2,x2 = 0 ) ) =: W . Conversely, let u = u E I d V . Then there are natural numbers a, b, r, s with 1 5 a, b 5 2 and r, s 1 and u l , . . . , u,, ul, . . . , us E X such that u = u ? u 2 . .. u, E I d W and v = v ! v 2 . . . vs E IdW. The case a = 1 is impossible. Otherwise we would get u l u 2 . . . u, = 0 by Lemma 7.2.4 and Lemma 7.2.5, respectively, a contradiction. For the remaining

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cases we can argue in the same manner as in case 2.1.1.

2.2.1 Clearly, x3 = 0 and x4 = x5 imply x%

x4.

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2.2.1.1.1 We assume that p is the least natural number r 3 such that 21x2.. . x, = 0. Then V C V ( { x 2 y= x y 2 , x2y = 0, 21x2.. . xp = 0 ) ) =: W. Conversely, let u = v E IdV. Then there are natural numbers a, b, r , s with 1 a, b 3 and r, s 1 and u1,. . . ,u,,vl, . . . , % E X such that u = u:u2.. . u , E I d W and u = u!u2...us E I d W . If a = b and { u l , . . . ,u,} = { u l , . . . , u s ) , then u = u E I d W . Let a # b or { u l , . . . , u,} # {ul , . . . , us). Without loss of generality let a b. By Lemma 7.2.4 and Lemma 7.2.5, respectively, we have u ? u 2 . .. u , = 0 and v ! v 2 . . .us = 0. Let r = 1. Since x2 # 0 we get a = 3. Then b = 3. Using x2y = 0 we get u"; u!u2.. .us E I d W , i.e. u = u E I d W . If s = 1 we get u = u E IdW in the same way. Suppose now that 2. Let r = 2. Since ulu2 = 0 implies x2 = 0 we get a 2 r, s (we assumed that x2 # 0 ) . Then b 2. Using x2y = 0 we get u?u2 = v ! v 2 . . .us E I d W , i.e. u = v E IdW. If s = 2 we get u = u E IdW in the same way. Suppose now that r, s 3. If a = b = 1 , then r, s p because of the minimality of p. Then from 21x2 . . . xp = 0 it follows that ulu2 . . . u , = ulu2 . . . us E I d W .

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7.2 Regular-solid Varieties of Semigroups

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If a = 1 and b 2 , then r p because of the minimality of p and u l u 2 . . . u , = ( u ~ ) ~ ~ ~ .u.us f u 2( .x 1 x 2 . . x p = 0 ) = ufu2 . . . u s ( x 2 y z = 0 ) , i.e. u = u E I d W . If a , b 2, then u ? u 2 . .. u , = v ! v 2 . . .us E IdW by x2yx = 0 , i.e. u = v E I d W .

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2.2.1.1.2 We assume that p is the least natural number r 4 such that ~ 1 x 2 .. .x , = 0. Then V c V ( { x 2 y= x y 2 ,x3 = 0 , ~ 1 x 2 .. .x, = 0 ) ) =: W . Conversely, let u = v E I d V . Then there are natural numbers a, b, r, s with 1 < a, b < 3 and r, s > 1 and u1,. . . , u,, u1,. . . , us E X such that u = ~ $ 2 . . u , E IdW and u = u f u 2 . .. us E I d W . If a = b and { u l , . . . , u,) = { u l , .. . , u s } , then u = u E I d W . Let a # b or { u l , .. . ,u,) # { v l , .. . , u s ) . Without loss of generality let a < b. By Lemma 7.2.4 and Lemma 7.2.5, respectively, we have u:u2. . . u , = 0 and v!v2. . .us = 0. Let r = 1. Since x2 # 0 we get a = 3. Then b = 3 and a+1 ( 2 3 Rz x 4 ) u.; = u1 = v ~ u ! u .2. . us ( x 2 y z= 0 by Corollary 7.2.6) = ufu2 . . . u s ( x 3 = x 4 ) ,i.e. u = u E I d W . If s = 1 we get u = u E IdW in the same way. Suppose now that r,s 2. Let r = 2. Since x2 # 0 and x2y # 0 we get a = 3 and thus b = 3. Using x2yx = 0 we get u?u2 = v ! v 2 . ..us E I d W , i.e. u = v E I d W . If s = 2 we get u = v E IdW in the same way. Suppose now that r, s > 3. Let r = 3. Since x2y # 0 we get a 2 and b thus b 2. Then u";2u3 FZ u1u2.. .us E I d W , i.e. u = u E I d W , using x2yz = 0. The same for s = 3. Let now r, s 4. Then we get u = v E IdW in the same way as in case 2.2.1.1.1.

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2.2.1.2 Then V C V ( { x 2 y = x y 2 , x 3 = 0 ) ) =: Wl if x2y # 0 and V c V ( { x 2 y = x y 2 = 0 ) ) =: W2 if x2y = 0. Conversely, let u = v E I d V . Then there are natural numbers a, b, r, s with 1 < a, b < 3 and r, s > 1 and u l , . . . , u,, v l , . . . ,us E X such that u = uu";u2... u , E IdWl n IdW2 and u = u$2. . . us E IdWl n IdW2. I f a = b a n d { u l , . . . , u,} = {ul , . . . , us},t h e n u = u E I d W l n I d W 2 . Without loss of generality let a 5 b. Then ~ $ 2 . . u , = 0 by Lemma 7.2.4 and Lemma 7.2.5, respectively. This shows that a > 2. If x2y = 0 , then we get u = v E IdW2 in the same manner as in case 2.2.1.1.1. If x2y # 0 , then we get u = u E IdWl in the same manner as in case 2.2.1.1.2.

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7 M-Solid Varieties of Semigroups

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2.2.2 We assume that p is the least natural number r 4 with 21x2.. . x, = 0. Then V C V ( { x 2 y= x y 2 , 21x2.. . x, = 0 ) ) =: W . Conversely, let u = u E I d V . Then there are natural numbers a , b , r , s w i t h 1 < a , b < 4 a n d r 7 s >l a n d u l , . . . , u,,vl, . . . , EX such that u = u ? u 2 . .. u , E I d W and v = v ? v 2 . ..us E IdW. If a = b and { u l , . . . , u,) = { u l , . . . , u s } , then u = u E I d W . Let a # b or { u l , .. . , u,} # { u l , . . . , us}. Without loss of generality let a b. Then ~ $ 2. . . u , = 0 by Lemma 7.2.4 and Lemma 7.2.5, respectively. This shows that r > 2 or a = 4. If a = 4, then b > 4 and we get u ? u 2 . .. u , = v t v 2 . . .us E I d W , i.e. u = v E I d W , because of x2yz = 0 (see Corollary 7.2.6). If q > 2, then we can argue that u = u E IdW as in case 2.2.1.1.2.

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The next step is to characterize all regular-solid varieties which are not commutative. If we apply regular hypersubstitutions to the associative law, we get identities which have to be satisfied in any regular-solid variety of semigroups, for instance the identities

which give us ( x y ) z = x ( y z ) , xyxzxyx = xyzyx, = x 2y 2z , and xy2z2= ~ ( y z ~Let) VRS ~ . be the variety defined by these identities, i.e. ~ x VRS := M o d { ( x y ) z = X ( Y X ) , X Y Z X X Y I ) : = Z Y X Y Z , ( x ~ ~=)x2y2z,

xy2z2= ~ ( ~ 2 ~ ) ~ ) .

Any regular-solid variety of semigroups is a subvariety of VRS.The following identities are also satisfied in VRS:

Lemma 7.2.8 Let a, k , 1 be natural numbers with a, k , 1 x ~ = x y~k ~ a ~zx a Ey zIdVRs. ~x ~ ~

> 1. Then

<

Proof: Without loss of generality let k I . Then x y k z y z x = (xy)kxa+z-kzxa (yx)' = x y k x a z x a y z using x x y x z x y z = xyzyx.

Lemma 7.2.9 F o r any integer a xyazyax is satisfied i n VRS.

>

2 the identity xyzyx

Proof: Using xyxzxyx = xyzyx we get xyzyx x y x y " - l z y " - ~ x y x= xyazyux.

=

xyxzxyx

= =

7.2 Regular-solid Varieties of Semigroups

Lemma 7.2.10 W e have x7

217

= x5 5 Idl'Rs.

Proof: Obviously, from

FZ

x2y2z it follows that x7 = x5.

Lemma 7.2.11 The identities xy%yx = xyzyx and xyzy3x = xyzyx as well as xyazybx = xyax2zybx= xyazx2y6xfor a , b 1 are satisfied i n V R S .

>

Proof: We have xy"yz = xy8zy6x(by Lemma 7.2.9) = x y 6 z y 6 x(by Lemma 7.2.10) = xyzyx (by Lemma 7.2.9). Similarly we get xyzy% n: xyzyx E IdVRs. We have xyazybx z ~ ~ z x x ~ ~ ~ ) :

N

N

X

~

~

X

~

Z

X

~

~

X

xy"x2zybxby Lemma 7.2.8. Similarly one gets xyazybxn: xy"zx2ybx. 2 r.

Lemma 7.2.12 The following identities are satisfied in V R S . (i) x2yx4 n: x2yx2 E IdVRS; (ii) x4yx2 = x2yx2.

Proof: (i) We have x 2 ~ x = 4 xxYx3x = x x p x (by Lemma 7.2.11). Similarly we get (ii). Lemma 7.2.13 Assume that a , b, c are natural numbers greater than 0. T h e n (i) xy"xbycx= X ~ " + ~ E+IdVRs ~ X if b is even; (ii) xy"xbycxn: xyxyn: E IdVRs if b is odd and a (iii) xyaxbyCxn: x y 2 x y x E IdVRs if b and a

+ c is even;

+ c are odd.

Proof: (i) Without loss of generality assume that a 5 c. Then a b c XY X Y X

= = = = =

N

xyax2xbx2ycx (by Lemma 7.2.8) xy"x2y2xby2x2ycx (by xyxzxyx FZ xyzyx) xy"x2y2x2y2x2ycx (by Lemma 7.2.12) x y a x 2 y 2 x y 2 x y C(by x Lemma 7.2.9) xyax2y2x2ycx(by ( X ~ Z J n) :~x2y2z) ~ xya+c+2x(by Lemma 7.2.8).

218

7 M-Solid Varieties of Semigroups

(ii) We have a b c X Y X Y X

-

Lemma 7.2.9) xzybyx (by Lemma 7.2.11) xyxyz (by xyzzxyz = zyzyx). (iii) We have a b c zYxYz xya+a zb y c+a z (by Lemma 7.2.9) = zy2zbyz(by Lemma 7.2.11) = xy2xyx (by Lemma 7.2.8). 7.2

= =

X Y a+a+lxbyc+a+lx (by

7.2

Lemma 7.2.14 T h e equation xy2zyx = xyzy2x is satisfied i n VRS.

Proof: We have xy2zyx = xy%y2x (by Lemma 7.2.9) = xyzy2x (by Lemma 7.2.11).

>

Lemma 7.2.15 For integers a ~ ~ = zayza+b. + ~ ~ z

2 and b ~

>

1 there holds

Proof: We have xa+byxa xayxb+b+a (by Lemma 7.2.11) -- zayza+b (by Lemma 7.2.9).

-

Lemma 7.2.16 For integers a, b, c, d x ~ + E ~IdVRS. ~ ~ + ~

>

2 we have zaybzCyd =

Proof: There holds

--

a b e d Z Y X Y

= =

-

Xa+c-2 Y 2 z 2Y d+b-2 (by Lemma 7.2.15) Xa+c 2 2 d+b-2 Y X Y (by Lemma 7.2.11) xa+cyx2yx2 yd+b-2 xa+cyx2yd+b-l

za+cyd+b using ( ~ ~ y=) z2y2z ~ z and ~ ( y z=~zy2z2, ) ~ respectively.

>

Lemma 7.2.17 If a 5 is a natural number, t h e n xyax = xyap2x is a n identity in VRS.

7.2 Regular-solid Varieties of Semigroups

Proof: Using xyxzxyx xyxya-4xyx = xy a-2 x.

= xyzyx we get

219

xyax

= xyxyya-4yxyx =

Lemma 7.2.18 If a, b > 3 are natural numbers, then (i) xyaxby= x y a x b ~ 2Ey IdVRS; (ii) xyaxby= xya-2xby E IdVRS.

Proof: (i) We have xyaxby xyaxyxyxb-2y (by Lemma 7.2.8) xya+2xb-2y(by Lemma 7.2.8) xyaxb-2y (by Lemma 7.2.11). The proof of (ii) is similar.

-N

2 r. 2 r.

Lemma 7.2.19 Let a

> 4 be a natural number, then

(i) x2yaz = x2yaP2zE IdVRS; (ii) xyaz2 = xyaP2z2E IdVRS.

Proof: (i) We have x2yaz x 2 y x 2 y x 2 y x 2 y a ~(using 3z (x2y)2z= x2y2z) = x2yx2x2x2ya-%(using xyxzxyx = xyzyx) x2yx2ya-% (by Lemma 7.2.12) -- x2yaP2zusing = x2y2z. The proof of (ii) is similar.

-

2 r.

Lemma 7.2.20 For any natural number a

> 1 we have

= xyxa+2yE IdVRs; xyax3y = ~ y " + ~ xEyIdVRs.

(i) xy%ay (ii)

Proof: (i) Using Lemma 7.2.8 we obtain xy3xay = xyxyxyxay ~ y x " + ~ The y . proof of (ii) is similar. Lemma 7.2.21 Let a

> 2 and b > 1 be natural numbers. Then

(i) xaybxyxFZ xa+lyb+lx E IdVRS; (ii) xyxybxa= xyb+'xa+l E IdVRS.

=

220

7 M-Solid Varieties of Semigroups

Proof:(i) We have xaybxyx -- xayb+1xy2x(by Lemma 7.2.9) -- xay2xyb+1x(by Lemma 7.2.15) = za+lyb+lz(using ~ ( y x=~zy2x2) ) ~ The proof of (ii) is similar.

-

Lemma 7.2.22 Let a, b, c, d > 1 be natural numbers. Then we have a b c d y ~ a + c y b + dE~ IdVRS. YX y x y x P r o o f : There holds a b c d YxYzYz yza+2yb+2~c+2 (by y dLemma + 2 ~ 7.2.9) y2x2yd+b+2x (by Lemma 7.2.15) YX yxa+cy2x2yd+bx (by Lemma 7.2.11) y d + b + 2 ~ (by Lemma 7.2.16) Yz y z a + c y d +(by b ~Lemma 7.2.19(ii)).

-7.2

7.2

7.2

7.2

Lemma 7.2.23 Let a have

> 4, b > 2

be natural numbers. Then we

(i) zaybz= zap2ybzE IdVRS; (ii) xybxa = xybxa-2 E IdVRS;

(v) xytxcycx c = 2,3.

=

~ y t ( z y ) ~ xxxcyctzy ,

=

x(xy)"txy E IdVRs for

P r o o f : (i) We have zaybz = x ~ ~ (by ~ Lemma + ~ z7.2.19(i)) = xayxybzyx(using zyzxxyz = zyxyz) = xayxybz3yz(by Lemma 7.2.11) -- xap2yxybx5yx(by Lemma 7.2.15) -- xap2yxybxyx(by Lemma 7.2.11) xap2 y b+2 x (using xyxzxyx = xyzyx) -- x ~ - (by ~ ~Lemma ~ z 7.2.19(i)). The proof of (ii) is similar.

--

7.2 Regular-solid Varieties of Semigroups (iii) We have

x ( x ~ ~= ~ 2x4y4 ) ~ (by Lemma 7.2.16)

= =

2x2y4 (by Lemma 7.2.14) zx2y4 (by (i)). Similarly we obtain (x2y2)2z= x2y2z. The proof of (iv) goes similarly. (v) We will give a proof for c = 3. For c = 2 one can proceed in a similar way. To show this, we have = xyx2tx3YxYxY~ = xyx2ytx3y2xyxyz(Lemma 7.2.8) = x y x 2 t x 3 y 2 x 2 y(Lemma ~ 7.2.8) = x ~ ~ x (Lemma ~ ~ ~7.2.11) x ~ ~ x = xytx3yx3yx2yx (Lemma 7.2.11) = xzytx3yx2y2,z(by = x2y2z) = xytx3y3z (by =~ ~ ~ ~ 2 ) . Dually we prove ~ ( x ~ ) ~=t zx3y3txy xy E IdVRS. Now using these identities we calculate all essentially binary terms over the variety VRS, i.e. the elements of the free algebra F ~ R S ({x, Y})'

7 M-Solid Varieties of Semigroups

222

We denote the terms which can be obtained by exchanging x and y by (65)-(128).Details of the calculation may be found in [30]. If V is a subvariety of VRS, then the essentially binary terms of V form a subset of the set of all terms (1)-(128). As we have already mentioned, any regular-solid variety of semigroups has t o be dual-solid and included in VRS. We start with the characterization of all varieties V having the additional property that the variety R B of all rectangular bands is included in V.

Lemma 7.2.24 A variety V of semigroups which contains R B and does not contain Z is regular-solid iff V E {RB, N B , RegB). Proof: If V is regular-solid, then the application of any hypersubstitution from Reg t o any identity from V gives again an identity in V. Since Z 9 V, then for some k 2 there exists an identity x = xk in V. If we apply the hypersubstitution a,a,a to this identity, we get x = x2' E IdV. But x5 = x7 and x = x2' implies x = x2.Every regular-solid variety of semigroups is dual-solid. If R B V VRS and V is dual-solid, then we can derive xyzx = xyxzx from the identities in V in the following way:

>

c c

xyzx

-

= = = = = = = =

xzyyzx (x = x2) xzyyzxyzx (x = x2) xyxyzyxyxx (xyxyx = xyxxxyx and x = x2) xyxxxyxx (xyxyx = xyxzxyx) xzyzxyzxzx (x = x2) xzyzxzyzxzx (xyzyx = xyxzxyx and x = x2) xzyzxyxzx (xyzyx = xyxzxyx) xyxxx (x = x2).

Thus V C RegB. Since N B and R B are the only dual-solid subvarieties of RegB containing R B , the variety V is one of R B , N B , and RegB. Conversely, it is easy to check that R B , N B , and RegB are regular-solid. This lemma shows that we have t o consider varieties of semigroups which contain the variety Z of constant semigroups. We will be able t o prove that all hypersubstitutions at where t is one of the terms listed in (1)-(128),are VRs-proper. We need some more technical lemmas.

7.2 Regular-solid Varieties o f Semigroups

223

I f u M u E IdV with ub(u) = ub(u) = 2, then l e f t m o s t ( u ) = le f tmost(t) and rightmost(u) = rightmost(u) ( R B C V ) ,so u = v . If vb(u) > 3 and vb(v) = 2 or conversely then x4 = x2 E IdV. Lemma 7.2.25 Let V be a self-dual variety of semigroups with R B C V C VRS. If V satisfies a non-trivial identity u M u with ub(u) = 2 and ub(u) 3 or vice versa, then

>

( i ) x4 M x2 E I d V ; (ii) xyuxy M xyax3y E IdV for a

=

1,2,3.

Proof: ( i ) Let u = v E I d V with vb(u) = 2 and vb(v) > 3. Replacing w E X by x we get x2 M xk for k = ub(u). But x2 M xk and x5 M x7 gives x2 M x4. I f ub(u) = 2 and ub(u) 3 dually we get x4 = x2 E IdV.

>

(ii) If a

=

1 then

XYXY

xyxyxyxy (by x4 M x 2 ) xYx3y ( b y xyxzxyx M x y z y x ) I f a 2 then XY " X Y X ~ " + (by ~ Xx4~M x 2 ) M xyax3y (by ~ y " + ~ M x yxyUx3y). M M

>

-

Lemma 7.2.26 Let V be a self-dual variety of semigroups with RB c V c VRS. Then for each identity ul . . . u , = vl . . . v, E IdV (2 m, n E IY,u l , . . . ,urn,ul, . . . , u, E X ) both u l u t . . . u k M ulu!j. . . u: E IdV and u! . . . u ~ _ , u r M n ulk . . . v:_,u, E IdV hold for 2
<

Proof: Let k

=

2, then we consider the following two cases:

< <

< <

-

Case 1: ul # ui for 2 i m and ul # uj for 2 j n. Then we get ulu,2 . . . u& = v l v ; . . . v: E I d V by the substitution ul ul and w w2 for w E X \ { u l ) .

-

Case 2: ul = ui for some 2 By the substitution ul

-

< i < m or ul = uj for some 2 < j < n. ul and w

w2 for w E X

\

{ul)

224

7 M-Solid Varieties of Semigroups

12 we get u1u;' . . . u&m % u1u2 . . . u$ E IdV where ki = 1 i f ul = ui and ki = 2 i f ul # ui for 2 i m and l j = 1 i f ul = uj and l j = 2 i f ul # uj for 2 j n. Moreover, by the substitution w w2 for w E X we get uf . . . u& % v; . . . v; € M IdV. Then we have ulu;'. . . . U & ~ U I U ~. U. .~. U ~ & ~ U. :. u ., 1 vlu{ . . . ~ u I V 1. .u. u$:: ~ . . . V: E IdV. Using z y z y z M z y z z q z E 2 2 2 2 2 IdV we get ulu; . . . uku:(u: . . . u & ) % ~ ulu22 . . . u,ul (21, . . . u,) E IdV. Moreover, we have zx2y2 % z ( x 2 y 2 ) 2E IdV. Thus we 2 2 2 2 v 1 v2 2 . .. v 2, v2~ 2v ~ . V: . € IdV. Since get u l u ; . . . urnulul. . .urn zz2y2 M zz2y4 E IdV and z 4 % z 2 if m = 2 or n = 2, we have 2 2 u l u22 . .. u,ul . . . u 2, % v l v22 . . . v 2 v f . . . v: E IdV. Now we have the following cases:

< < < <

-

Case 2.1: ul = ui for some 2 < i < m and ul = v j for some 2 j n. Using xyzyx % ~ y z x ~E ~IdV x we get 2 2 2 2 ) ulu,2.. ~ . u,u, 2 . . . u,2 u 1 u 2 . .. u r n u l . .. u g % u U 1 ( u.;..~ 2 and 2 2 2 u1(v, . . . v , ) , i.e. u ~ ( u ; . . . u ~M) ~v ( v . . . v ) ~E IdV and u l u22 . .. u k M v l v ; . . . V: E IdV since zz2y2 M z ( z2 y 2 ) 2 E IdV.

<

<

-

< <

< <

Case 2.2: ul # ui for 2 i m and ul = uj for some 2 j n. By the substitution ul ti z y 4 and w ti y4 for w E X \ { u l ) we get zY4 M (zy4)Pfor some natural number p 2 using z y 4 % z y 6 . Then there holds xy4 % (xy4)(xy4)p-I = (~y4)p(zy4)p-~ M (~y4)~p-l = ( x y 4 () ( ( ~ y ~ ) ~ ) " - ' % ( x y 4 () x 2 y 4 ) ~ -(since ' zx2y2% ~ ( x y ~ ) ~ ) % ( x y 4 (x2y2)p-'(since ) zx2y2% zx2y4) % xy4x2y2(sincezx2y2% z ( x2 y 2 ) 2 ) M xy2x2y2(sincez 2 y z 2 M z 4 y z 2 ) . Thus xy4 M z y 2 z 2 y 2E IdV and we have u1u;ul.u; % u l u i % ulu; in the case m = 2 and for m 3 we get

>

>

Similarly we get v l u ; . . . u:u?. . . u:

%

ulu,2... u; E IdV.

7.2 Regular-solid Varieties of Semigroups

225

< <

< <

n. Case 2.3: ul = ui for some 2 i m a n d u l # uj for all 2 j Similarly we get u l u ; . . . u g M uvlu,2. . . u: E IdV. Let k = 3 then we have the following two cases:

< <

< <

i m and ul # v j for 2 j n. Then Case 1: ul # ui for 2 we get that u l u g . . . u& M vlv,".. . v i E IdV by the substitution ul Hu 1 and w ti w v o r w E X \ { u l ) .

< <

< <

Case 2: ul = ui for some 2 i m or ul = uj for some 2 j n. Then u l u ; . . . u k M ulu,2... u: E IdV (see case k = 2), u : . . . u k M v: . . . v: E IdV and u: . . . u; M v; . . . v i E IdV and thus 2 2 2 2 3 u1u2. . . U,Ul . . . u,ulul . . . u& vlv22 . . . v:vf.. . v:vlv;. . . v i E IdV.

N

Using x2yx2 M x3yx3 we get 3 3 2 3 3 3 u1u2. . . u,u1u2. . . u,ulul . . . u; 3 = vlv2 . . . viv;v,". . . v i v l v ~. . v i E IdV. Now we have the following cases:

< <

i m and ul = uj for some Case 2.1: ul = ui for some 2 2 j n. Then 3 3 2 3 3 3 u1u2. . . u,u1u2. . . u,u1u1 . . . u; M u u l ( u g ...~ L ) ~ ( s i n xyzyx ce MX ~ ~ X ~ ~ X ) u l u g . . . u&(since xx"' = x ( ~ " ~ ) % n d x2 M x4 if m = 2).

< <

N

. . u i u l u ~... u i In a similar way one gets ulu,3.. .uiu~u,3. ulug. . . u: E IdV. Thus u l u ; . . . u; M u l u ~. . u: E IdV.

< <

M

< <

Case 2.2: ul # ui for 2 i m and ul = uj for some 2 j n. In the case k = 2 we have shown that xy4 = xy2x2y2E IdV. If m = 2 2 2 2 2 3 then ulugufugulu~u:= ulu2ulu2ulu2 = uluiu:u!j M ulu; = ulu: since x2yx2 M x3yx3, xy4 M xy2x2y2,and x2 M x4. Let now m 3. Then we have 3 3 2 3 UlU2. . . u,u1u2. . . u;ulu:. . . u;

>

226

7 M-Solid Varieties of Semigroups

3 3 2 3 3 3 4 3 3 3 u1(u; . . . urn) u1(u2 . . . urn) u1( u 2. . . urn) (since xz3y3 = ~ ( z ~ y ~ ) ~ ) = u l ( u ; . . . U,)3 2 u 2l ( u32 . . . u ,3) 2 u 4l ( u32...U,)3 3 (since z 2 y z 2M z 3 y z 3 ) M u l ( u i . . .u,)3 4 u 4l ( u32 . ..u,)3 3 (since xy4 M xy 2 x 2 y 2 ) 3 4 2 3 3 3 M ul (u; . . . urn) ul (u2. . . urn) (since zx2y2 M xx 2 y 4 ) 3 3 2 2 3 3 3 M ul ( u 2. . . urn) ul (u2. . . urn) (since z 2 y 2 x = z 4 y 2 x ) M u1(u:. . . since z y 4 = z y 2 z 2 y 2 ) 3 u1u2.. .&(since xz3y3 M x ( z3y 3 ) 3 ). 3 2 3 3 3 In a similar way we get vlv23 . . . v,vlv2 . . . v,vlvl . . . v,3 vlv,".. . v i E I d V . This shows that ulu:. . . u; M v l v i . . . v: E I d V . M

7 2 .

-

< <

< <

Case 2.3: ul = ui for some 2 i m and ul # uj for all 2 j n. Similarly we get u l u ; . . . u 2 M ulu,3... u: E IdV. Let k = 4 then from ulu,2 . . . u k M ulu,2 . . . u: E I d V using x2y2x M x4y2x and xz2y2 = 2z2y4 ( z 2M z 4 in the case m = 2 or n = 2) it follows that u l u42 . .. u 4, M v l v t . . . v: E I d V . Let k = 5 then from u l u g . . . u& M vlu,".. . u i E I d V using x2y2x M x4y2x and xx2y2 M xx2y4 ( x 2 M x4 in the case m = 2 or n = 2) it follows that u l u z . . . u k M ulug. . . u i E IdV. k Dually we can show that uf . . . u&-,u, = v1k . . . v,-,v, E IdV.

Lemma 7.2.27 Let V be a self-dual variety of semigroups w i t h Z V RB c V c VRS T h e n for all a, b E { 1 , 2 , 3 , 4 , 5 } the hypersubstitution axaxb i s V-proper. 1 2

>

Proof: Let u M v E I d V with u # v and vb(u),vb(v) 2. Then there are natural numbers m, n 2 and u l , . . . ,urn,ul, . . . , u, E X such that u = u l . . .urn and u = u l . . .u,. If a = 1 then we b have irXlx;[u] M ulu2 . . . u h and 6xlxb[u] M ulu2b . . . u: and by Lemma 7.2.26 we get 6 x l x ; [ u ]= 6,,,;fv] E I d V . If b = 1 then we have 6,:,,[u] M uy . . . u k p l u , and 6,:,,[v] = vf . . . vgplv, and by Lemma 7.2.26 we get 6 x y x 2 [ uM] 6 x y x 2 [ uE] IdV. Thus the cases where a, b E { 2 , 3 , 4 , 5 } are left.

>

2 2 Case 1: Both a and b are even. If m = 2 then 6,:,;[~] = u1u2 is clear by x2 M x4. Let now m 3 and we want t o prove by induction that e X a1 x b2 [u]M u: . . . u g E IdV. Let m = 3 then

>

7.2 Regular-solid Varieties of Semigroups X

X

I

= =

a b a b ( w 2 ) u3

(u&;)"u; (x2y22= x4y22 = x 2 y 2 2 and z y 2 z 4 = z y 2 z 2 ) 2 2 2 = U,U2U3 (x2y22= ( I ) : 2 y 2 ) 2 2 ) . Suppose now that the statement holds for m = p. T h e n for m = p+l we have

= =

b ( u f . . . u2p a)up+, (by hypothesis) 2 a 2 (Uf . . . 'Up) ( 2 . ~ M ~ ~2 ' ). ~ ~ ~ ~ ) 2 2 2 2 2 2 = (I): y ) 2 ) . (x2y22 u , . . . upup+, In a similar way one gets 6 x y x [u] ; = u: . . . u: E IdV. Moreover, from u , . . . u, = ul . . . u, E IdV it follows that uf . . . u& = u; . . . u: E IdV by the substitution w Hw2 for w E X, i . e A,:,; [u]= A,:,; [u] E IdV.

Axax;[u]

Case 2: Both a and b are odd. I f m = 2 then iixYxi: [u]= u;ui( follows from x2 = x4. Let now rn > 3 and we want to prove by induction that AxYxa[u]= u: . . . U & E IdV. Let rn = 3 then L

u

Suppose now that the statement holds for m = p. T h e n for m = p+l we have 6,.,;[u] = (u; . . . 43 )a b (by hypothesis) 3 = (21;. . . u3p a)up+1 (2x2y2= 2x2y4) 3 3 = u; . . . upup+, ( x 3y32 = ( x3y 3 ) 3 2). In a similar way one gets A,a,b1 2 [u]M u; . . . u i E IdV. Moreover, from ul . . . u, = ul . . . u, E IdV it follows that u; . . . u: = u; . . . u i E IdV by the substitution w Hw3 for w E X , i.e. 6xTx;[u]= 6xTx;[u] E IdV. Case 3: a is even and b is odd. If m = 2 then A,,,;[u] = u:u: is clear because o f x2 = x4. Let now m 3 and we want t o prove 2 a b [u]= U , . . . ug-,u; E IdV by induction. Let rn = 3 then oxl"a

>

A

k:.;["I

= = = =

a b a b ( w 2 ) u3

( u f u ; ) " ~ (;x 2 y x 2= x3yI):3) ( u ~ z L ; ) (~2z L 1 ;) :=~2~1~) : ~ 2 2 3 2 2 2 U,U2U3 (x2y22= ( I ) : y ) 2 ) .

~ ~ )

7 M-Solid Varieties o f Semigroups

228

Suppose now that the statement holds for m = p. Then for m = p+l we have 2 3 a b A,;,; [u] = ( u . . . up-,up) up+,(by hypothesis) 2 b M (u:...U ~U-2 )~a Up+1 ( x ~ M~x x~ ~~ x ~ ) 2 a 3 M ( U . . . up) (2x2y2M 2x2y4) 2 3 2 2 2 M U: . . . UPUP+, ( X 2 Y 2 2 M ( X ) 2). 2 2 Similarly one gets A,a,ti[v] M v, . . .V,~,V;E IdV. B y Lemma 1 2 7.2.26 from ul . . . urn M ul . . . u, E IdV it follows u: . . . ukplurnM uf . . . U ; - ~ V , E IdV and thus u: . . . u:-,uL M ul2 . . . v ~ - , uE~IdV

(since RB i V ) ,i.e. 6,+..[u]M 6,?,; [v]E IdV.

Case 4: b is even and a is odd. Similarly we get A,y,; [u]M A,y,; [v] E IdV. Lemma 7.2.28 Let V be a self-dual variety of semigroups with Z V RB c V c VRS.Then for all a , b, c E { 1 , 2 , 3 , 4 , 5 ) the hypersubstitution ~ , a1X .b 2x c1 is V-proper.

>

Proof: Let u M v E IdV with u # v and vb(u),vb(v) 2. Then there are natural numbers m , n 2 and ul, . . . , u,, ul, . . . , u, E X such that u = ul . . . urnand u = ul . . . u,. We have to consider the following two cases.

>

Case 1: b is even. If m = 2 then A,,,;,y[u] = u;u;uE is clear by x2 M x4. Let now m > 3 and we want t o prove by induction that a 2 2 2 2 2 c 6,azbzc [u]M Ulu2.. . Urn-lUrnUrn-l.. . U2u1 E IdV. 1 2 1 Let m = 3 then X X X I = u;L ( ua2bwc2 )bu1c a+l b c+l b c U ; (u2 u3u2 ) U ~ ( I ) : Y X Z J I ) : I ) : ~ ~ x ~ ~ I ) : )

Suppose now that the statement holds for m we have

= p.

Then for m

= p+l

%;,;,: ["I 2 c b c u ? . ( u ~.u. u:t~:+~ug. ~. . . u3u2) u1 (by hypothesis) 2 2 2 2 c+l b c U;(U;+~U:. . . upup+lup. . . %u2 ) u1 (by xyzy~): MI ) : ~ ~ . Z ~ ~ I ) : ) 2 2 2 2 2 b c M u ? . ( u ~. .u. u ~ p ~ p + l u. .pu3u2) . u1 (by I ) : ~ ~ I M ) : ~I ) : ~ z I ) : ~ ) a 2 2 2 2 2 2 2 c 2 2 -- u1u2u3. . . u p ~ p + l u. .pu3u2u1 . (by I): y x M ( I ) : 2 y 2 ) 2 2). M

-

Similarly one can show that 2 [u]M uTug . . . ~,-~u:u:-~ . . . uzuf E IdV. By Lemma 7.2.26 1 . 2 1

AxaXbxc

7.2 Regular-solid Varieties of Semigroups

229

from ul . . .urn M ul . . . u, E I d V it follows ulu;. . . u: M u , u ~ ...u: E I d V and thus ug . . . u;ul M u: . . . u?jul E I d V ( V is self-dual). Since RB C V we have u";;. . . uk M uv;";.. . u: E I d V 2 2 and u2r n . . u;uT M urnv,-, . . . v?jvtE IdV. This gives a 2 2 2 2 2 c U1U2... U,plU,u,pl.. . u2u1 , 2 2 2 2 c , ua1 u2 2 .. u,plu,v,pl.. . v2v1 , 2 2 2 upJ; . . . unp,ununp, . . . u22 uc1 , . 1.e. a,a,b ,c [u]M 62 1a2 2b2 1 c [u]E IdV. 1 2 1

A

Case 2: b is odd. If m = 2 then it is easy t o check that 6 x y x 1[u] x ~M u?ugu; M u?ug2uyb y x2 M x4. Let now rn 3 and we want t o prove . . . u? E IdV. Let by induction that ii,a,~,c [u]M U? . . . 1 2 1 m = 3 then

>

M u ~ u ~ u ~ (since u ~ u xyxzxyx f M xyzyx). Suppose now that the statement holds for m = p. Then for m = p+l we have ~z;,;,: [ul M u?(u;... u ~ u ~ + ,. .uU~; ) .~ Z L (by E hypothesis) M u?@.. . u ~ u ~ + ,. .uu;u? ~ . (since xyxzxyx M xyzyx) a b2 M uyu;. . . upup+,u~. . . uiuy (since x5 M x 7 ) . Similarly one can show that c i i [ I M up. . . V a, - ~ V ,b 2v,-I . . . vf E IdV. B y Lemma 7.2.26 from ul . . . u, M vl . . . v, E I d V it follows u? . . . ukp,u, ,vy . . . ~ : ~ , v E ,I d V , uy . . . uLplu, M vy . . . vgplv, E I d V and thus u,u&~, . . . uf M uv,u~pl . . . uf E I d V ( V is self-dual). Since RB C V "b a b" we have uy . . . ukplu, ,- vf . . . vnplv, E I d V and v,b%umpl. . . u; M C v,b2 v,-, . . .v$ E IdV. This gives u? . . . u&-,ugu',,-, . . .US a b2 c M U? . . . V,-,U, Urn-,... uf a M U T U ~. . . V,-,V, b 2U,-,c . . . u ; u ~ , . 1.e. ax 1ax^,, [u]M 6x 1ax 2Lx 1 [v]E IdV. 2 1 N

A

Lemma 7.2.29 Let V be a self-dual variety of semigroups with Z V RB V VRS Then for all a, b E { 1 , 2 , 3 , 4 , 5 ) the hypersubstitutzon (~.,~,~,b 2s V-proper.

>

Proof: Let u M u E I d V with u # u and ub(u),ub(u) 2. Then there are natural numbers m , n 2 and ul, . . . ,urn,ul, . . . , u, E X

>

7 M-Solid Varieties of Semigroups

230

such that u = u l . . .urn and u = u l . . .un. have xyaxy M xyax3yby Lemma 7.2.25 and ulu~ulub,M uU1u$u:u~. We want to prove a 3 3 6x1113111;[u] M ul . . . um-lurnul. . . uln-lugi. Let 6.1.;.l.$

--7 2 .

7 2 .

7 2 .

7 2 .

I f m = 2 then we thus 6x1z.x1x; [u]M by induction that m = 3 then

[ul b b u1u;u1u;u~u1u;u1u2u3 U ~ a U 3 ~ bU a~ 2U a~ 2U b~ bU ~(by U X~Y ZUI J X ~ U M ~x U~~ ~ z ~ ~ x ) u1u2ulu2u3u1u2u1u2u3 a 3 b a 6 a 2 b b (by x 2 z x 2 M ~a +UUb ~~a U 3 a~ 2Ub ~b U (by ~ UX Y~Z Y UX ~M X ~ X ~ Z X ~ ~ X ) l + b a 3 1 2 b b ulu2 u3ulu2ulu2u3(by xyaxyaxM xyzyx) t ~ ~ u ~ + ~ u $ (by u ~ x2yx2yx u ~ + ~=ux~2 y 2 z ) b+l u3 b u1u2 l+b U a ~ 2 U 2 ~ U ~ U ~ (by U ~zxy2xy2M z x 2 y 2 ) (by M~ y ~ x y ~ + ~ x ) u ~ u ~ + ~ u $ u ~ u ; +(by ~ ux2zx2 ~ u ; uM~x42x2) a 2 3 I L ~ ~ L ~ I L ~ ~ L ~ I(by L ~ xy ~ Ll +~b z Iy l L+ b~x ~ L ~ Z Y ~ Y ~ ) u 1 u 2 u ~ u ~ u ~ u(by 1 u x3zx2 ~ u ~ M x2zx3) U

U ~ U a ~ 3 U3 ~6 U (by ~ UZ ~ X U ~~ M ~

z xX2 y 2~).

~

Suppose now that the statement holds for rn = p. Then for rn = p+l we have ~ x l x ~ x l [ul x; 3 UbUa 3 b b u1 . . . up-1uau3 p 1 . . .up-1 p+lul...up-1u;u.:. . .up-lupup+l by hypothesis.

-

We put r := ul . . . up-1 and get

--

["I

rupr a 3 ~b ~ a u a ~3 ub+~b u~ ~r(since +u ~ ~xYtx3y3z ~ M~yt(xy)~z) r u ~ r u ~ ~ ~ +b , br u ~ r(since u ~ uxyzyx ~ + ~M xy3zy3x)

rupui+,r3 upup+l 3 6 (similar to the case m = 3) U l . . . u p ~ l u p u ~ +. .up-lu;u;+l l u3 ~ . (since xytx3y3zM ~ y t ( x y ) ~ z ) . Similarly one can show that

B y Lemma 7.2.26 from ul . . . urn M ul . . . un E IdV it follows 3 u;... U ~ - ~ M U ,u:. . .U,-~U, E IdV and thus u:. . .u2-,ugi M u;. . . u ~ - , u E~ IdV (since RB C V ) . Moreover, since RB C V

7.2 Regular-solid Varieties of Semigroups

-

231

from ul . . . urn M ul . . . u, E I d V it follows that ul . . . urn-luk ul . . . u,-lu~ E I d V . This gives ul . . . urn-lu&u;. . . uL-,u& M a 3 3 ul . . . q--lu,ul . . . U,-~U: IdV.

t I d V , i.e

6z1z.z1z6 [u]

M

6z1z1z1z;[u] t

Lemma 7.2.30 Let V be a self-dual variety of semigroups w i t h Z V RB c V c VRs. T h e n for all a , b E { 1 , 2 , 3 , 4 , 5 ) the hypersubstitution O z a1z Z z b1z z i s V-proper. Proof: The proof is similar to that of Lemma 7.2.29. Lemma 7.2.31 Let V be a self-dual variety of semigroups w i t h Z V RB C V C V R S . T h e n for all a , b E { 1 , 2 , 3 ) w i t h a b 3 the hyperswbstitution o ~ + i s~V - p~r o p~e r .~ ;

+ >

>

Proof: Let u M u t I d V with u # u and ub(u),ub(u) 2. Then there are natural numbers rn,n 2 2 and ul, . . . ,urn,vl, . . . , v, E X such that u = ul . . . urnand v = vl . . . v,. b Let m = 2. Then we have 6 x ~ x ~ x l x=; [ulu2ulu2 u2 ]a M u,fu$u:u; by Lemma 7.2.25. We want to prove by induction 2 a b 6x:x:xlx; [u]M u1u2 . . . u ~ u3 lbu 2. ..urn E IdV.

Let m

3 then

=

[ ~ l

~z:z;zlz; M M

u~ugulu;u~ugulu;u~u~ugulu~u~ ufu$ulu",~u$ulu",$~~u$~lu~~~ (by x%x3 = x2xx2) u,f u$u~u~u~u$u~u~u$u~u$ulu~u~ (by xyxyx M x y ~ x ~ ~ x )

(by xyxyz = x y x x ~ y x ) f u ~ u l u ~ u $ u ~ u ~ u(by l u "xyxyx ,~ xy3xyx) - uulu2ulu2u3u1u2ulu2u3 (by xY2zY2x xyxyx) u ~ u ~ b u ~ u ~ (by u ~ x2yx2yx u ~ u ~ ux2y2x) ~ ~ u ~xy3xyx) u b , ~ ~ -- u ~ ~ ~ ~ u ~ u(by~(byuxyxyx xyaxyax= zyxyx) ulu2 u3ulu2 u3 (by x2yx2yz x2y2z). a+ b >3 a >2 b > 2. a > 2 --(by 2x2y2= M

u2U2a+b I 2 ~ L b~ aI L3 ~a ~ L ~ U ~ ~ L ~ ~ L ~ I L ~ I L ~

M

M

2 b 2 b a 3 a 2 b b

M

M

M

M

M

u2U1+a ~aL3~ I L , ~ L ~ u , ~ ~ L ~ I L ~ 2

Since

l+a

a 3 b+l

b

we have

U2Ul+a a 3 l+b b 1 2 u3u1u2 2 a-l 2 2 a 3 l+b b U l U 2 U3U2U3U1U2 U 3 u2Ua-l 2 a 3 b b

M

M

or

If

then

u3u2u3ulu2u3(by xy2xy2x= xyxyx) u f u $ u ~ u : u ~(by u ~ xx2y2M ~ ( x y ~ ) ~ ) .

232

7 M-Solid Varieties of Semigroups

>

If b 2 then U2Ul+a a 3 l+b b 1 2 u3u1u2 u3

7 2 .

= =

U2Ua+l a 3 2 2 b-1 b 2 2 1 2 u3u1u2u1u2 Y (x~Y u2UaUaU3U U 2 U b - l 7-43 b (by X Y ~ Z Y ~=XX Y Z Y X ) 1 2 3 1 2 1 2 2 2

)~')

u & ; u $ u ~ u ! u ~(by x y z = ( ~ ~ y ) ~ z ) . Suppose now that the statement holds for rn = p. Then for rn = p+l we have a 3 6 [u]= ( u : ~ ;... upu1u2 . . . Ubp )2 ~ a~ + ~ u :. .l .lUqaP U3l UP2 . . . b b by hypothesis. We put r := u2 . . . up. Since x y ~ ( x y )M ~ txyzx2y2t as well as z ( ~ y ) ~ t = x yzx2y2txy and x y ~ ( x y ) = ~ txzyzx3y%and z ( ~ y ) ~ t=x yzx3y3txy we have "2,.s,.1

-

[ul

,u 2l r a u1r 3 b 2 a 3 b a 2 a 3 b b u1r u l r Up+lU1r u1r Up+l ,u 2l r a u l r bu 2l r a

(since x y 2 z y 2 x= x y z y x ) ~ 3+b b~ u (similar ~ ~ t o the case m = 3) a a 3 b b b ,U 21 Ua2 . . . UpUp+1U1U2 UPUP+l (since ~ y z ( x y )= ~ xyzx2y2t, t ~ ( x y ) ~=t zx2y2txY, x ~ x y z ( x y ) %t xYzx3y% and z ( ~ y ) ~ t=x yzx"'txy). a 3 b Similarly one can show that 6,2,a1,b2 2 [u] M u:ui . . . u,u1u2 . . . u,b E IdV . B y Lemma 7.2.26 from ul . . . u , = ul . . . u, E IdV it follows b u l $ . . . u& = ulu!. . . u i E IdV and u l u ! . . . u; = u1u2.. . u,b E I d V . Since R B C V we have u?u; . . . u& = u&v,". . . u; E IdV and b , u;u~, . . . U b, M ufu;. . . uk E I d V . This gives u&;. . . uhu;ub,. . . urn ufug.. . u ~ u ~ u !.uk j . . E I d V , i.e. 6,.,;,l,;[u] = 6,:,;,l,;[u] E I d V . ,u 2l r aua

2

2

2

-

Lemma 7.2.32 Let V be a self-dual variety of semigroups with Z V R B C V C V R S . Then for all a, b E { 1 , 2 , 3 } with a b 3 the hypersubstitution gSa,2S.,; is V-proper.

+ >

Proof: The proof is similar t o that of Lemma 7.2.31. Lemma 7.2.33 Let V be a self-dual variety of semigroups with Z V RB c V c VRS. Then for all a E { 1 , 2 , 3 ) the hypersubstitution 0,3, ,, is V-proper. l ; l 2

>

Proof: Let u = u E IdV with u # u and ub(u),ub(u) 2. Then there are natural numbers m, n 2 and u l , . . . ,urn,ul, . . . , u, E X such that u = ul . . . urn and u = ul . . . u,.

>

7.2 Regular-solid Varieties of Semigroups

233

2 then we have xyaxy M xyax3y by Lemma 7.2.25. Hence ~ x ; x ; x l x z[ul If m

=

M

U : U ~ U ~ U ~

M

U:U$U~U~

M u ~ u $ u ~ u(by 2 x3zx3 M x 2 z x 2 ) . We want to prove by induction that

Let m

=

ex:x2xlx2

= =

3 then

bl

u~ugu1u2u~ugu1u2u:ugu1u2u~u~ugu,u2u~ u:u;u;u2u:u;u;u2u:u;u;u2u;u~u;u~u2u3 (since x y z y z = x y z y % ) U3U3a+3 u3u2u,u2u3 a a 3 (since xyzyx M x y x z x y x ) u1u2 3 2a+4,pU2 U3U 2 3 (since xyazyax M x y z y x ) u1u2 2 2a+4UaU 2 u2U (since z 3 z z 3 M z 2 z z 2 ) u2U2a+4u3ulu2u:u2u3 a 2 (since z y z y x = z y z z 2 y z ) u2U2a+4u3ulu2u3 a 2 2 (since z 2 y z 2 y z = z 2 y 2 z )

- ,, -- , - , 2 r.

, ,

x4zx2). Suppose now that the statement holds for m = p. Then for m we have ~ x 3 x ; x x 2[ul 2 2 = (u: . . . up1 u;u: . . . up- 1 u p ) 2 2 u;+116:. . . up-116;u:. . . 16p-,upup+1 by hypothesis. We put r := u: . . . u:-,and get M

-

U ~ U ~ U ~ U (since ~ U ~ x2zx2 U ~ M

= p+l

a 2 a 2 a 2 upr upr2 upr upr2 upr upug+lr2 upr UpUp+l (since z 2 y 2 zM ( z2 y 2 ) 2 z )

7-

2 a 2

T 3 ~ p " T ~ p T 3 ~ ~ T ~ p ~ 3 ~ p " ~ ~ p ~ ~ + 1 ~ 3 ~ p " ~ ~ p ~ P

(since z y 2 z y 2 zM z y z y z ) 2 2 a to the case m = 3 ) T u p u p + , ~ 2 u ~ u p (similar +~ 2 2 = u: . . . up-,u;ug+,u:. . . up-,u;up+l (since z 2 y 2 zM ( z2 y 2 ) 2 2 ) . 2 Similar one can show that 8 x ~ x 2 x[v] l x= 2 v12 . . . v,-,v~v: . . . ~2, - ~ v E, IdV. By Lemma 7.2.26 from ul . . . urn M ul . . . u, E I d V it follows u : . . .ug-,urn M u . . u 2n 1 u E IdV. Since RB C V we have u: . . . uk-,u& M u: . . . u ~ - , u ~E IdV. This gives

234

7 M-Solid Varieties of Semigroups

2 u1 ... 2 &x:x;x,x,[ul

"

2 2 2 2 RZ u, . . . u n p l u ~. u . .~unplu, E I d V , i.e. . . . ump,u, ~x:x;x,x,[ul E I d V .

Lemma 7.2.34 Let V be a self-dual variety of semigroups w i t h Z V RB c V c VRS.T h e n for all a E {1,2,3) the hypersubstitution ax1x2x~x3 i s V-proper. Proof: The proof is similar t o that of Lemma 7.2.33. Lemma 7.2.35 Let V be a self-dual variety of semigroups w i t h Z V RB C V C V R S . T h e n a x , x ~ x i~s xV-proper. 2

>

Proof: Let u = v E IdV with u # v and vb(u),vb(v) 2. Then there are natural numbers m , n 2 and ul, . . . , u,, vl, . . . , v, E X such that u = ul . . . u, and v = vl . . . v,. Clearly, if m = 2 then Az1z;z:z2[u] FZ u1u~u:u2. We want to prove by induction that

>

Let m = 3 then 6z1z;z:z2 [ul 2 2 2 2 u1u;u~u2u~u1u2u1u2u1u2u1u2u3 RZ u l u ~ u ~ u 2 u ~ u 1 u ~ u2~ u3 2 u 1 u(since 2 u 1 ux%x% 2u3 x2zx2) RZ u l u ~ u ~ u 2 u ~ u 1 u ~ u(since ~ u 2 xyzyx u3 FZ xyxzxyx and xyzyx xyx2zyx) 2 2 2 ulu2ulu2u3u~u2u~u2u3 (since xyzy% = xyzyx) 2 2 2 3 3 ulu2ulu2u3u,u2u,u2u3 (since xyzy% x xyzyx) 2 2 2 2 2 M U ~ U ~ U ~ U ~ U ~ U ~(since U ~ U x%x3 ~ U ~ =Ux2zx2) ~ = ~ l ~ ~ u ~ u ~ u 2(since u ~ uxy2x2zyx 2 u 3 FZ xy 2 zyx) 3 2 2 2 RZ ~ l ~ 2 ~ 3 u 1 (since u 2 u 3x2yx2yzRZ x2y2z) 2 2 2 3 ulu2u,u,u2u3 (since xy2zy3x= xy3zy2x) 2 2 3 ulu2uiu2u3u,u2u3 (since zxy2xy2= zx2y2) 2 2 2 ulu2u3u2u3u,u2u3 (since xyzyx = xyzy%) RZ ulu;ugu:u2u3 (since zxy2xy2RZ zx2y2). Suppose now that the statement holds for rn = p. Then for rn = p+l we have

7.2

-

"

7.2

7.2

. . upu;+1u1u2. 2 . . u2 u2 712. . . upu116;. .. u1u2. 2 . . upu1u2. 2 2 P 1 2 2 upu1u2... upup+,by hypothesis.

7.2 Regular-solid Varieties of Semigroups

235

We put r := u 2 . . . u p and get

[ul 2 2 2 2 2 u l r 2 ulrup+lulr ulrulr 2 ulrup+l (since zx2y2txy= z ( ~ y ) ~ tand x y z ( x 2 y 2 ) 2= z x 2 y 2 ) (similar to the case m = 3) u l r 2 up+lll~rup+l 2 2 2 2 2 UlU2... UpUp+lU1U2.. . UPUp+l (since zx2y2txy= z ( ~ y ) ~ tand x y z ( x 2 y 2 ) 2= z x 2 y 2 ). 2 2 2 Similar one can show that dx,,zxZx,[u] = u1u2 . . . u,ulu2 . . . u, E I d V . B y Lemma 7.2.26 from ul . . . u , = vl . . . v, E IdV it follows 2 ulu2 . . . u 2, = v l v z . . . v: E I d V . Since RB c V we have u ? u 2 . .. u , = v f v 2 .. . v, E I d V . This gives

--

'xlx;xfxa N

-

Lemma 7.2.36 Let V be a self-dual variety of semigroups with Z V RB C V C VRS. Then for all a E {1,2) the hypersubstitution ~ x ~ x ~ x is ~V x- P~ ~x O~ P ~ ~ .

>

Proof: Let u = u E IdV with u # u and ub(u),ub(u) 2. Then there are natural numbers m, n 2 and u l , . . . , u,, v l , . . . , v, E X such that u = ul . . . u , and v = vl . . . v,. Clearly, if rn = 2 then

>

We want to prove by induction that

6,1,.,l,2,1 [u]

ulu; . . . u ~ u l u ., . . u1 E IdV.

Let rn = 3 then 6zlz:zlzazl [u] u1u;u1u2u1u~u1u;u1u2u1u3u1u;u1u2u1 u1u~u1u2u1u~u1u2u1u2u1u3u1u2u1u2u1

--

(since xyzyx

= xy2zy2x)

(by using of z y z y x = z y z z x y z ) . Suppose now that the statement holds for m = p. Then for m we have

= p+l

236

7 M-Solid Varieties of Semigroups

[ul u1u;. . . u;ulup... U ~ U ; + ~ U . .~ u;ulup. U ; . . . ulup+lulu; . . . u;ulup. . . ul (by hypothesis) u l u ; . . uap u ~2 u p. u . .; ~ ~ u ; + ~ u.~. .uU ;; + + ~~~ ~ U : , 2 . . . u2u~uP+lu&+'. . . u ~ + ' u l u P..u1 . (by xy2zy2xr xyzyx) a 2 2 2 11111;. . . upulup... u ; ~ l u ; + ~ u l. .uu~2p.u ~2U p. u;ulup+lulu2 .. 2 . . . u p u ~ u .P. ul . (by x2yx2M x3yx3) u1u;. . . upau;+lulup+lup. . . u1 (by using of xyzyx M xzyzzzy). Similarly one can show that 6,,,;,,,,,, [v] vlv,".. . vgvlv,. . . vl E IdV. By Lemma 7.2.26 from ul . . . urn M vl . . . v, E I d V it follows ulu;. . . uk M ulu,".. . ug E IdV. Since V is self-dual we have u,. . .ul M u,.. .ul E IdV. Moreover, RB C V implies ul M ul E IdV. This provides u l u $ .. . u&ulurn... ul M vlv,".. . vivlv,. . . vl E IdV, i.e. 6,,,;,,,,,, [u]= &,,;,,,,,, [v]E IdV.

-

6xlx;xlxaxl 7.2

7.2

7.2

7.2

Using the previous lemmas we can characterize all regular-solid varieties V with Z V RB C V.

Proposition 7.2.37 Let V be a variety of semigroups with Z V RB C V. T h e n V is regular-solid ifl V C VRS and V is dual-solid.

c

Proof: Because of Lge, Lg any regular-solid variety is a subvariety of VRS and is dual-solid. Conversely, let V C VRS be a dual-solid variety. For all essentially binary terms of VRS we have t o check that the application of any hypersubstitution ot t o an identity in V is again an identity in V. Since V is dual-solid we have to take into consideration only such terms t with leftrnost(t) = x, i.e. the terms listed before. In the previous lemmas for all the listed terms t we have checked that the application of at to an identity in V is again an identity in V. This shows that V is regular-solid. We consider now the case that a non-outermost identity holds in V, i.e. RB V. If such a variety is regular-solid then we will show that it is a subvariety of the variety

7.2 Regular-solid Varieties of Semigroups

237

Proposition 7.2.38 For each regular-solid variety V of semigroups with R B g V there holds V C V z y . Proof: Since V is regular-solid we have V C VRS. Since V is self-dual we have only t o show that x2y3 = y2x3 E I d V . Since R B g V there is an identity u = u E IdV such that le ftmost(u) # le f tmost ( v ) or rightmost ( u ) # rightmost(v). Without loss of generality we assume that le f tmost ( u ) # lef tmost ( v ) . We substitute in u = v the variable l e f t m o s t ( u ) by x4y4x4 and w E X \ { l e f t m o s t ( u ) ) by y4x4. SO we get x4(y4x4)"= ( y 4 x 4 ) bE IdV for some natural numbers a, b 1. Because of x4y4x4y4= x4y4 (fol= lows from x2yx2 = x4yx4 and x4y4 = x 2 y 2 x 2 y 2 ) from (y4x4)E b I d V it follows that x4y4x4= y4x4 E IdV. Since V is selfdual we have also x4y4x4= x4y4 E IdV. Thus y4x4 = x4y4 E I d V and then y2x4 = x2y4 E IdV (because of x4y4 = x2y4).In particu) x ( x ( y ( y ( y y ) ) )E) I d V . We apply the lar, we have y ( y ( x ( x ( x x ) ) )= hypersubstitution ox:,,t o this identity and get y4x7 = x4y7 E I d V and using x4y4 = x2y4 we get y2x7 = x2y7 E IdV. Using x2y5 = x2y3 we get y2x3 = x2y3 E IdV. Since V is self-dual we have also y3x2 = x3y2 E I d V .

>

Corollary 7.2.39 For each variety V C V$yc holds x3y2z x2y2z = x2y3z = y2x2z E IdV and zx2y3 = zx2y2 = zx3y2 z y 2 x 2E I d V .

= =

Proof: We have y2x3 = x2y3 E IdV. We substitute y by y2 and get y4x% x2y6 E IdV. Using x 4 y b x2y3 and x2y4 = x2y6 we get y2x3 = x2y4 E I d V and thus zy2x% zx2y4 E IdV. We have zx2y4 = zx2y2 and thus zy2x3= zx2y2 E I d V . Moreover, we have zx2y2 = zx2y6 (zx2y4= z x 2 y 2 ) zy2x3y3 (y2x3= x2y3) zy3x3y2 (x2yx3= x3yx2) z ~ " ~ (y3x2 x ~ = &y2) z y 2 x 2 (zx4y2= z x 2 y 2 ) . This shows zx2y3 FZ zx2y2 = zy2x2. Dually one obtains x3y2z FZ x2y2z= y2x2z.Moreover, z y 2 x 2 = zy2x3 zy"3 (x3y2z= x 2 y 2 z ) ( z y 2 x 2= zy2x3). Dually we get zx3y2 = z y 2 x 2E I d V .

7.2

7.2 7.2

7.2

7.2

7.2

7 M-Solid Varieties of Semigroups

238

Lemma 7.2.40 Let V C V$Tc. If t E W ( X 2 ) with ub(t) term containing x2y2 as a subterm, then t M x2y3 E IdV.

> 5 is a

Proof: We give a proof by induction of the length of a term. If t is a term of the length 5 with x2y2 as subterm, t is one of the following terms: t = x 2 y 3 or t = yx 2 y 2 or t = x3y2 or t = x2y2x. If t = yx2y2 then t = y3x2 (by zx2y2M zy2x2) x"' (by y3x2 M x3y2) ,-.2 x3y3 (by 2 ~ M =:2x2y2) ~ 7 ~ ~ ,-.2 x2y3 (by x3y22 M x2y22). If t = x3y2then we get similarly t M x2y3.If t = x2y2xthen we have t M y2x3 (by x2y2zM y2x2z)and finally t M x2y3 (by x2y3 M y2x3).

-

Suppose that r M x2y3 for all terms r of the length p with x2y2 as subterm. Let s be a term of the length p 1 with x2y2 as subterm. Then there is a subterm r of s of the length p with x2y2as subterm. Then s E {xr, r x , yr, ry} and we have:

+

xr

--M

xx2y"by hypothesis) x2y3 (by x3y2z = x2y2z) and x2y3x (by hypothesis) y2x3x (by x2y3 M y2x3) y%4 (by x3y2z M x2y2z) y%3 (by zx2y3 M zx2y2) y2x3 (by x3y2z M x2y2z) x2y3 (by x2y3 M y2x3) and yx2y3 (by hypothesis) yy2x"by x2y" y2x3) x2y3 (as in the previous case) and x2y3y (by hypothesis) x3y4 (by x3y2z M x2y2z) ~ " 3 (by X X M ~zx2y2) ~ ~ x2y3 (by x3y2z = x2y2z).

Corollary 7.2.41 Let V C V$yc with x2 M x3 E IdV. If t E W ( X 2 ) is a term with x2y2 as subterm, then t = x2y2. Proof: If t is a term of length 4, then t = x2y2. If t has a length greater than 4, then we have t M x2y3 by Lemma 7.2.40. Using

7.2 Regular-solid Varieties of Semigroups

x2 = x3 we get t

FZ

239

x2y2.

We mentioned already that an identity u = v is called regular if both terms u and v use the same variables and that the regular identities are precisely the identities of the variety SL.

Lemma 7.2.42 For each variety V of semigroups with S L V z y there holds xy4 = zy4 E IdV and y4x = y4z E I d V .

V

C

Proof: We show xy4 = zy4 E I d V . The proof for y4x = y4z E IdV V there is an identity u FZ u E IdV and is dually. Since S L an element w E X with w E uar(u) and w $ uar(u) or conversely. Without loss of generality we can assume that w E uar(u) and w 6 var(v). In u = v we substitute a E X \ {w) by x6 and w by y6 and get u:. . . u: = xP6 E IdV for some natural numbers n , p 1 and u l , . . . , u , E { x ,y ) where ui = y for some i E ( 1 , . . . , n). Then zu:. . .u: FZ zxp6 E IdV and using zx2y3 FZ zx2y2 = zy2x2 we get zx2y2 = zx4 E I d V . If we exchange x and y, we obtain zy2x2= xY4 € IdV. Altogether we have xx4 = zx2y2 = xy2x2 = zY4. Dually we get x4z = y4z E I d V . Finally, we have

>

xy4

= FZ FZ FZ

=

N

zz4 ( z x 4 = zy4) z4z4 (2x2y3= zx2y2implies x6 = x 5 ) z4x4 (2x4 = z y 4 ) x4x4 ( x 4 z = y4z) xx4(x6= x 5 ) ~~4 ( z x 4 = xy4).

If only regular identities are satisfied in V , then S L have:

c V and we

Lemma 7.2.43 Let S L c V c V z y c . Is there an element u E W ( X 2 )with vb(u) > 3 such that u = xy E IdV then xyx = x2y2 E IdV. Proof: Since S L c V the term u contains both variables x and y and from u FZ xy it follows that xP FZ x2 for some natural number p 3. Together with x6 FZ x5 E IdV we get x2 FZ x3 E IdV and we have the following four cases:

>

7 hd-Solid Varieties of Semigroups

240 Case Case Case Case

1: u contains a subterm 2: u contains a subterm 3: u = x"' with k l 4: u = ykxl with k l

x y k x with k yx" with k 3. 3.

+ > + >

> 1. > 1.

Case 1: If we use the identity u w xy we get ux w w E I d V where w has xyaxbycxwith a, b, c 1 as subterm. Moreover, we have zyaxbycz= z y a z 2 y 2 ~ b y 2 zbecause 2 y c ~ of xyaxycx = zyaz2xx2yCz and z y 2 x y 2 z = z y 2 z 2 x z 2 y 2 zThus . w = v E IdV where v has zyax2y2xby2x2ycx as subterm. Since u has a subterm x2y2by Corollary 7.2.41 we get that u w x2y2. Consequently, ux w w w u w x 2 y 2 . But from u w xy it follows that ux w xyx and thus xyx w x2y2 E

>

IdV. Case 2: In a similar way one gets yxy w y2x2 E I d V , i.e. xyx w x2y2E I d V .

>

Case 3: Without loss of generality let k 2. Then from u w xy it follows that yu w yxy, i.e. yx"' w yxy. Now we have YXY yx"' w yx2x%2y1 (by x2 w x 3 ) w yx 2 y 2 x k y 2 x2 y1 (by xy2zy2xw xy2x2zx2y2x). Since y ~ 2 y 2 ~ k y 2 ~contains 2y' y2z2 as subterm, by Corollary 7.2.41 we get yx2y2x"2x2y1w y2x2, i.e. yxy w y2x2. This shows that xyx w x2y2E I d V .

>

Case 4: Without loss of generality let k 2. Then from u w xy it follows that xyx w ykxl+l.Since ykxl+l has a subterm y2x2we have ykxl+' w y2x2 by Corollary 7.2.41, i.e. xyx w y2x2 E I d V . From z2y% y2z3and z 2 w z 3 follows z2y2 = y2z2, i.e. z y z w z2y2 E

IdV. Now we are able t o characterize all regular-solid varieties of semigroups:

Theorem 7.2.44 Let V be a variety of semigroups. T h e n V i s regular-solid i f l V i s self-dual and one of the following statements i s true:

7.2 Regular-solid Varieties of Semigroups

g Mod{(xy)z M x(yz),xy4 M y2xy2);

(2) V

C Vzyc a n d

(3) V

C V$yC n Mod{(xy)z M x(yz),xY4M x2y3} a n d

V

241

V

g C;

(5) V E {RB, N B , RegB). Proof: Let V be regular-solid. Then V is self-dual. Further, V is a subvariety of VRS. If Z V R B C V then we have case ( I ) . If Z V in the case that R B V, then by Lemma 7.2.24 we have (5). We consider now the case that R B V. Then V C V$yC by Proposition 7.2.38. If V g Mod{(xy)z M x(yz),xy4 M y2xy2) then we have case (2). In the case V C Mod{(xy)z M x(yz),xy4 M y2xy2) we have that xy4 M y2xy2 is a Reg-hyperidentity in V. In particular, bzlz; [xy4]I 6z1z2[y2~y2] E IdV. This provides xy8 M y3x2y4E IdV. Then we have xY4 M xy8 (by zx2y2M zx 2 y4 )

c

=

---

3 2 4

Y X Y

x2y"by Lemma 7.2.40). This shows that V C Mod{(xy)z M x(yz),xy4 M x2y3), i.e. V V z ~ c n M o d { ( x y ) zM x(yz), xy4 = x2y"). If in addition, V g C then we have case (3). Finally, if V C C then we get V C VRC by Theorem 7.2.7 and we have case (4).

c

Suppose now that V is self-dual and one of the statements (1)-(5) is satisfied. In case (1) V is solid by Proposition 7.2.37

c

In case (2), i.e., if there holds V VzFc and V g Mod{(xy)z = ~ ( y z )xy4 , = y2xy2), we have to show that V is regular-solid. Let u M u E I d V be a non-trivial identity. Then there are u1,. . . , u,, ul, . . . , u, E X (1 m, n E N) such that u = ul . . . u, and u = ul . . . u,. Without loss of generality we can assume that n. We consider the cases 1) (rn = l ) , 2) (m = 2), and 3) m (m 2 3).

<

<

>

Case 1) From m = 1 there follows x M xn E I d V where n 2. From x M xn and x6 M x5 (which follows from zx2y3 M zx2y2E I d V z y )

7 M-Solid Varieties of Semigroups

242

we get x z x 2 . Moreover, x 2 y 3 z y2x3 and x z x 2 gives the commutative law. Thus x y 4 z y 2 x y 2 E IdV, a contradiction. Hence such an identity u z u cannot be satisfied in IdV. Case 2 ) If 2 = m < n,then we have the following two cases: 2.1) u z v is the commutative law or 2.2) u = v gives an identity z2 z zr E I d V for some natural number r > 3. Case 2.1) Then V C C and in particular x y 4 z y 2 x y 2 E IdV, a contradiction. Thus the commutative law is not satisfied in V. Case 2.2) From x 2 z x r and x6 z x 5 we get x 3 z x 2 E IdV. Then we have the cases 2.2.1) S L g V and 2.2.2) S L C V . Case 2.2.1) Now we have zy4 = xy4 E I d V and y 4 z z y4x E I d V by Lemma 7.2.42. Then there holds z2 z z5 ( 2 3 = z 2 ) z y4x4 ( x y 4 z x y 4 ) z y4y ( y 4 x z y42) z y2 ( x 3 z x 2 ) . Now we have the cases 2.2.1.1) Z C V and 2.2.1.2) Z g V.

>

3 implies Case 2.2.1.1) If ul # u2 then ulu2 = v with v b ( v ) together with x 3 z x 2 z y2 the identity x y z xt (this is a well known fact in semigroup theory), so V = Z. We mentioned already that Z is regular-solid. If ul = u2, i.e. u = ulul, then i [ u ]z u: for all o E Reg. Further, 6[ululul]= uf for all o E Reg. This shows that we get the result in both cases by the application of any a E Reg t o u z u and t o ululul z u . In case 3 ) we will examine that ululul z u is a Reg-hyperidentity in V. If this is the case, so i [ u ]z i [ u l u l u l z ] i [ u ]E I d V for all a E Reg. Case 2.2.1.2) Then there is an z z zr E I d V for some natural number r 2. As in case ( I ) we can show that such an identity u z v cannot be contained in V.

>

Case 2.2.2) We have 2.2.2.1) u contains only one variable and 2.2.2.2) u contains two different variables, x and y.

7.2 Regular-solid Varieties of Semigroups

243

Case 2.2.2.1) Because of x% x2 E I d V for all a E Reg we have i [ u ] = x2 and i [ u ] = x2 and thus 6[u] FZ i [ u ] E IdV.

>

Case 2.2.2.2) We have xy = v E I d V where vb(v) 2. Since S L V and xy = yx Sf I d V we have vb(v) 3. Then x occurs two times in v or y occurs two times in v. Without loss of generality we can assume that x occurs two times in u. Now we substitute y by y4 and get xY4= w for some w E W(X2). Since x occurs two times in w one of the terms x2y2or xy4x is a subterm of w. But we have xy4x = xyx 2 y 2 x 2 yx since xyzyx = xyxzxyx E IdV. Thus there is a term t with xy4 = t where t contains the term x2y2 as subterm. By Corollary 7.2.41 we get that t = x2y2 and thus xy4 = x2y2.SO we have xy4 = x2y2 x2y6 (by x2 = x3) = y2 x 2 y4 (by x2y2zFZ y2x2z) FZ y2xy6 (by xy4 FZ x2y2) FZ y2xy2 (by x2yx2= x2yx4). Thus V C Mod{(xy)z FZ x(yz), xy4 = y2xy2}, a contradiction and I d V cannot contain such an identity u FZ u.

>

c

-

Case 3) Let II be the class of all varieties W for which there is an u = v E I d W with the following two properties: a) the variable leftmost(u) occurs only one time in u; b) the variable le ftrnost(u) occurs in v at least one time where this occurrence is different from the leftmost.

n.

Assume that V E Then there is an identity u FZ u E I d V such that le ftmost(u) occurs only one time in u and at least one time in v where this occurrence is different from the leftmost position. In uy = vy we substitute w E X \ {leftrnost(u)) by y4 and le ftmost(u) by x and get xY4 = w' for some w' E W ( X 2 ) (using xy4 M xy6). If x occurs once in w, then we have xy4 FZ w w y 2 xy 2 (using x2yx2FZ x4yx2 FZ x2yx4),a contradiction and this shows that such an identity u = v cannot be satisfied in V. If x occurs at least two times in w, then the term w contains x2y2or xy4x as subterm. But we have xy4x = xyx 2 y 2 x 2 yx since xyzyx FZ xyxzxyx E IdV. Thus there is a term t with xy4 FZ t where t contains x2y2 as subterm. By Lemma 7.2.40 we get t = x2y3 and thus xy4 = x2y3. SO

7 M-Solid Varieties of Semigroups

244 we have

xy4

=

-

x2y3

(by x2 = x 3 ) y2x2y"by x2y2x= y2x22) y2xy4 (by xY4 = y2xy2 (by x2yx2 = x 2 y x 4 ) .

,x2y5

= = =

c

Thus V M o d { ( x y ) x = x ( y x ) ,xy4 = y 2 x y 2 ) ,a contradiction. Altogether this shows that V Sf II. Since the essentially binary terms t with l e f t m o s t ( t ) = x are of the form xayb,xaybxc,xaybxcyd,xaybxCydxe for a , b, C, d , e 1, we consider the hypersubstitutions at where t is a term which has one of these forms. Cearly, 6,,,, [u]= 6x,x,[u]E I d V . For 2 5 a E N we have ezlz;[u] = [ ~. .l. urn] = u l u $ . . . u& and &z; [v] = 6z1z;[v1. . . v,] = vlv,. . . .v;. Since V Sf II there holds: If ul E { v 2 , .. . , v,) then ul E { u 2 , .. . ,urn). Otherwise we have ul E { v 2 , . . ,v,) and ul Sf { u 2 ,. . . , urn) and u = u gives that V E a contradiction. Similarly we have: if ul E { U Z , . . . ,urn), then ul E ( ~ 2 ,. . , un). In u = u we substitute w E ( ~ 2 ,. . ,urn,u2,. . . , u,} by wa and get uTu$. . . u& = v f v ; . . . v; E IdV where r = 1 if ul Sf ( ~ 2 ,. . , urn,v2,. . . , v,), r = a if ul E ( ~ 2 ,. .. ,urn,v2,. . . , v,), s = 1 if vl Sf { u 2 , .. . ,urn,v2,. . . , v,), s = a if ul E { u 2 , .. . ,urn,u2,. . . , u,}. If T = a, then ul E { u 2 , .. . ,urn,212,.. . , u,). We notice that if u 1 E { v 2 , .. . , v,) then ul E { u 2 , .. . ,urn). Thus ul E ( ~ 2 ,. . ,urn) and we get u;u; . . . u& ,u 3l ua2 . .. U& (x3y22= x2y22)

>

ezlz;

n,

-

. . u& ( ~ " ~=2 x2y2x = y2x2x and xx2y3 = xx2y2= xy 2 x 2 ). If s = a so we obtain in a similar way uSug. . . ug = ulug. . . ug. This shows that 6z1z;[u1 = u l u ; . . .u; = u;u; . . . u; = vsv;. . . v; ,ulu;...u; ,u l u ; . 7 2 .

-

,

~x,xz["l.

7.2 Regular-solid Varieties of Semigroups

245

Dually we can show that 6 x y x 2 [ u=] 6 x y x 2 [ uE]IdV. For 2 5 a, b E N

6q.; [v]

7.2

^.

M

6q,; [v1. . . %I v; . . . v i (by ~ " =~z 2 y2 2 z M y 2 z 2 z and zz2y3 M z z 2 y 2M z y 2 z 2 ) .

From u = u E I d V it follows that u: . . . u g = u; . . . u: E substitution (we replace w E X by w 2 ) .Thus bZ7,;[u]= BX7,; [a]E IdV.

>

Now we consider for any natural numbers a, b, c 1 the hypersubstitution axaxpX,. We have 1 2 1 xyzyx = x z y 2 z z 2 y z ( z y z y z = z y x z z y z ) x ~ x ~ ( z~y z y z~ z y~z z z ~ yz) ~ x ~ ~ z 3 y 3 z y 3 ~ " z 3 y 2 z = z 2 y 2 z= y 2 X 2 z and zx2y3 = zx2y2 = z y 2 x 2 ) x2y2zy2x2 (x3yx3= x 2 y x 2 ) .

=-

=

^.

^. 7.2

Using this identity and x"2z = z 2 y 2 z M y2z2z and zz2y% zx2y2 = z y 2 x 2it is easy t o check that for t = xUybxcthere holds 2 Gt[u] = G t [ u l . ..urn] = u 2l . . . U , ~ ~ U & .U . u 2~ l and ~~. 2 6,[u] = 6t[u1. . . u,] = u: . . . u,-lu;u:-l . . . u;. Now we have I ) b = 1 and 11) b

> 2.

Case I ) Now we have the cases 1.1) urn # u, and 1.2) urn = u,.

Case 1.1) Here we have to consider the cases 1.1.1) S L 1.1.2) S L g V.

C

V and

Case 1.1.1) In this case, { u l ,. . . , u,) = { v l ,. . . , v,). From u, # v, there follows urn E { u l , . . . , v,-l} and u, E { u l , .. . ,urn-1). Thus urn E { u l , . . . , urn-1} and u, E { u l ,. . . , u , - ~ ) since V $ Then u: . . . ~ g - ~ u ~ .u. .gu:- ~

n.

x

7 M-Solid Varieties of Semigroups

246

u 4 . . . ~ k - ~ u k -,... u k 4 ( x3 y 2 Z = x 2 y 2 Z M y 2 x 2 Z and urn E { u l , . . . ,urn-,}) U21 . .. U ; - ~ U ; (xx2y3= n 2 y 2= z y 2 x 2 ) 2 2v1 . . . v: (from u = v it follows that uf . . . urn-,u; by the substitution w H w 2 ) uf . . . U ; - ~ V ; U ; - ~ . . . uf (2x2y3 = zx2y2 = zy2x2) ~ f. . v.2~ - ~ v ~ v .;. v1 - 2~ .( X 3 y 2 z = x 2 y 2 z = y 2 x 2 z and v, E { v l ,. . . , v,-1)). This shows that et[u]= e t [ v ]E IdV.

=

-7 2 .

= v12 . . . v,2

-

Case 1.1.2) Here we have z y 4 = xy4 E IdV and y4z = y4x E IdV by Lemma 7.2.42. Then u 21 . . . U 2r n ~ l ~ r n ~. .; .~U l2, u1 2 . . . u 4r n ~ 22 u r n ~ 1 u r n.u. .;u: ~ 1 (x3y22= x2y22)

= -

u 4 . . . ~ h - ~ u. .fu,-,u,u;-,.. .2 . u; ( y 4 x= y42) u; . . . . . . uf ( x y 4= q4) 2V 1 . . . V2 n - l V n ~ ~ - l . 2. . V ( I )1 : 3 y 2 Z = X 2 y 2 2).

This shows that et[u]= e t [ v ]E IdV.

Case 1.2) Since V $ n there holds: If urn E { u l , .. . ,u,-I}, then urn E { . . . , - 1 ) and if v, E { ~ i. .,. , urn-,}, then v, E { v 1v } . In u = v we substitute w E 2 { u l , .. . , u r n - l , u ~ ,. .. ,u,-1) by w2 and get u f . . . u r n - , u ~ uf . . . u:-,ui E I d V where T = 1 if urn $ { u l , . . . , urn-,, u ~. .,. , un-I}, r = 2 if urn E { u ~. .,. , ~ ~ - l , ~. . 1, Un-l), , . s = 1 if v, $ { u l ,. . . , urn-1, v1, . . . , v , - ~ ) , s = 2 if v, E { u l ,. . . , urn-1, v1, . . . , v,-1). If T = 2 then urn E { u l , . . . , urn-1, u l , . . . , un-l}. We notice that if urn E { u l , . . . , u,-I}, then urn E { u l , . . . , urn-1). Thus urn E { u l , .. . , urn-1} and we get 2 u 21 . . . Urn = u . . .u 1 (2x2y3= Z X 2 y 2 ) (x3y2z= x2y2z= y 2 X 2 Z U 2 1 . . . um,-lurn 2 and zx2y% m 2 y 2= z y 2 z 2 ) . If s = 2 , so we get in a similar way v : . . . v 2, - 2v , . . . v 2, - , v w 2 2 2 This shows that uf . . . urn-,urn ul2 . . . urn-,uL u,2 . . . u,-,u~ = 2 u . . u u n Since V is self-dual we have u,u:-, . . .u; = urnug-, . . . u? E IdV. Thus

--

-

--

---

7.2 Regular-solid Varieties of Semigroups 2 2 2 u1 . . . u,plumu~pl . . . u1 2 M 'LL? . . . U,pu,'Un'U~pl.. . 'Uf z u: . . . ~ g - ~ u ~ u .;. u-: ~ . z u:. . .~:-~u,u:-~. . .u:,i.e. Gt[u] z Gt[u] E IdV. Case 11) Using zx2y3 z zx2y2 z zy2x2we get 2 2 U 1 . . . U m - l ~ ~. .~. U? ~= - Ul21. . . U,2 and 2 2 u: . . . un-lu;u:-l . . . u: u1 . . . u,2.

=

But from u z u by the substitution w H w2 it follows that u: . . . ug z u: . . . u;. This shows that Gt[u] z Gt[u]E IdV. Now we consider the hypersubstitution bers a, b, c, d 1. We have

>

oxaybxcyd

for natural num-

xyzyzzyxyz z z y 2 z 2 y z z 2 y 2 x y z( z y z y z z y 2 z y 2 z ) z z 2 y 2 z y 2 z 2 y 2 z( z 3 y 2 zz z 2 y 2 z y 2 z 2 z and zz2y3 = zz2y2 z z y 2 z 2 )

=

=

and zz2y3 = zz2y2 z z y 2 z 2 ) . Then it is easy to check that for t = xaybxcydholds 2 Gt[u] z Gt[ul . . . u,] z uu, . . . u,2 and Gt[u] z Gt[ul...u,]z u: ..A;. From u z u it follows that uf. . . u : z u; . . . u: by the substitution w H w 2 , i.e. Gt[u] z Gt[u] E IdV. Finally, we consider the hypersubstitution o x a y b x c y d x e for any natural numbers a, b, c, d, e I. We have xyxyxzxyxyxzxyxyx z xyxy2x2zxz2yxyz2xzx2y2xyx (xyzyx z xyxzxyx) 2 2 2 2 2 3 2 x y z (xy z z x y zzy2x2z and zz2y3 = zz2y2 z z y 2 z 2 ) . Then it is easy to check that for t = xaybxcydxeholds 2 et[u]z Gt[ul . . . u,] z uu, . . . u,2 and Gt[u] z Gt[ul...u,]z u: ..A;. From u z u it follows that uf. . . u : z u; . . . u: by the substitution w H w 2 , i.e. Gt[u] z Gt[u] E IdV.

>

N

7 M-Solid Varieties of Semigroups

248

( 3 ) In this case we have V C V$Tc n M o d { ( x y ) z M x ( y z ) ,xy4 M x2y3) and V C. Let u M u E IdV be a non-trivial identity. Then there are u l , . . . ,urn,ul, . . . , un E X ( 1 m, n E N ) such that u = u l . . . u , and v = vl . . . v,. Without loss of generality let rn 5 n. Then we have the following cases:

<

a) S L

c) m

c V , m = 2 and ul # u2;

< 2 and ( S L g V or m # 2

or ul = u 2 ) .

a) Without loss of generality let ul = x and u2 = y. Since S L C V and xy M yx $ IdV we have n 3 and { u l , . . . , u,} C { x ,y}. By Lemma 7.2.43 we get xyx M x2y2 E IdV and xyx M y2x2 E IdV since V is self-dual. Further from xy M vl . . . v, it follows that x2 M xn, and n 2 y 2M zx2y3 implies x5 M x6. Thus x2 M x3 E IdV. Moreover, from x2 M x3 and xy4 M x2y3 we get x y 2 M x2y2 and x2y M x2y2 since V is self-dual. Because of x2 M x 3 , xyx M x2y2 M xy2 M x2y we have only t o consider the hypersubstitutions a,,,, and ox:,;. Clearly, 6 ,,,, [u] M 6 ,,,, [v] E I d V . Moreover, we have [ u ]M x 2 y 2 and 6,2,2[~] M x2y2 (by x2y2 M x y 2 M x2y M xyx M 1 2 y2x2).Thus 6,:,z[u] M 6,.,; [u]E I d V .

>

b) Since the essentially binary terms t with le ftmost(t) = x are of xaybxCydxe for a , b, C, d , e 1, we the form xayb,xaybxc,xaybxcyd, consider the hypersubstitutions at where t is a term in one of this forms. We have xY4 M x2y3 E IdV and y4x M y3x2 E I d V . Clearly,

>

6x1x2 [u]= 6 x l x ,[u]E I d V . For 2 a E N we have 6,,,; [u] M 6 x l x ;[ul . . . urn] M u l u $ . . . u& and 6xlx;[~M ] 6 x l x ; [ .~.1. un] M u1'U!.. . u:. From u M v E IdV it follows that uy . . . u& M vf . . . vg E IdV by substitution (we replace w E X by w a ) .Then we have ulu; . . . u; 2a a (x3y2zM x2y2z and zx2y3 M zx2y2) ulu2 u , . . . u; Uaua+l I 2 U a, . . A & ( z y 4M x2y3)

<

-7.2

= u~;"u;u; . . . U; Similar we get v l v i . . . vg gives 6,,,; [u]M u l u ; . . . u; 6x1s; [ul.

= x2y2z= ~ ~ ~ 3 ~ ) . vFvi. . . vg E I d V . Altogether this M uu;". . . u; M uf . . . ug M ulu,".. . ug M

M

7.2 Regular-solid Varieties of Semigroups

249

<

Dually, we can show that Gxyx2[u]= Gxyx,[v]E I d V . For 2 a, b E N the followina identities are valid: G X a x b [u] = G X a x b [ul . . . u,] = U: . . . and 1 2 1 2 G x a x b [a] Gxlxb2[vl. . . vn] = . . . vn2 1 2 (by 2 y 2 z = z2y2z = y2z22 and zz2y3 = zz2y2 = z y 2 z 2 ) . From u = v E IdV it follows that uf . . . u k = v; . . . v: E IdV by substitution (we replace w E X by w 2 ) .Thus ir,ax;[u] = kaXq [v]t

-

IdV.

~2

If t E {xaybxcI 1

< a,b,c E

I

N } U {xaybxcyd 1

<

a , b,c,d E W) U { z a y b z c y d zIe1 5 a, b, C, d , e E N) then both terms [u]and et[v]have subterms of the form wrsrw with w E X and r, s E W ( X ) .Then wrsrw = wrw4sw4rw.Using z y 4 = z2y3 E IdV and y4x = y3x2 E I d V , x3y2z = x2y2z = y2x2z E IdV and zx2y3 = zx2y2= zy2x2E IdV it is easy t o check that 6, [u] = Gt [ul. . . u,] = u: . . . ug and 2 Gt[v] = et[vl. . . v,] = v12 . . . vn. But from u: . . . ug = vf . . . v: E IdV it follows that Gt[u]= Gt[v]E

et

IdV. c) We get that u = v is a Reg-hyperidentity in the same manner as in the cases 2.2.1), l ) , and 2.2.2.1), respectively.

( 4 ) We have V

C VRC.Then V

is regular-solid by Theorem 7.2.7.

( 5 ) V is solid by Lemma 7.2.24. Theorem 7.2.44 shows that VRS is regular-solid. As we have already mentioned, VRs is the greatest element in Lgeg,since each regularsolid variety of semigroups is a subvariety of VRS. This gives us a characterization of the Reg-hyperequational class defined by the associative law. Since each regular-solid variety of semigroups must satisfy the associative law as a Reg-hyperidentity, the regular-solid variety HRegMod{(xy)z= x ( y z ) } of semigroups is the greatest element in Lgeg. So we have

Proposition 7.2.45 HRegMod{(xy)z= x ( y z ) } = VRS. The variety T ( 2 ) is the least element in L Z g . The atoms in the lattice Lsg of all semigroup varieties are SL, Z , L Z , R Z , and the p-groups defined by the commutative law and xPy = y for prime

250

7 M-Solid Varieties of Semigroups

numbers p. We are interested in the atoms of the complete sublattice -c"Rgeg.

Theorem 7.2.46 The varieties R B , SL, and Z are the atoms in Lgeg . Proof: R B , S L , and Z belong to Lgeg.Since S L and Z are atoms in Lsg they are also atoms in the sublattice Lgeg.It is well known that the varieties R Z and L Z are the only non-trivial subvarieties of RB. Since neither R Z nor L Z belong to Lgeg (this is easy to check by Theorem 7.2.44), R B is an atom in Lgeg,i.e. R B is the least element in the interval between R B and VRS. Let now V be a non-trival subvariety of Vzyc which does not contain the variety Z of all zero-semigroups. Then there is an identity u M u with u E X and u $ X. By substitution we get x M xz"E I d V for k = ub(u) 2. From x2y3 = y2x3 E I d V we can derive x7 M x8 E IdV. Both iden2 provide the idempotent law tities x = xk and x7 M x8 with k x M x2, SO V is a variety of bands and x2y3 M y2x3 E I d V means that the commutative law holds. Thus V = SL. Further, S L and Z are the only atoms of Lsg which belong t o the subvariety lattice of VRC. Altogether this shows that R B , S L , and Z are precisely the atoms in Lgeg.The atoms S L and Z of the lattice LSg are also atoms in its complete sublattice Lgeg.The other atoms of LSg does not belong to L"Rgeg.

>

>

We now look at varieties of bands. Each variety of bands is defined by the associative law, the idempotent law (x = x2), plus one additional identity. There are a countably infinite number of varieties of bands. The collection of all varieties of bands forms a complete lattice, the subvariety lattice of the variety B = Mod{(xy)z M x(yz), x = x2) of a11 bands. The structure of the lattice they form has been completely determined (independently by Birjukov ([5]), Fennemore ([44]),and Gerhard ([46])).We will be particularly interested in several of these varieties. The trivial variety is the least variety of bands, SL, L Z , and R Z are the atoms in this lattice. The join of L Z and R Z is the variety R B of all rectangular bands. R B is dual-solid. The variety N B of all normal bands and the variety RegB of all regular bands are dual-solid, too. N B and R B are the only non-trivial dual-solid subvarieties of RegB.

7.3 Solid Varieties of Semigroups

251

Using Theorem 7.2.44 we will determine all regular-solid varieties of bands. Let V be a variety of bands. If V C VRC, then V must be commutative. Thus V C SL. Since S L is an atom, V is equal to S L or t o T(2). If V VzFc, then x2y3 M y2x3 E IdV. Using the idempotent law, from this identity we can derive the commutative law, i.e. V S L and thus V = S L or V = T(2). Altogether, T(2), SL, R B , N B , and RegB are precisely the regular-solid varieties of bands by Theorem 7.2.44.

c

c

In the next sections we characterize complete sublattices of C?,, namely the lattice of all solid varieties of semigroups as well as the lattice of all pre-solid varieties of semigroups using Theorem 7.2.44.

7.3

Solid Varieties of Semigroups

We are now ready to consider solid varieties of semigroups. It is clear that the lattice of all solid varieties of semigroups forms a complete sublattice of Lsg and we will investigate some of the properties of this complete lattice. We note that a variety of semigroups to be solid must satisfy the associative law as a hyperidentity. Such varieties of semigroups were called hyperassociative by Wismath in [104]. The greatest solid variety of semigroups is the hyperequational class HMod{(xy)z FZ x(yz)} of the associative law. We begin with some necessary conditions for a semigroup variety, t o be hyperassociative. Proposition 7.3.1 Let V be a hyperassociative variety of semigroups. T h e n V VRS.

c

Proof: Since Reg C Hyp, the associative law must be a Reghyperidentity in V. Since VRS is the Reg-hyperequational class defined by the associative law, V is a subvariety of VRS.

Proposition 7.3.2 A n y hyperassociative variety V of semigroups

satisfies x2 M x4. Proof: The application of the hypersubstitution a,: t o the associative law gives an identity in V, so x2 FZ x4 E I d V since

7 M-Solid Varieties of Semigroups

252

4

6x:[x(yz)] = x2 and 6x:[(xy)z] = x . These both propositions give two necessary conditions for a semigroup variety t o be hyperassociative. They are also sufficient. Let VHS be the collection of all semigroups from VRS satisfying x2 M x4, i.e. VHS = Mod{(xy)z M x(yz), x2 M x4, xyxzxyx M xyzyx, (x2y)2z M x2y2z, xy2z2 M ~ ( ~ 2 ~Our ) ~goal ) . is t o show that a variety V of semigroups is hyperassociative iff V C VHS making VHS the largest hyperassociative semigroup variety.

Proposition 7.3.3 A variety V of semigroups is hyperassociative ijf V c VHs. Proof: By Propositions 7.3.1 and 7.3.2 a hyperassociative variety is a subvariety of VHS Let now V C VHS Because of V C VHS C VRS = HRegMod{(xy)z M ~ ( y z ) the ) application of any a E Reg t o the associative law gives an identity in V. For a E Hyp \ Reg 1 such that a = a,? or a = a,$. there is a natural number k The application of oz, (ox:) t o the associative law x(yz) M (xy); gives x b xk2 (zk M 2")). Note that from x2 M x4 it follows that 2% xk2.Thus 6[x(yz)] M 6[(xy)z] E IdV. Altogether V is hyperassociative.

>

Proposition 7.3.3 shows that VHS = HMod{(xy)z M ~ ( y z ) )VHS ; is the hyperequational class defined by the associative law. By Theorem 2.5.5, VHS is solid. The same identity basis for the variety HMod{ (xy)z M x(yz) ) was independently determined by L. PolRk ([MI) and by G. Paseman ([72]). Not every subvariety of VHS is solid. For instance, the variety Z of all zero-semigroups is a subvariety of VHS, but not solid as we have already shown (the application of a projection hypersubstitution t o xx M yy E I d Z gives x M y $ IdZ). Conversely, as we also observed earlier, any non-trivial solid variety V of semigroups must be hyperassociative, i.e. V VHS, and has t o satisfy only outermost identities, i.e. RB C V. Moreover, each solid variety of semigroups is dual-solid since Lgeg L g . This proves the following result:

c

Proposition 7.3.4 For any non-trivial solid variety V of semigroups, RB C V C VHS and V is dual-solid.

7.3 Solid Varieties of Semigroups

253

Note that R B is also the least non-trivial solid variety of semigroups. Now let L(VHs) be the subvariety lattice of the variety VHS As intersection of the complete lattice L(VHs) and the complete lattice of all solid varieties of type 7 = (2), the collection Cgvp of all solid varieties of semigroups forms a complete sublattice of the lattice Csg of all semigroup varieties. This lattice has VHS as its greatest element, the variety T(2) of all trivial algebras of type (2) as its least element, and R B as its only atom. Now we characterize all solid varieties of semigroups.

Theorem 7.3.5 A non-trivial variety V of semigroups is solid iff V E {RB, N B , RegB) or Z V R B V VHS and V is dual-solid.

c c c V c VHS and

Proof: If V is solid, then R B V is dual-solid V, then V E {RB, N B , RegB) by by Proposition 7.3.4. If Z Lemma 7.2.24. Conversely, let Z V R B V VHS and V be dual-solid or V E {RB, N B , RegB). Because of VHS C VRS, V is regular-solid by Theorem 7.2.44. Thus the application of a regular hypersubstitution t o any identity of V gives again an identity in V. Let o t Hyp \ Reg. Then o = ox: or o = ox; for some natural number R 1. The application of ox: (0,:) t o any identity

c

c

>

>

s = t t I d V gives wFa = w!b for some natural numbers a,b 1 where wl = le ftmost(s) (= rightmost(s)) and w2 = le ftmost(t) (= rightmost(t)). Note that from x2 M x4 it follows that xk = x k m for 1 5 rn t N. Since R B V, s = t is outermost, so wl = w2. Consequently, wta= w, t IdV, i.e. B[s] = B[t] t IdV. Altogether this shows that the application of any a E Hyp t o any identity in V gives an identity in V.

"

c

In [89], L. Poliik gave another proof of this important characterization of all solid varieties of semigroups. His proof does not use the characterization of the lattice LZe, of all regular-solid varieties of semigroups. In section 7.2 we mentioned that SL, R B , N B , and RegB are precisely the non-trivial regular-solid varieties of bands. Since each solid variety is regular-solid, the solid varieties of bands are among the regular-solid ones. Since R B S L (SL is an atom in Lsg), S L is not solid by Theorem 7.3.5. Thus R B , N B , and RegB are precisely the solid varieties of bands.

7 M-Solid Varieties of Semigroups

254

In chapter 5 it was shown that the lattice of all solid varieties of type 7 contains infinite chains. This is also true for the lattice of all solid varieties of semigroups as the following example shows. Let us consider for 3 k E N the varieties Wk := M o d ( ( x y ) z = ~ ( y z )x4 , = X 2 , X Y Z Y X = X Y X Z X Y X , 2 x 1 . . . X k - 2 Y M Xy1.. . Yk-2?J} and Uk := M o d { ( x y ) z = x ( y z ) , x4 = x 2 , x y 2 z x = x"2zx, xyzt = xzyt, xxl . . . xkP2y = xyl . . . ykP2y}. By definition, W1 c W2 c ..Wic and Ul c U2 c - - - U ic Clearly, the varieties Wk and Uk ( 3 k E N) are dual-solid and contain RB.We can derive x2y2z = from the identities in Wk and U k ,respectively, as follows:

<

0

.

.

0

-

0

<

.

x2y2z = x2YY2k-2Yz( x 4 = x 2 ) -- x2yx2k-2yz( q . . . xk-2y -- x ~ .1. . ~ k - a ~ ) = x2yx2yz ( x 4 = x 2 ) . Dually one can show that x y 2 z 2= ~ ( ~holds 2 in ~ both ) ~varieties, in Wkand in Uk.This shows that the varieties Wk ( U k )( 3 k E N ) are dual-solid and in the interval between Z V RB and VHS, SO by Theorem 7.3.5 these varieties are solid. N

N

<

7.4

Pre-solid Varieties of Semigroups

In the previous section we showed that the variety VHS

=

M o d { ( x y ) z = x ( y z ) , x2 = x 4 , x y x z x y x = x y z y x , = x2y2z, x y 2 z 2= x ( y ~ ~is )the ~ }greatest solid variety of semigroups. As a solid variety VHSis certainly pre-solid. Since the application of the projection hypersubstitutions a,, and a,, to the associative law provides a trivial identity, the hyperequational class generated by the associative law is also the Pre-hyperequational class generated by the associative law. Thus Hp,,Mod{(xy)z = x ( y z ) ) = VHS and VHSis the greatest pre-solid variety of semigroups. Since D C Pre C Hyp, each solid variety has t o be pre-solid and each pre-solid variety is dual-solid. This shows that any pre-solid variety of semigroups in the interval between Z V RB and VHS is solid. Therefore a pre-solid variety of semigroups which is not solid must be outside of the interval between Z V RB and VHS. For the greatest pre-solid variety of semigroups which is not solid

7.4 Pre-solid Varieties of Semigroups

255

let us consider the variety VpS = M o d { ( x y ) z M x ( y z ) , xyxzxyx M xyzyx, x2 M y2,x3 M y3). It is a subvariety of V z y n M o d { ( x y ) z M

~ ( y z xy4 ) , M x2y3). Lemma 7.4.1 VpS

C Vzyc n Mod{(xy)x M x ( y z ) , x Y 4M x2y3).

Proof: Clearly, we can derive x2 M x3 from x2 M y2 and x3 M y3. Thus x2y3 M x2y2 M y2x2 M y2x3, x3y2 M x2y2 M y2x2 M y3x2,and xY4 M x ( ~ M ~ )xx2 ~ M x2x3 M X 2 Y 3 using x2 M y2 and x3 M y3. Moreover, x2y2z M x2x2z M M ( x 2 y ) 2 zand xy2x2 = xy2y2 = = ~ ( ~ 2 ~ ) ~ . Altogether this shows that the identities of Vzyc as well as xy4 M x2y3 hold in VpS. Lemma 7.4.2 A regular-solid variety satisfying x2 M y2 and x3 M y3 is pre-solid. Proof: Let V be a regular-solid variety satisfying x2 M y2 and x3 M y3. Then x2 M x3 holds in V . We have to show that the application of any Pre-hypersubstitution a t o an identity in V gives again an identity in V. If o E Reg, all is clear. If o E P r e \ Reg then there is a natural number k > 2 such that o = o,! or o = 0,;.The b application of o t o any identity s M t E IdV gives wFa M w! for some natural numbers a, b 1 and some w l , w2 E X . Because of x2 M y2 and x3 M x2 we have wFa M wf M w; M w2k , so wta M w!jb E IdV and thus the application of a t o any identity in V gives again an identity in V. Consequently, V is pre-solid. Using these both lemmas we have:

>

Proposition 7.4.3 The variety VPS is pre-solid but not solid. Proof: By Lemma 7.4.1, VpS C V z y c nM o d { ( x y ) z M x ( y z ) ,xY4 M x2y3}. It is easy to see that VpS C. Hence VpS is regular-solid by Theorem 7.2.44. Then VpS is pre-solid by Lemma 7.4.2. Since x2 M y2 is not outermost, VpS is neither in the interval between Z V R B and VHSnor it belongs t o {RB, N B , RegB), so VpS is not solid.

256

7 M-Solid Varieties of Semigroups

Any non-trivial subvariety of VpS is not solid. Thus any non-trivial pre-solid variety V of semigroups is not solid if V C VPS The converse is also true, i.e.

Proposition 7.4.4 For any non-trivial pre-solid variety V of semigroups, V C VpS i&f V is not solid.

c

Proof: If V VpS then V is not solid. Suppose now that V is not solid. Then V is not in the interval between Z V RB and VHS. But since V C VHS, we have Z V RB V. This means that Z 9 V and R B C V or R B 9 V. If Z 9 V and R B C V, then V E {RB, N B , RegB} by Lemma 7.2.24 and V is solid by Theorem 7.3.5. Thus this case is impossible. Hence, R B 9 V, i.e. there are natural numbers m, n and variables uo, . . . , urn,vo, . . . , v, E X such that uo # vo or urn # v, and u o . . .urn = vo . . . v, E IdV. Without restriction of the generality assume that uo # vo. The application of the pre-hypersubstitutions a,: and a,: t o uo . . . u, = vo . . . v, gives ui M vi and ui = vi, respectively, so x2 M y2 E I d V VHS, the associative law and and 2% y3 E IdV. Since V xyxxxyx = xyxyx hold in V, too. Altogether, V C VpS.

c

This proposition shows that VPS is the greatest pre-solid variety of semigroups which is not solid. For the least pre-solid variety of semigroups which is not solid, let us consider the variety Z of all zero-semigroups. We mentioned already that an identity s M t is called normal, if both s and t are the same variable or neither s nor t are variables and that the normal identities are precisely the identities in the variety Z of zero-semigroups.

Proposition 7.4.5 A n y pre-solid variety V of semigroups which is not solid, does contain Z . Proof: Assume that Z V. Then there is an identity s M t in V with s E X and t $ X. Substituting w E X by x we get x = xk for k := vb(t) 2. Since V is pre-solid but not solid, V C VPS, SO x2 = y2 and x3 = y3 are identities in V by Proposition 7.4.4. These identities provide x2 = x h n d x2 = x together with x = xk. But x2 M y2 and x2 M x gives y = x, so V = T ( 2 ) and V is solid, a contradiction.

>

7.4 Pre-solid Varieties of Semigroups

257

We know now that any V E Li!, \ Lgypis in the interval between Z and V P S The variety LN := Mod{(xy)z M x ( y z ) , x2 M x , xyz M x z y ) of all left-normal bands belongs t o this interval but is not presolid. The application of o,,,,E Pre t o xyx M xxy gives xyx = yxx which does not hold in LN. In particular, LN is not dual-solid. We have the following characterization of Life \ Lgyp:

Proposition 7.4.6 A variety V of semigroups belongs to Li!, Lgyp ifl V is dual-solid and in the interual between Z and V p S .

\

Proof: Let V E Ci!,\Cgyp. Then propositions 7.4.4 and 7.4.5 show that Z V Vps. V is dual-solid because of Ci!, Cg. Conversely, if V is in the interval between Z and VpSand dual-solid then V C V$yc n Mod{(xy)z M x ( y z ) ,xy4 M x2y3)by Lemma 7.4.1. If V C then V is regular-solid by Theorem 7.2.44. If V C C then xy = yx E IdV and one can derive x2y M xy2 from x2 = y2 and x3 = y3 ( x 2 y= y3 = x3 M x y 2 ) ,SO V C VRCand V is regular-solid by Theorem 7.2.7. Altogether, V is regular-solid and it satisfies both identities x2 M y2 and x3 M y3 ( V C VRS).Hence V is pre-solid by Lemma 7.4.2. Because of V C V P S ,V is not solid.

c

c

c

The collection Life \ Lgyp of all pre-solid but not solid varieties of semigroups forms a sublattice of the lattice Ci!, of all pre-solid varieties of semigroups since both, join and meet of two dual-solid varieties in the interval between Z and VpSare again dual-solid and between Z and VpS. Any solid variety V of semigroups must contain the variety RB. Therefore the join of a solid variety V and the pre-solid variety Z can be written as V V Z = ( V V R B ) V Z = V V ( R B V Z ) where RB V Z = Mod{(xy)x M x ( y x ) , xyx = x x ) is well known. In particular, RB V Z is solid by Theorem 7.2.44. This example motivates the following question: Is the join of a solid variety and an arbitrary pre-solid variety solid? This is the case, since the join of a solid variety and an arbitrary pre-solid variety is a dual-solid variety in the interval between Z V R B and VHs.Thus the sublattice C g y p of Li!, is a filter in C&. On the other hand, the meet of a pre-solid but not solid semigroup variety and an arbitrary pre-solid variety is a subvariety of VpS again and thus not solid. This shows that the lattice L;!, \ L$yp is an ideal in L$,. We remark that the varieties

258

7 M-Solid Varieties of Semigroups

R B and VpS have certain splitting properties for the lattice L&. Indeed, if a non-trivial variety V is pre-solid and contains R B , then V is solid. If the variety R B is not contained in V then V belongs to C$, \ Cgyp. If V is non-trivial, pre-solid and contained in Vps If V is not contained in Vps then V then V belongs t o Li!, \ Cgyp. is solid. Together with Proposition 7.4.6 this proves the following characterization theorem:

Theorem 7.4.7 A variety V of semigroups is pre-solid ijf one of the following statements is satisfied: (i) V is solid. (ii) V is dual-solid and in the interval between

Z and VpS.

Corollary 7.4.8 A non-trivial variety V of semigroups with Z C V is pre-solid ifl V is dual-solid and is contained either in the interval between Z V R B and VHS or in the interval between Z and VPS . Any solid variety of bands is also pre-solid, so T(2),R B , N B , and RegB are pre-solid. The variety S L of all semilattices is not pre-solid by Theorem 7.4.7. A pre-solid variety of bands must be regular-solid. Since the solid ones plus the variety S L are precisely the regular-solid varieties of bands, T(2), R B , N B , and RegB are precisely the pre-solid varieties of bands. Since the commutative law is not outermost, there is no solid variety of commutative semigroups. But Z is a pre-solid variety of commutative semigroups. Our goal now is t o describe the greatest pre-solid variety of commutative semigroups which by Section 3.3 means t o determine the Pre-hyperequational class generated by the associative law and the commutative law. Let us consider the variety VPC := Mod{(xy)z = x(yz), xy = yx, x2 = y2, x2y = xy2}.

Proposition 7.4.9 VpC is the greatest pre-solid variety of commutative semigroups. Proof: It is easy to check that the identities x3 = y3 and xyxzxyx = xyzyx can be derived from the identities of VpC, SO we have VpC C VPS. Moreover, VpC is dual-solid and contains the

7.5 Locally Finite and Finitely Based M-solid Varieties

259

variety 2.The variety VpC is pre-solid by Theorem 7.4.7. Let V be a pre-solid variety of commutative semigroups. Then the application of the Pre-hypersubstitution ox:t o the commutative law gives an identity in V, i . e x2 = y2 E IdV. The application of ox:,,E P r e t o the commutative law, i.e. x2y = y2x, is also an identity in V. This shows that V C VpC. Altogether, VpC is the greatest pre-solid variety of commutative semigroups. Corollary 7.4.10 VPC = HPreMod{(xy)z = x(yz), xy

FZ

yx).

Each pre-solid variety of commutative semigroups must be a subvariety of VpC. It is easy t o check that VpC C VRC,SO all subvarieties of VPC belong to C(VRc) and are regular-solid by Theorem 7.2.7. This motivates the following question: Are all subvarieties of VPC pre-solid? This is satisfied by Lemma 7.4.2 since in VpC C VpS there holds x2 FZ y2 and x3 = y3. Therefore we have the following proposition: P r o p o s i t i o n 7.4.11 C(Vpc) is the collection of all pre-solid uarieties of commutative semigroups. In [26] the complete lattice L(Vpc) is described. For its characterization we fix the following notation: Pn := Mod{(xy)z = x(yz), xy = yx, x2 = y2, x2y = xy2, x0 . . . xn = yo . . . yn) for n E N.Clearly, here Pois equal t o T (2) and PI is equal t o 2.There holds L(Vpc) = {PnI n E N} U {VpC) (see [26]). This shows that the lattice of all pre-solid varieties of commutative semigroups forms an infinite chain with T(2) as least element, the uniquely determined atom Z and the greatest element VpC. There are no dual atoms. The lattice C(Vpc) is a complete sublattice of C!,; \ Czyp.So both parts Czypand C& \ Czypof C!,; contain infinite chains.

7.5

Locally Finite and Finitely Based M-solid Varieties

A variety V is finitely based if it admits a finite basis of identities, i.e. if there is a finite set C of identities such that V

=

ModC.

260

7 M-Solid Varieties of Semigroups

In the previous sections we proved that the semigroup varieties HnnMod{x(yz) = ( x y ) ~ }for M E {Reg, P r e , Hyp) are finitely based. We may ask for a condition on the monoid M which forces HMMod{z(yx) = (xy)x) to be finitely based by identities. Surprisingly, there are varieties of semigroups which are generated by a finite semigroup but are not finitely based. Perkins proved that the variety generated by the five-element Brandt-semigroup

is not finitely based ([77],[78]).It is interesting t o remark that there are varieties V having a finite hyperidentity basis C but no finite basis for its identities. As an example we consider the type T = ( 2 , l ) with a binary operation symbol t o be indicated by juxtaposition and a unary operation symbol f . Let C be the following finite set of equations:

We consider the monoid M of hypersubstitutions of the form a(k), for any natural number k, where maps the binary operation symbol to itself and the unary symbol f to x:, for k 1, together with the identity hypersubstitution aid. Then the set XM [C] = {(z122)x3 FZ z l ( ~ 2 ~ 3X)l X, 2 X 3 X 4 XlX3X2X4, x 2 ~ : ~ 2 x ~ x ~ Ux {~x )~ x ~ x := x ~X ~ X ~ X ~I kX ~ I) X ~ does not have a finite basis, as shown by Perkins in [78]. Another similar example was given by Paseman in [72] using the type T = (1,1,1).

>

>

We will show that the problem of which varieties of the form HMMod{(zy)x = x(yx)) are finitely based by identities is closely connected with the following one: For which monoids M of hypersubstitutions of type T HMMod{(zy)x = x(yx)) locally finite ?

=

(2) is

If we want t o check whether an identity s = t is satisfied as an M-hyperidentity in a variety V we can restrict our checking to one representative from each equivalence class of the quotient set

7.5 Locally Finite and Finitely Based M-solid Varieties

261

M/-vlM. A complete set of representatives can be selected by a choice function 0 : + M . From the class of the identity hypersubstitution the choice function 0 has t o select aid. We will use the notation M g ( V ) and call the elements of this set normal form hypersubstitutions. (In section 3.1 this set was denoted by N$'(V).) In [2] was proved that ModxMIC] = M o ~ x ~ N ( ~ ) [ C ] for every variety V, for every set C of equations and every choice function 0 . Now we return t o varieties of semigroups and prove (see e.g. [33], [20]):

Theorem 7.5.1 If V is an M-solid locally finite variety of semigroups which is definable by a finite set C of M-hyperidentities, then V is finitely based by identities. Proof: As an M-solid variety, V is the class of all semigroups which satisfy all equations from C as M-hyperidentities, i.e. V = HhfModC. Using the equations HhfModC = ModxMIC] = ModxMg(,)[C], we see that x ~ ( ~ ) [isCa ]basis for the set of all identities satisfied in V. Clearly, the cardinality of M is equal to the cardinality of all images of f under hypersubstitutions from M , i.e. 1M1 = I{a(f)lo E M)1. From this and from 01 -V a 2 iff a l ( f ) % a 2 ( f ) t I d V we obtain M ~ ( v ) I = M w V l A I= l I { [ ~ ( f ) ] i n vt~ M)1 IFv({x, y}) 1. Since V is locally finite, Fv({x, y)) is finite and then M g ( V ) is also finite. Therefore XMN [C] is finite and V is finitely based by identities. rn

<

M. Sapir defined in [92] identities built up by terms of a specific form. These identities have the form Z, = w where the left hand side is called a Zimin word and is defined inductively by

z1:= XI

Zn+l ..= Znx,+lZ, for n E w+. Then the sequence of Zimin words can be produced by the hypersubstitution a,,,,,, which maps f to the word zlx2xl(or by a,,,,,, mapping f to the word x2x1x2).

Definition 7.5.2 Let f be a binary operation symbol. Then we define

f l ( ~ 12,2 )

:= 21

f n ( ~ 1 , x 2.,..,x,) := f ( f n p l ( x l , . . . ,x,-~),x,), 2 5 n E N+.

7 M-Solid Varieties of Semigroups

262

Proposition 7.5.3 For every n xn)]= Zn. Proof: For n

=

> 1 we have i,,,,,,[ f,(xl, x2,. . . ,

1 we have 6,,,,,,[xl]

= xl =

Z l . Assume that

t o the terms apply the hypersubstitution a,,,,,, f ( " ) ( x l2, 2 , . . . , x,) inductively defined by f ( l )( X I , X 2 ) := X I , f ( X I , . . . , 2,) := f (X l , f ( n - l ) ( ~ 2 . ,. . , & ) ) , 2 5 (If we

n E N+, then we obtain a sequence of words from which we get the Zimin words by exchanging of variables.) Then we have

Proposition 7.5.4 If o,,,,,, (),o,, is a proper hypersubstitution of a variety V of semigroups, then for every n 3 there is a term u, such that Z, M u, is an identity in V .

>

>

Proof: Clearly, for every n 3 the equations f ,(zl, . . . , x,) M f ( , ) ( x l , .. . , 2,) are identities in V since these equations are consequences of the associative law. Then using a,,,,,, we obtain all Zimin words on the left hand side and if a,,,,,, is a proper hypersubstitution of V , we obtain the Zimin words on the right hand side after exchanging of variables. We recall the following definitions:

Definition 7.5.5 A semigroup S is said t o be periodic if for every a E S there exist two different natural numbers ma and na such that ama = ana. A variety V of semigroups is called periodic if every member of V is periodic. A zero 0 in a semigroup S is an element from S with x0 = Ox = 0 for every x E S. A semigroup S with zero is called a nil - semigroup if for any a E S there is a natural number n with an = 0. For periodic semigroups we have

7.5 Locally Finite and Finitely Based M-solid Varieties

263

Proposition 7.5.6 ( [ 6 0 ] )A finitely based periodic semigroup uariety is locally finite i f l all its groups and all its nil-semigroups are locally finite. Olshanski ( [ 7 0 ] )gave sufficient conditions for groups to be locally finite.

Proposition 7.5.7 Every group satisfying the identity x x 3 (or x M x 2 ,x M x 4 , x M x 5 , x M x 7 ) is locally finite.

M

In [60]the following result was proved:

Theorem 7.5.8 Let V be a variety of semigroups given by a (possibly infinite) set C of identities. Assume, the number of variables occurring i n words of E is n. T h e n the following conditions are equivalent: (i) All nil-semigroups from V are locally finite. (ii) V satisfies a non-trivial identity with one side equal ZnP1.

(iii) All semigroups from V with a n identity x 2 finite.

M

0 are locally

-

( i v ) There exists a n identity s M t t E and a substitution : Xn F V ( X n ) (where F V ( X n ) is the free semigroup with contains a value respect to V generated by X n ) such that 6 ( s )(respectively 6 ( t ) )where 6 ( s ) # 6 ( t ) . Now we prove:

Theorem 7.5.9 Let V be a variety of semigroups and assume that ox1x2x1(orox,x1x,)is a proper hypersubstitution of V . T h e n V is locally finite. Proof: Since a,,,,,, (or a,,,,,,) is a proper hypersubstitution of V , for every n > 3 the identities 6 x l x , x l [ f n ( ~ l. ,. ., z ,)1 M oxlx2xl [ f ( n ) ( x l ,... , x n ) ] are satisfied in V . For n = 3 this gives ~ 1 x 2 ~ 1 ~ 3M ~ 1~ ~1 2x ~2 ~1 3 (~f o2r ~axaxlxa 1 we obtain t h e identity ~ 3 x 2 ~ 3 ~ 1 ~M 3~ ~3 2x ~2 3~ 1 ~and 2 ~b 3y exchanging x l and x3 we obtain t h e first identity). Therefore V is a subvariety o f t h e variety

264

7 M-Solid Varieties of Semigroups

We prove that the last variety is locally finite. The variety Vl is finitely based and periodic since by identification of variables from the second identity of the basis we obtain x: = 2:. We can apply Proposition 7.5.6 and show that all groups and all nil-semigroups in VI are locally finite. If G is a group in Vl, then xf has an inverse and by multiplication of x: = xy with this inverse we obtain x: = XI. Then by Proposition 7.5.7, G is locally finite. Now we apply Theorem 7.5.8. In the basis of VI we have words containing the three variables x l , x2,x3 From x1x2xIx3x1x2x1= x1x2x3x2x1we obtain the iden= x1x2x3x2x1x~x1x2x3x2x1 tity x1x2xlx3x1x2x1x~x1x2x1x3x1x2x1 which is also satisfied in Vl. The left hand side is the Zimin word Zq and therefore every nil-semigroup from Vl is locally finite. By Proposition 7.5.6 the variety Vl is locally finite and as a subvariety, the variety V is also locally finite. Now Theorem 7.5.8 and Theorem 7.5.9 give the following result:

Corollary 7.5.10 Let V be an M-solid variety of semigroups and assume that o,,,,,, (ox,,,,,) is a proper hypersubstitution of V. If V is definable by a finite set C of M-hyperidentities, then V is finitely based by identities. Now we prove that varieties of the form V = HhfMod{x(yx) (xy)z} satisfy also the converse of Theorem 7.5.9, i.e.

=

Theorem 7.5.11 Suppose that V is a variety of semigroups for which there is a monoid M of hypersubstitutions such that V = HhfMod{x(yx) = ( x y ) ~ ) . Then V is locally finite ifl ozlzazl (ozazlz,) is a proper hypersubstitution of V. Proof: Because of Theorem 7.5.9 we have t o show that locally finite varieties of the form HhfMod{F(xl, F ( x 2 ,2 3 ) ) = F ( F ( x l ,x2),2 3 ) ) admit a,,,,,, or as proper hypersubstitutions. All nil-semigroups from V are locally finite and we may assume that the condition Theorem 7.5.8 (iv) is satisfied. Since HnnMod{x(yx) = (xy)x) = Modxnf[{x(yx) = ( x y ) ~ ) ] the , set C = xlcf[{x(yz) = ( x y ) ~ } ]is an identity basis of V. The set xM[{x(yz) = (xy)z)] contains only three variables, therefore we have t o find an identity s = t t xM[{x(yz) = ( x y ) ~ ) ]such

7.5 Locally Finite and Finitely Based M-solid Varieties

-

265

that there is a substitution @ : X3 FSEICI(X3)into the free semigroup and @(s)or @(t)(@(s) # @(t)) occur in the Zimin word Z4 = x ~ x ~ x ~ x ~ x ~ x ~ x ~ x ~ The x ~ xidentity ~ x ~ xs ~= xt ~ x ~ x ~ arises from the associative law by hypersubstitution. Applying a,,, a,,, a,,,,, a,,,, on both sides of the associative law gives equal (X3). If we apply a hypersubstitution a which maps words of FsEnr the operation symbol F t o a word which contains a power of a word with an exponent > 1 built up by two variables, then the image of the associative law contains also a power with an exponent > 1. Since the extension 5 : FsEnr (X3) i FsEhr (X3) of the substitution : X3 i FSEM(X3) is an endomorphism, the image of a word containing a power of the image contains a power of the image: $(uwZu) = $ ( u ) $ ( w ) ' ~ ( u ) .But Z4 contains no power of a word. Therefore only the hypersubstitutions a,,,,,, and a,,,,,, can produce the identity s = t. It follows that a,,,,,, E M or a,,,,,, E M . or a,,,,,, is a proper hypersubstitution with reBut then a,,,,,, spect to V. As a consequence of Theorem 7.5.1 and Theorem 7.5.11 we have

Corollary 7.5.12 A variety of semigroups of the form V HhfMod{x(yx) = (xy)x) is finitely based if a,,,,,, or a,,,,,, a proper hypersubstitution of V.

=

is

8 M-solid Varieties of Semirings Semirings are algebras of type r = (2,2) with two binary associative operations, usually written as and respectively, where two distributive laws are satisfied. Single semirings as well as classes of semirings form important structures in Automata Theory and in the theory of Formal Languages ([62], [52], [57]). First, we want to determine all solid varieties of semirings and will give some necessary conditions for varieties of semirings to be solid.

+

8.1

0

,

Necessary Conditions for Solid Varieties of Semirings

In this section we will prove that every solid variety of semirings has t o be medial, idempotent and distributive.

Definition 8.1.1

+, .) is called medial if

(i) A semiring S = (S;

(ii) A semiring S = (S;+, .) is said t o be idempotent if

(iii) A semiring S = (S;+,

0

)

is called distributive if

(iv) A variety V of semirings is called medial if all algebras in V are medial. In a similar way one can define the varieties of distributive semirings and of idempotent semirings, respectively.

268

8 M-solid Varieties of Semirings

For abbreviation we call idempotent and distributive semirings I D semirings (see e.g. [76]) and the variety of all medial ID-semirings will be denoted by VMID. From now on, the associativity will be used in our calculations but for simplicity reference will not be made t o their use on each occasion. For every term s, let shdbe the term arising from s by exchanging the binary operation symbols F and G. As usual, sometimes we will skip the symbol for the binary operation and will write simply xy instead of x y. Corresponding t o 1.3 we will use different letters for the elements of our alphabet: x, y, 2 , . . . or xl, x2,x3, . . ..

Definition 8.1.2 (i) Let C C W T ( X ) 2 Then . C is said to be hyperdualizable if for every identity s z t t E , the equation shd= thdbelongs also t o C. (ii) A variety V of type 7 = (2,2) is said to satisfy the duality principle if the set I d V is hyperdualizable.

Lemma 8.1.3 Let V be a variety of type (2,2) such that V = Mod C and C i s hyperdualizable. T h e n V satisfies the duality principle.

Proof: Let

a 0 := a ~ ( x l , x a ) , ~ and ( x l ,let x aMo ) := (00, aid). Since a. o h a. = a i d , Mo is a submonoid of Eyp(2,2). Also io[s]= s hd , so we have the following equivalences: V satisfies the duality principle u V is Mo-solid u Xgo[IdV] I d V (by Theorem 2.5.6 or Theorem 3.3.3) u Xgo[C]C I d V (by Theorem 2.5.6 or Theorem 3.3.3).

c

By hypothesis, C is hyperdualizable and C C IdV, that is, Xffo [C] = C IdV. Therefore, by the previous equivalences, the variety V satisfies the duality principle.

c

Every M-solid variety of type T = (2,2), with a ~ ( ~ ~ E~ M~ ~ ) , satisfies the duality principle. As a consequence, every solid variety of semirings satisfies the four distributive laws. Moreover, we have

Proposition 8.1.4 If V i s a solid variety of semirings, t h e n V i s a variety of medial I D - s e m i r i n g s ; t h a t is, V C VhfID.

8.2 The Minimal Solid Variety of Semirings

269

Proof: We show that all defining identities of VhIID are satisfied in V. Clearly, all four distributive laws are satisfied in V. We apply the hypersubstitution a = a ~ ( ~ to the ~ distributive , ~ ~ ) identity ~ ~ ~ G(zl,F(x2,2,)) = F(G(xl,x2),G(zl, 2,)) E I d V and get in V the following identities:

superposition operation. By the duality principle we have also zl G(zl,xl) E IdV. Applying the hypersubstitutions ~ G ( ~ ~ , ~ and o ~ ( x a , x l ) , ~t(oxthe 2 , xdistributive l) identity

= ~ ) , G ( ~

gives in V the identities

i.e. 212223

= ~ 1 ~ 2 x and 1 ~ 233 2 2 2 1 = ~

~ 1 ~ 2 x 1

Using the previous identities we obtain in V:

The duality principle gives the second medial identity. The next point to discuss is connected with the minimal solid variety of semirings.

8.2

The Minimal Semirings

Solid

Variety

of

Definition 8.2.1 An equation s = t is called regular if both terms s and t contain the same variables. A variety V is called regular if all identities satisfied in V are regular.

270

8 M-solid Varieties of Semirings

The set of all regular equations of a given type T is an equational theory and a variety V of type T is regular iff its generating system of identities is regular. In [go], a generating system of the variety RAT (see 5.1) was given. In the case of type T = (2,2) the variety RA(2,2) is defined by RA(2,2) := M o ~ { ( x I + x ~ ) +=x X~ I + ( X ~ + X ~ )= X I + X ~ , ( X I X ~ ) X= ~ xl(x2x3) = xlx3, x1xl = = (XI x2)(~3 ~ 4 = ) x1x3 x2x4). The equation (xl x2)(x3 x4) = x1x3 x224 is called entropic law.

+

+

+

+

+ +

+

Theorem 8.2.2 The variety RA(2,2) is the least non-trivial solid variety of semirings and every non-trivial solid variety of semirings diflerent from RA(2,2)is regular. Proof: Let V be a non-trivial solid variety of semirings. Then V is a non-trivial solid variety of type (2,2). Therefore RA(2,2)c V (Proposition 5.1.2). Now, we show that RA(2,2)is a variety of semirings. We have only t o show that (xl x2)x3 = 21x3 22x3 E IdRA(2,2)and x1(x2 2 3 ) = ~ 1 x 2 21x3 E IdRA(2,2).Indeed, if we substitute in the entropic law the variable x3 for x4 and using the idempotency, we obtain (xl x2)(x3 x3) = (xl x2)x3 = Xlx3 22x3 E IdRA(2,2). In a similar way, we obtain (XI X I ) ( X ~ x4) = x1(x3 xq) = 21x3 ~ 1 x E 4 IdRA(2,q. Altogether, RA(2,2)is the least non-trivial solid variety of semirings.

+

+

+

+

+

+

+

+

+

+

+

+

For the second part, let us assume that there exists a non-regular identity s = t in V. We will show that V = RA(2,2). The first part of this theorem ensures that RA(2,2)C V. It is left to show that RA(2,2) 2 V, i.e. IdRA(2,2) IdV. Applying the hypersubstitution O G ( ~ , , ~ , ) , G ( ~ ,to , ~ ~the ) identity s = t E IdV, we obtain an identity of the form xilxi2. . . xil = xjlxj2 . . . xjm with {il,22, . . . , il}U{jl, j 2 , . . . , jm}C (1, . . . , n} if t and s are n-ary terms. The application of a,,,,, and a,,,,, t o the previous identity shows that xi, = xj, and xi, = xjm otherwise V would be the trivial variety. Since s = t is not regular, there exists a variable xir which occurs on one side, but not on the other side. Substituting for all variables which are different from xi7. the variable xl and using the idempotent and the medial laws, since V is a solid variety of semirings, one has xlxirxl = XI E IdV. This gives

c

8.3 The Greatest Solid Variety of Semirings ( X I X ~ ) XM ~ ( X I X ~ X I )M~ 21x3 ~

get (21

+ + 22)

2 3 % 21

+

271

E IdV. By the duality principle, we 23

E

IdV

(*>.

The variety V satisfies also the following identities:

Having determined the minimal solid variety of semirings, we are now going t o determine the greatest one.

8.3

The Greatest Semirings

Solid

Variety

of

Proposition 8.1.4 shows that every solid variety of semirings is a subvariety of the variety VMrD. Therefore, the variety VMIDis a good candidate for the greatest solid variety of semirings. The subvariety lattice of VMIDwas fully described in [73]. This will be helpful in solving our problem: determination of the greatest solid variety of semirings. Let us consider the following two-element algebras: A = ((0, I}; ef , ef ), (e: is the binary projection (0, (0, I ) on the first input); (0,l) .A" = ({0,1}; e;, e;), (e; is the binary projection (0, 1)2 on the second input); B = ( ( 0 , l ) ; e:, A ) , ( A denotes the conjunction), B0 = ((0, 1); ei, A ) ; B' = ((0, I ) ; A, e f ) ; Boo= ({O,l}; A , e;); 3== ({o, 11; 4 7 4); F = {O, I}; e;, e:); 3 = ((0, I ) ; A, v), ( V denotes the disjunction); L = ({O, I}; A , A). The algebra 3 generates the variety D of all distributive lattices and L generates the variety S of bi-semilattices. This notation corresponds t o the notations in [73]. Then we have

-

2 72

8 M-solid Varieties of Semirings

Lemma 8.3.1 ( [ 7 3 ] )The subvariety lattice of V h f I Dis a Boolean lattice with 20 atoms and 10 dual atoms, i.e., with 2'' elements. The atoms are exactly the following varieties:

V ( 4 7 V ( 4 ,V ( B ) , V ( B O )V, ( B D )V, ( B D OV) (, F ) V , ( F ) ,D and S . Furthermore, the variety V M I Dis the join of these 20 atoms. Now we want t o prove that the variety Vhflo is solid. To do this, we will show that V M I D = H M o d ( F ( F ( x 1 ,Q ) , 2 3 )

F ( x l , F(x2,~ 3 ) ) G 7 (xl,

F ( ~ 2 > ~ 3 ) ) F(G(x17

~ 2 ) G(x17 >

~ 3 ) ) ) '

we have Lemma 8.3.2 The variety V M I Dis a subvariety of the variety

Proof: Since V,, is the class of all algebras of type ( 2 , 2 ) in which the associative equation F ( F ( x l ,x 2 ) ,2 3 ) % F ( x l ,F(x2,2 3 ) ) and the distributive equation G ( x l ,F ( x 2 ,x 3 ) )% F(G(x1,x2), G(x1,2 3 ) ) are satisfied as hyperidentities, we have to prove also that the equations F ( F ( x l ,x 2 ) ,2 3 ) = F(x1,F ( x 2 ,2 3 ) ) and G(x1,F(x2,23)) F ( G ( x l ,x 2 ) ,G ( x l ,x 3 ) ) are satisfied as hyperidentities in V M I D . It is well-known that RA[2,2) = V ( A ) V V ( P ) V V ( F ) V V ( F ) . Thus, by Lemma 8.3.1 the variety V M I D is the join ) , ( B ), V ( B O )V, ( B D, )V ( B D O )D, and S . Therefore, we of R A ( 2 , 2 V have only t o prove that the equations F ( F ( x l 7 x 2 )2,3 ) M F(x17F ( x 2 , ~ 3 ) and ) G ( x i ,F ( x 2 7 2 3 ) ) M F(G(x1,x2), G(x1,2 3 ) ) are satisfied as hyperidentities in V ( B ) ,V ( B O )V, ( B m )V, ( B m OD) , and S since RA(2,2)is solid. The normal forms of binary terms (i.e. a choice of 'representatives of the classes of the free alge bra) over the variety D of distributive lattices are x1,x2,x1 A x2 and x1 V 22. Then the D-normal forms of hypersubstitutions are

ozlAz2,z2,o z l A z 2 , ~ 1 A z 2~, Z ~ A Z ~ , Z ozlvz2,z1 ~ V Z ~ , 7 oz1vz2,z2, o z 1 ~ z 2 , zand 1 ~ozlvz2,zlvz2. z2 Obviously, applying these hypersubstitutions to the associative equation ( x l A x 2 ) A 2 3 = X I A ( x 2A x 3 ) , we get again associative equations or equations of the form xl M X I which are satisfied in D. For the distributive equation xl A ( 2 2 V 2 3 ) % ( x l A x 2 ) V ( x l A x 3 ) ,we built up the following table: ozlAz2,z17

8.3 The Greatest Solid Variety of Semirings

2 73

The table shows that the distributive equation xl A (x2 V 2 3 ) = (xl A x2)V (xl A 2 3 ) is hypersatisfied in D. Let V be one of the vari-

eties V(B), V(BO),V(Bo),V(BoO)and S. Then the V-normal forms of binary terms are 21,x2 and xl A 2 2 . Hence, the V-normal forms of hypersubstitutions are g ~ l , ~ g ~l 1 , ~ g2 ~ l , ~ l g ~~ a~ , a~ glx 2 , x l A x a g ~ a , ~ ga ~ l ~ ~ ga ~, ~l ~l ~ and a , ~ g

~

l

~

~

2

,

~

l

~

~

a

.

By similar calculations as we did in the case of D we can show that the associative equation F ( F ( x 1 ,x2),2 3 ) = F(zl,F ( x 2 ,Q)) and the distributive equation G ( z l , F ( x 2 ,2 3 ) ) = F ( G ( x l ,x2),G(xl, 2 3 ) ) are satisfied as hyperidentities in each of the varieties V(B), V(BO),V(B"), V(BoO)and S. Now all the tools are in hand t o show the main result of this section. Theorem 8.3.3 The variety VMID of all medial ID-semirings is the greatest solid variety of semirings. Moreover, VMID = Vgs. Proof: Lemma 8.3.2 shows that VMID C Vgs For the converse inclusion we need to show that Vg, is a solid variety of semirings.

a

2 74

8 M-solid Varieties of Semirings

Indeed, by definition, Vgs is a solid variety of type (2,2). Since

and

we conclude that the equation

is satisfied as an identity in Vgs. Applying also the hypersubstitu,,,Ito the associative identity F(F(xl, x2),2 3 ) = we see that the second associative identity holds in Vgs. Hence, Vgs is a solid variety of semirings; moreover, we obtain VMID V,, (Proposition 8.1.4). Consequently, the variety VhIID is equal t o the variety Vgs and it is the greatest solid variety of semirings.

>

We recall the definitions of a semilattice ordered system of algebras and of the sum of such systems, usually called Plonka sum.

Definition 8.3.4 A semilattice ordered system of algebras of type r is a triplet: A = (J, (Aj I j E J),(hjl I j 5 I; j, 1 E J))such that (d.1) J = (J;5) is a semilattice for which any two elements have the least upper bound (1.u.b.). (d.2)

(4I j E J) is a family of algebras of the same type r.

(d.3) (hjl I j 5 1; j, 1 E J) is a family of homomorphisms, hjl : Aj i Al and hjj is the identity mapping, moreover hjl o hlk = hjk for all j, I, k E J with j 5 1 5 k in J .

Definition 8.3.5 ([84]) Let A = (J, (Aj I j E J),(hlj I I 5 j; 1, j E J))be a semilattice ordered system of a type r, where A Aj = (Aj; (fi ' ) i , ~ ) . Assume that the carriers of the algebras Aj are mutually disjoint. Define an algebra A = S(A) which will be called the sum of a system A in the following way: The carrier set of the algebra S(A) will be equal t o u{Aj I j E J} and the fundamental operations of S(A) are defined by

f?@) k

=

= f ) (bilk (al), ' ' 7 hintk (ani)) l.u.b.(il,---,in% and ) a, E Ail_, for r = l , . . - , n i . * * ' 7

*

8.4 The Lattice of all Solid Varieties of Semirings

2 75

Pastijn and Romanowska have proved

Lemma 8.3.6 ([76]) Every ID-semiring is the Ptonka sum of a semilattice ordered system of ID-semirings which satisfy the generalized absorption law x l ( x l 22 x l ) x l M X I .

+ +

Since every algebra in a solid variety of semirings is an ID-semiring, we have obviously Corollary 8.3.7 Every algebra in a solid variety of semirings is the Ptonka sum of a semilattice ordered system of ID-semirings which satisfy the generalized absorption law xl ( X I +x2 + x l ) x l M X I . The determination of the minimal and the greatest solid variety of semirings leads to the question: Are there more non-trivial solid varieties of semirings?

The Lattice of all Solid Varieties of Semirings

8.4

In [73],Pastijn provided a list of identities each of them determines one of the dual atoms in the lattice of all subvarieties of VMID.Each of these identities is satisfied by all but one of the ten atoms in the lattice of all subvarieties of VMID. (i) (ii)

+ +

+

~ 1 x 2 22x1 M ~ 2 x 1 ~ 21x2 2 ~ 1 x 2 22x1 M 21x2

+

+

+ ~ 2 x 1is not satisfied in V(A),

X ~ X I + ~ 2 x 1 is ~ not 2

satisfied in V(A"),

= xlx2xl+ x1 + x l x 2 is not satisfied in V(B), (iv) ~ 1 x + 2 x~ = xlx2 + xl + x I x 2 x Iis not satisfied in V(BO), (v) x I x 2 + x I x 2 x I= x I x 2 + x 2 x l+ x l x 2 x l is not satisfied in V(B*),

(iii)

xl

(vi)

X I X ~ + X ~ X M ~ Xx ~ ~ x ~ + x 2 x l + x 2 x lis x not 2

~1x2

+ +

+ ~ 1 x +2 (viii) ~ 1 x 2 + ~ 1 x+2 2 + ~ 1 x 2 ~ 1 x+2 (ix) ~ 1 x+ (vii)

~ 1 x 2 ~ 2 xM 1 ~ 22x1

~

M 22x1

XI

FZ

2

x

1

~2x1 is

satisfied in V(BeO),

not satisfied in V ( F ) ,

~ 2 x 1 is ~ not 2

X I X ~ X I + ~ 1 x is 2

satisfied in V ( F ) , not satisfied in D,

276

(x)

8 M-solid Varieties of Semirings XI+

+ = xl

~ 1 x 2 ~ 1 xl

is not satisfied in S.

For a variety V of type r and for a set C of equations of this type we denote by V(C) the subvariety defined by C. If s = t is one of these ten equations then by VhfID(s = t ) we denote the subvariety of VMID generated by s = t ; that is, if VMID = Mod Co, then VhfID(s = t ) = Mod( Co U {s = t}). Then the equation s = t will be called the defining equation of VhIID(s t). is the least non-trivial solid variety of Since the variety RA(2,2) semirings (Theorem 8.2.2), our strategy is t o check all intervals between RA(2,2)and each of the dual atoms of the subvariety lattice of VMID (see the picture below in which V,,i = 1 , . . . , 10, is the subvariety of VMID generated by the identity (i) in the previous list of identities). It is important t o mention that our picture is not a representation of the algebraic structure of the subvariety lattice of VMID. VAT I 11

8.4 The Lattice of all Solid Varieties of Semirings

277

Lemma 8.4.1 The variety RA(2,2)is the only non-trivial solid subvariety of the variety VMID(xl ~ 1 x 2 ~ X1I = x l ) =: VIO.

+

+

Proof: Clearly, RA(2,2)c VMID(xl +xlx2xl +xl = xl) since RA(2,2) satisfies the equation xl x2 xl = xl. If there is a solid variety V with RA(2,2) c V C VMID(xl ~ 1 x 2 ~ 1XI = xl), then V satisfies the identity X I xlx2xl x1 = XI which is not regular. This contradicts Theorem 8.2.2.

+ + + + +

+

Proof: Let V' be a subvariety of V. Assume that V' is solid. Then V' has to satisfy the defining equation of V as a hyperidentity. Applying the hypersubstitutions a,,,,, , a,,,,, , a,,,,, , and a,,,,, we obtain the identity xl = x2 which holds only in the trivial variety.. Lemma 8.4.3 If V is one of the four varieties: VMID(~I~ +1x2 = xlx2xl+ X I + x1x2) =: V3, VhfID(xlx2 21 = X l X 2 f 21 X1X2X1) =: K , ) I+, or VMID(XIX~~ 2 ~ 1 x = 2 ~ 1 x 2 22x1 ~ 2 ~ 1 x 2 =: VhfID(xlx2 xlx221 ) &, xlx2 22x1 ~ 1 ~ 2 x 1=: then RA(2,2)is the only non-trivial solid subvariety of V.

+ + +

+ +

+

+ +

+

Proof: By using the identities ~ 1 ~ 2 = x 3x123 E IdRA(2,Z)and xl x2 x3 = xl 2 3 E IdRA(2,2)as well as xlxl = xl xl = xl, it is clear that RA(2,2)is contained in each of these varieties. Assume that there is a non-trivial solid subvariety V' of V.

+

+

+

In the first case for V, if we apply the hypersubstitution a,,,~(,,,,,) t o the defining equation of V and o,,,~(,,,,,)in the second one, we get xl = xlx2xl E IdV'. Therefore, V' = RA(2,2) (see the proof of Theorem 8.2.2). In the third case, using that V' is a solid variety of semirings and satisfies the duality principle, one has: (xl x2)(x2 X I x2) = (XI 22) (22 21)( 2 2 + X I 2 2 ) E IdV' and then ( X I + X I 22) (22

+

+

+

+

+ + +

+

2 78

8 M-solid Varieties of Semirings

x1+x2) = x1x2+xl+x2 = (x1+x2) (x2+x1 (x1+22)) = (x1fx2)(x2+ x1 x1x2) E IdV' (by idempotency and the distributive laws) and also xlx2 XI x2 = ~ 1 x 2 X I 21x2 x2 x2xl+ ~ 2 x 1 ~ E 2IdV' (by the distributive and the idempotent laws). Since V' is solid, this equation is satisfied as a hyperidentity in V' and using the hypersubstitution a,,,~(,,,,,), we have 2 2 = ~ 2 x 1 ~ € 2IdV' and therefore V' = RA(2,2).

+

+ +

+ +

+ +

In the fourth case, a similar calculation gives the result.

+

It remains to check the interval between RA(2,2)and VhIID(x1x2 X I X ~ X I + 21x2 = ~ 1 x 2 XI x1x2) =: &. We define:

+ +

Lemma 8.4.4

1. Let V be a variety of medial ID-semirings. Then V satisfies the following identities:

2. The variety VBE satisfies the following identities: (i) (21 + x2) ( 2 3 + x1) FZ x1x3 + x2x1, (ii) xlx3 x2x1x3= x1x3 x2x3 x2x1x3.

+

+

+

Proof: l.(i) By idempotency and the distributive laws, we have x xy y = x(x y) y = (x + Y ) ( X + Y) = x + y .

+ +

+ +

(ii) This can be proved in a similar way as (i). (iii) By idempotency and the distributive laws together with (ii), we get x + x y x y = X ( X + y x ) + y = (X + Y ) ( X + Y X + Y ) =

+

8.4 The Lattice of all Solid Varieties of Semirings

279

(iv) This can be proved in a similar way as (iii). (v) Using the idempotency, the medial laws and the distributivity, ~ 4 M (x1x2)(x1x3x4) 22x3 M (x1x2 we also obtain ~ 1 x 2 ~ 3 22x3 ~ 2 x 3( )X I X ~ X ~ + Z ~ X ~~ ) 1 ~ 2 ~ 3 ~ 4 ++ x~ ~1 x~ ~2 +x~ x~3 ~x x E ~ IdV. ~

+

+

+

2. (i) Substituting 22x3 for 2 2 in ~ 1 x 2 ~ 2 x 1M (xl ~ 1 x 2 X I 2 2 22x1 E IdVBE, we obtain

+ + +

(ii)

~

1

+ 22~1x3 ~

3

M

= M

+ x2)(x2+ xl)

+

M

+

~ 1 x 3 ~ ~3 2 ~ 3 x by 1 ~the 3 idempotency and the medial laws (2123 ~ 2 x 3() ~ 3 21x3) by 2. (i)

+

X1X3

+

+ X2X3 + X2X1X3

by the distributivity, the idempotency and the medial laws.

Theorem 8.4.5 solid.

(i) The variety VBE is equal to VENT and it is

(ii) There is n o solid variety diflerent from RA(2,2)and VBE between the solid varieties RA(2,2)and VBE.

280

8 M-solid Varieties of Semirings

+

+

Proof: We show first that V9 = VMID(xlx2 xlx2xl x1x2 = 21x2 X I x1x2)= VBE. Since & is a dual atom in the subvariety lattice of VhIID and VBEis a proper subvariety of VMID, it is enough to show that Vg VBE. Indeed, we have in Vg:

+ +

c

+ x2)(x2 + 21)

(21

= = = = =

+ 21 + 22 + X2Xl by the distributive laws XlX2 + 21 + XlX2 + 22x1 + 22 + X2Xl by the idempotent and the medial laws 21x2 + 21x221 + 21x2 + 22x1 + 2221x2 +z2xlby its defining identity XlX2 + XlX2Xl+ + 22x1 (x1x2+ ~ ~ 2 by1 the ) ~idempotency 21x2

x2x1x2

and the distributive laws 21x2 22x1 by the idempotency.

+

Altogether, we have

V9 = VBE = VENT.Since

+ x2)(x3 + x4) = ~ 1 x 3+ x2x4, ~ 1 x 2 ~ =3 ~~ 41

(XI

the equations x 3 ~ and 2 ~ 4X I

+

8.4 The Lattice of all Solid Varieties of Semirings

+ x3 + xq

281

+ x3 + 22 + xq

are satisfied as hyperidentities in every variety in which they hold as identities and since VhIID is solid, then the variety VENTis solid. The same goes for VBE and V9. is It is left to show the second part of the theorem. Clearly, RA(2,2) solid (Theorem 8.2.2). Assume that there is a solid variety V different from RA(2,2)and VBE between the varieties RA(2,2)and VBE. If S is not contained in V, then the identity X I + xlx2xl+ X I M X I is satisfied in V. But this cannot be a hyperidentity since it is not regular (Theorem 8.2.2). Assume that S is contained in V and at least one of the varieties V(B), V(BO),V(B') and V(BD0)is contained in V. Let oi,i = 1 , 2 , 3 be hypersubstitutions defined by 22

M XI

e: H 22 02: e: H x1 A 22 A H z1Az2 A H ~1 a3 : e: H z1 A z2 A H 22. (We remark that here we made no difference between operation symbols and symbols for concrete operations.) Then BO = al (B),B' = a2(B) and Boo = 0 3 (B). In a similar way, we conclude that each of the algebras B, BO,Bmand Boocan be derived from one of them using a hypersubstitution and since the solid variety V has t o contain all derived algebras obtained from algebras in V, then all of the varieties V(B), V(BO),V(B') and V(BD0)are contained in V. Altogether V contains RA(2,2), S, V(B), V(BO),V(B') and V(BD0). Therefore, V = VBE (since VBE = V9). This contradicts V # VBE. If S is contained in V and none of the varieties V(B), V(BO), V(B0) and V(BmO)is contained in V then V = RA(2,2)V S since D VBE. Thus xl ~ 1 x 2= xlz2zl ~ 1 x 2E IdV. Moreover, zl = z1x2z1E I d V using the hypersubstitution ~ x l , ~ ( x l , x This a ) . contradicts V # RA(2,2).Altogether, V cannot be solid. 0 :

+

+

Now all the tools are in hand to present the main result of this chapter.

Theorem 8.4.6 The lattice of all solid varieties of semirings is the &element chain represented b y the following picture:

8 M-solid Varieties of Semirings

Proof: Let V be a solid variety of semirings. Then V is either trivial or V is solid and RA[2,2) C V C VMID using Theorem 8.2.2 and Proposition 8.1.4. Moreover, by Lemma 8.3.1 there exists a dual atom V' of the subvariety lattice of VMID such that V V' C VMID. Lemma 8.4.1 together with Lemma 8.4.2 and Lemma 8.4.3 show that V C VBE c VMID. Altogether, using Theorem 8.3.3 and Theorem 8.4.5 we conclude that V is either trivial (V = 7 ( 2 , 2 ) ) or is one of the varieties RA(2,2),VBE and VMrD.

c

Now, we are going t o present a method t o construct a hyperidentity basis for a given non-trivial subvariety of VMID. For every variety V of type T and for an arbitrary monoid M C Hyp(r) of hypersubstitutions of type r, we may define an associated M-solid variety as follows:

c

V'), called the M-solid XhfV := n{V' I V' is M-solid and V closure of V. For M = H y p ( r ) , XA4V =:X V and this will be called the solid closure of V. To achieve our goal, we recall the following result: T and let M C IFtyp(~)be an arbitrary monoid of hypersubstitutions of type T. Then Xhf V = HhfModHMIdV.

Lemma 8.4.7 ([4]) Let V be a variety of type

8.5 Generalization of Normalizations

283

Theorem 8.4.6 shows that the solid closure of a variety V of semirings is again a variety of semirings iff V C VhfID. Each nontrivial subvariety of VhfID has one of the following solid closures: RA(2,2),VBE and VMID. Therefore, by using Lemma 8.4.7, a hyperidentity basis of a given non-trivial subvariety V of VMID is a hyperidentity basis of its solid closure, i.e. a hyperidentity basis of one of the three non-trivial solid varieties of semirings. For instance, for the variety D of all distributive lattices, we have XD = VMID since D VBE = V9 and V9 is a dual atom of the subvariety lattice of VMID. But VMID = HMod(F(x1, F ( x ~ , x=~ ) ) F ( F ( x 1 7x2), x3), G ( x 1 7 F(x2, ~ 3 ) ) F(G(xl, ~ 2 ) G > ( x l >~ 3 ) ) ) Theorem 8.3.3) and XD = H M o d H I d D (by Lemma 8.4.7). Therefore, the set {J'(x~,F(x27 2 3 ) ) = J'(F(x1, x2), x3), G ( x 1 7 J'(x2, 2 3 ) ) = F (G(xl,x2),G(xl, x3))) is a hyperidentity basis of the variety D. This shortens a hyperidentity basis of D (given in Corollary 2.10 of [71]) because in that corollary, a hyperidentity basis of the variety D contains the idempotent and the medial laws.

8.5

Generalization of Normalizations

An equation s = t is called normal if either both terms s and t are equal to the same variable or none of them is a variable, that is, if s = t or the complexities op(s), op(t) (number of occurrences of operation symbols) of s and t are greater or equal to 1. A variety in which all identities are normal is called a normal variety. The concept of normalization was first studied by Mel'nik ([65]) and Plonka ([85]) and later by Graczyriska ([54] and [55]) and Chajda ([I11 and [12]).This chapter will present a generalization of normalization using the occurrence of operation symbols of a term. This is a first step towards studying a much more general valuation theory of terms which is described in the paper [40]. An equation s = t is called k-normal (k E N+) if s = t or the complexities op(s), op(t) of both terms s and t are greater or equal to k. A variety in which all identities are k-normal is called k-normal. First of all, in the same way as it was generally done by the previous authors, we will construct some operators on the lattice C ( 7 ) of all varieties of type T and on the power set lattice P(W,(X)2) of all equations of type T and will point out some properties of these operators. Finally,

284

8 M-solid Varieties of Semirings

we will give a method to construct a generating system of some varieties which are closed under the operators on C(T). Since we will see later in Section 8.6 that the normalization of a solid variety of semirings is Pre(2,2)-solid, we are primarily interested in the normalization and its generalization. Let k E N+ and let N;(T)

be the set defined by

We consider the operators Nf and N k which map for C W, (X)2and W E C(7) in the following way:

>

c

We want to show that, for any k 1, the set N E ( r ) is closed under the five derivation rules. Clearly, for any k 1, Nf (r) is an equivalence relation on W,(X). For every equation s FZ t t N; (T), the equation 5 FZ t", where t" and 5 are the terms obtained from t and s, respectively, by replacing every occurrence of a given variable z E X by a given r E W,(X), belongs t o Nf ( r ) . We now show the compatibilj ni for i E I. If for all ity. Let sj FZ tj E N p ( r ) , 1 j, 1 j ni, we have sj = t j then we obtain f i ( s l , . . . , Sn,) = fi(t1,. . . , i n , ) and f i ( ~ 1 ,... , sni) FZ f i ( t l , . . . , t n , ) E N:(T). Otherwise, there exists at least one jo E { I , . . . , ni) such that ~ ~ ( s j o op(tj0) ) , k . Thus7 we have o ~ ( f i ( s 1.,. . 7 sn,)) 0p(sjo) k and op(fi(tl,. . . , tn,)) op(tjo) k . Therefore, we obtain f i ( s l , . . . , Sni) FZ f i ( t l , . . . ,t,,) E N:(T). We denote by E the operator which assigns t o each set C of equations the smallest set closed under arbitrary applications of the five derivation rules for identities and containing C. Then altogether, E (NE (7))= N F (r). That is, N p ( r ) is an equational theory. If C is an equational theory then the set NF(C) is also an equational theory because it is an intersection of two equational theories. Obviously, a variety V of type T is called k-normal if IdV C N;(r). From

<

< <

>

>

>

<

>

>

>

8.5 Generalization of Normalizations

285

> N:(IdV) > N ? ( I ~ V )> . . . > NP(1dV) > . . ., it follows that V c N ~ ( v c ) ... c N ~ ( v c ) . . .. IdV

Let Ck(-r) be the variety of type -r defined by Ck(-r) = ~ o d N E ( - r ) . Then for every variety W C Alg(-r), we get I ~ N ~ ( w=) N,E(IdW) = IdW n N:(-r) = IdW n IdCk(r).That is, N ~ ( w =) W V C k ( r ) .The variety N ~ ( w is ) called the k-normalization of W and for k = 1 it is its normalization in the usual sense and will be denoted by N A ( W )or W V C ( T ) . Now we will prove some useful properties of the operator NE as Graczynska did for k = l ([55]). Let r E W , ( X ) . We denote by r ( x ,. . . , x ) the term arising from r by replacing each variable occurring in r by the variable x.

Lemma 8.5.1 Let (Cj)j,J be a family of sets of equations of type T and let C and C' be two elements of this family. Then

If C

c C' then NE ( C ) c N,f ( 2 ) .

and

By (i), (ii), (iii) the operator NE is a kernel operator.

286

8 M-solid Varieties of Semirings

Proof: (i) N,E(C) = C n N P ( r ) (ii) Assume that C NP ( 2 ) .

C C.

c C'. Then NE(C) = C n N F (7) C C'n NE(7) =

(iii) NE(N,E(C)) = N,E(r) n NE(C) N E ( r ) n C = IV;(C).

=

N,E(r) n ( N E ( r ) n C)

=

(iv) N,E (C u C' ) = N F (7)n (C u C') = (Nf (7)n C) u (N: (7)n C') = NE(C) U NE(Ct). Similarly, the result can be proved for an arbitrary family of sets of equations.

NP

(NF

(v) N,E (C n ct) = (7)n (cn 2)= (7)n c)n ( ~(7)fn c') = NE(C) n N:(Ct). This can be generalized t o an arbitrary family of sets of equations.

c

NP(E(C)). (vi) Since C C E (C), by (ii) we have N:(C) Moreover, since NF(E(C)) is an equational theory we have E(NE(C)) C NE(E(C)). If E ( C ) = C then NE(E(C)) = NE(C) = EWE (C)). (vii) The proof will be provided for C and C'. For an arbitrary family of sets of equations one can do it in a similar way.

c Clearly, E ( C ) U E(Ct) c E ( C U C'). By (ii) we have N: (E(c)U E(Ct)) c Nf (E(CuCt)).Hence, we obtain E ( N F ( E ( C ) u E ( C r ) ) )c N P ( E ( C U C')) since Nf (E(C U C')) is an equational theory.

2 Let s = t E NE(E(C U C')) = N,f(r) n E ( C U C') , we have to show that s = t E E ( N f (E(C)u E(Ct))).If s = t then obviously we have our result. Otherwise op(s), op(t) k. Since x = r ( x , . . . , x) E C U C', without loss of the generality we can assume that x = r ( x , . . . , x) E C. Replacing each variable x occurring in s and t by r ( x , . . . , x), we obtain the terms s* and t*, respectively. Clearly, we have s = s* E Nf(7) n E ( C ) = N:(E(C)), since op(s) k. Also, we have t = t* E N f ( E ( C ) ) , so from N f ( E ( C ) ) E ( N F ( E ( C ) )U IVP(E(Ct))),we get s = s*, t = t* E E ( N F ( E ( C ) )U IVP(E(Ct))). Let el, . . . , em with em = s = t be a sequence of equations for which m, belongs t o the set C U C' or follows each equation ej, 1 j

>

c

< <

>

8.5 Generalization of Normalizations

287

from the previous equations el, . . . , ej-1 by using the five rules of equational logic. Clearly, the equations el, . . . , em with em := s = t are those which are needed t o derive s = t starting with the j m, we denote the identity obtained set C U C'. By e;, 1 from ej by replacing each variable x occurring in ej by the term r ( r ,. . . , r ) . Hence, for every j, 1 j m, e; t N,'(E(C)) U N,f(E(C1)) or e; it follows from e;, . . . , e;-, by using the five rules of equational logic. That is, e;, . . . , e; := s* = t* is a sequence j m, is from the of equations for which each equation eg, 1 set N,'(E(C)) U Nf (E(2)) or e;, 1 j m, follows from the previous equations e;, . . . , e;-, by using the five rules of equational logic. Hence s* = t* E E ( N E ( E ( C ) ) U N;(E(C1))). Altogether s = s* = t* = t t E(N;(E(C)) U N,f(E(C1))). Consequently, we get s = t E E(N,f(E(C)) U N;(E(C1))) = E ( N E ( E ( C ) U E(C1))) (by using (iv)).

< <

< <

< < < <

>

Corollary 8.5.2 For every k 1, t h e operator Nf i s a closure operator o n the lattice C ( r ) of all varieties of type r and the class of all fixed points with respect t o N: forms a complete sublattice of t h e lattice C ( r ) . Proof: Let V and V' be varieties of type 7. From I d ( N f ( V ) ) = NE(IdV) C IdV, it follows clearly that V C N ~ ( v ) .Furthermore, we have I ~ ( N , ~ ( N ~ (= v )N) ,) ~ ( I ~ ( N ~ ( v=) N,f(N,f(IdV)) )) = N,f(IdV) = I ~ N , ~ ( (by v ) Lemma 8.5.l(iii)). We show the monotonicity. Let V and V' be varieties of type 7. Then we have V

C V'

IdV' C I d V ==+ N,f(IdV1) N,f(IdV) by Lemma 8.5.l(ii)

c

==+ I ~ ( N ~ ( v 'c) )I ~ ( N ~ ( v ) ) ==+ N f ( V ) c Nf (V').

Therefore, Nf is a closure operator on C ( r ) . Now we show that Nf is a join-preserving operator. Let (V,)jEJ be a family of varieties of type r .

=

n N;(I~Y) j€ J

by Lemma 8.5.1 (v).

288

8 M-solid Varieties of Semirings

Hence Nf (

V V,) = V Nf (V,).Altogether, Nf is a closure oper-

. i tJ

. i tJ

ator which preserves t h e join for an arbitrary family of elements of the lattice C ( r ) . Therefore, Theorem 2.2.1 (i) shows that the class of all fixed points with respect t o N; forms a complete sublattice of the lattice C(7) of a11 varieties of type 7. It is well-known that the normalization N A ( v ) of any non-normal variety V of type r covers V in the lattice of all varieties of type r, this means that I d V = E ( I ~ N ~ (Uv{e)) ) for any non-normal identity e E I d V ([65]). This result can be generalized as follows:

Lemma 8.5.3 Let V be a non-normal variety of type r , such that x = r ( x , . . . , x) E IdV, where I- is not a variable. Then for any natural number k, 1 k op(r), I d V = E ( N F ( I d V ) U {x = ~(x,...,x)>).

< <

Proof: Let kt := op(r). It is enough to prove that IdV

=

E(N;(I~v) u {z = r ( z , . . . , z)))

<

(1)

<

Indeed, if (1) is true, then for 1 k kt, we have I d V = E(N;(I~v) u {X = I - , . . . , x)}) E(N:(I~v) u {X = r ( x , . . . , x))). Therefore, we obtain I d V = E ( N F ( I d V ) U {x = r ( z , . . . , x))), since E(NE(IdV) U {x = r ( x , . . . , x))) IdV. Clearly, we have E(N$(IdV) U {z = r ( z , . . . , z))) IdV. Conversely, let s = t E IdV. Assume that s and t are mary terms. Replacing each variable x j (1 j m) occurring in s and t by r ( z j , . . . , xj), we obtain s* and t*, respeckt. Then we have s* = t* E tively, where op(s*),op(t*) IdNfi(V) E(IdNfi(V)u{z = r ( x , . . . , z))) = E(NZ(IdV)u{z = T ( X , . . . , x))), and also s = s*,t = t* E E({x = I-(x,. . . , x)}) C E(N;(IdV) U {x = I-(x,.. . , x))). Altogether, we obtain s = s* = t* = t E E(Nf(1dV) U {z = r ( x , . . . , x ) ) ) . That is, s = t E E(N$(IdV) U {z = r ( x , . . . , x))). Therefore, we have proved the equality (I).

c

c c

<

<

>

c

>

Theorem 8.5.4 Let k 1 and let V be a non-normal variety of type r . Let C(V) be the subvariety lattice of V. Then the operator

8.5 Generalization of Normalizations

is an injective lattice homomorphism. (Actually, Nf denotes here and in the following considerations the restriction of the operator N,f onto L(V).)

Proof: First, we show that Nf is a meet-preserving operatlor. Since V is not normal, then there exists an identity x = r ( x , . . . , x) E IdV, with op(r) k. Let Wl, W2 E L ( V ) . Then x = r ( x , . . . , x) E IdWl U IdW2 and we have the following equalities:

>

I~(N,A(w~

A

w2)) = N , E ( I ~ ( w ~ A w2))

Nf(E(IdWlUIdW2)) E(Nf(IdW1 U IdW2)) by Lemma 8.5.1 (vii) and by the fact that IdWi, i = 1 , 2 , are equational theories = E (Nf (IdWl) U Nf (IdW2)) by Lemma 8.5.1 (iv) = E ( I d i l i f ( ~U ~ )1dNf(W2)) = I ~ ( N , A ( w ~ ) A IV,A(W~)). Hence N;(w~ A W2) = Nf(w1) A N f ( w 2 ) . In the proof of Corollary 8.5.2, we have shown that ~f is a join-preserving operator. Therefore, we conclude that N k is a latt8icehomomorphism. =

=

It is left to prove the injectivity. Assume that Nf(w1) = Nf(W2). That is, we have NE(IdWl) = NE(IdW2). So NF(IdWl) U {x = r ( x , . . . , x)) = NE(IdW2) U {x = r ( z , . . . , x)). It follows that E(NF(IdWl) U {x = r ( x , . . . , x))) = E(N:(IdW2) U {x = r ( x , . . . , x ) ) ) . Consequently, by Lemma 8.5.3 we obtain IdWl = IdW2. That is, Wl = Wz. In a similar way as we did in the proof of Theorem 8.5.4, we can show

>

Remark 8.5.5 Let k 1 and let V be a non-normal variety of type r. Let V' be a variety of type r. Then Nf (V A V') = Nf (v)A N,A (V') . Now we will present a method t o construct a generating system of the k-normalization of some varieties of type r = (ni)i,I. Let I3 be

290

8 M-solid Varieties of Semirings

an algebra of type T and let k be a natural number different from 0. We define:

{ t E WT( X ) I W ( t )> k ) ,

Tk ( X ) := T k ( B ) :=

IJ

I

{ t B ( b l , . . . , b,)

b l , . . . , b, E B ) ,

t€Tk(X)

T o ( B ) := B and T k ( B ):= ( T k ( B )(;f p ( B ' ) i e r ) . (Here fp(B)is the restriction of f: onto T k ( B ) . )Then we have Lemma 8.5.6 Let B be an algebra of type r = (ni)iE1.Then each Tk+l( B ) is a subalgebra of Tk( B ) ,Tk-1 ( B ) ,. . . , To( B ) = B for k E N.

> >

Proof: It is enough to show that for any k 1, T k ( B )is a subalgebra of B since B = T o ( B ) T I( B ) T2(B) . . . . For each i E I , let fi be the ni-ary corresponding operation symbol. Let b l , .. . , b,, E T k ( B ) ,then for each 1 < j < ni there exist an rnj-ary term t j E Tk( X ) and bil, . . . , bim3 E B such that bj = ( 1 , . . . , b ) . Let m = x72, r n j and let t i := t l ( x l , .. . , X m 1 ) , t; := t2(xrnl+1,... , xml+m2),. . . 7 t;, := tni(xml+...+mni-1+l;.. . r xml+...+m,,i-l+m,,i)and t := fi(t',, . . . , tL,). Considering the terms t i , . . . , tLi as m-ary terms, it follows that t is

>

>

Now we will give a description of the algebras of the variety N ~ ( v ) , where V is a non-normal variety.

Theorem 8.5.7 Let V be a non-normal variety of type r and let B be an algebra of type r . Then B E N k ( V ) , with k > I , if and only if the following conditions are satisfied: (i) Tk( B ) E V . (ii) There exists an identity r ( x ,. . . , x ) = x E IdV (with op(r) k ) such that the mapping cp : B + B , b H ( b , . . . , b ) , is an endomorphism of B which is the identity mapping on Tk( B ).

>

291

8.5 Generalization of Normalizations

Proof: Since V is a non-normal variety, there exists an identity r ( x , . . . , x) FZ x E IdV, where op(r) k.

>

Let B E Nf(V). First, we show that Tk(B) E V, that is, Tk(B) satisfies all identities in V. Since Tk(B) is a subalgebra of ) 8.5.6), then Tk(B) satisfies all identities of V B E I V ~ ( V(Lemma for which both sides have complexities (number of operation symbols) k. Hence, by Lemma 8.5.3 it is enough t o show that Tk(B) satisfies the non-normal identity r ( x , . . . , x) = x E IdV, where k. That is, we have to show that for any b E Tk(B), b = op(r) ~ ~ k ( ~ ). (. ,bb). , .Indeed, b E Tk(B)means that there exists an m-ary term t E T k ( X ) and there exist bl, . . . , b, E B such that ==+

>

>

b = t B (bl, . . . , b,)

(*>.

>

because of op(t) k. Since B E N ~ ( v )using , the previous identity and using the fact that Tk(B) is a subalgebra of B, we get

and

That is, rTk(B)(b,.. . , b) = b by (*). This proves also that cp is the identity mapping on Tk(B).We will now show that cp is an endomorphism of B. For each i E I, we remember that fi is the ni-operation symbol of type 7 = (ni)i,I, we have to show that for any elements . . ,bnJ = f;(cp(bl), . . . ,cp(bnJ). That bl,. . . , h i E B : (~(fiB(bl,. is, we have to show that rB(f;(bl, . . . , bnz), . . . , f;(bl, . . . , bnz)) = B B fi (r (bl,...,bl),...,rB(bq,...7bn2.)). Since r ( x , . . . , x ) RZ x E IdV, we obtain in V the identities r(fi(x1, . . . 7 xn,), . . . , fi(x1, . . . , xni))

8 M-solid Varieties of Semiriugs

292

Therefore, we get the identity r ( f i ( z l , . . . , zni),. . . 7 fi(zl, . . . 7 zn,)) = fi(r(z17.. . , X I ) , . . . , r(zni7 . . . , X J ) E IdV. % Moreover, we have ( f ( x 1 . . . , x i . . . , f ( x . . . , x i f r ( r ( x l 7 . . , X I ) , . . . , r ( x n i ,. . . , x n z ) ) t I ~ N ~ ( since v ) op(r) k. Then for any bl, . . . , bna E B , we get ~ ~ ( f ? (. .~. ,lb > n i ) , . . . 7 f?(bl> . . . , b n i ) ) = fiB (rB ( b l , . . . , bl), . . . , r B ( b n z , . . . , bnz)) since B E N ~ ( v ) . +== Assume condition (i) and condition (ii) are satisfied. We will show that B E N ~ ( v ) ,that is, that for any identity s = t t I d V (where s and t are m-ary terms, m 1) for k , we have s = t E IdB. Indeed, which op(s), op(t) let bl,. . . , b, E B. Then p(bi) E Tk(B),1 5 i 5 rn, since k. By condition (i), we have Tk(B) E V. It follows that op(r) s ~ (p(bl), ~ ( ~. . ). , ~ ( b , ) ) = tTk(B)(cp(bl),. . . , cp(b,)). Hence, we obtain the equality sB(cp(bl), . . . , p(b,)) = tB(cp(bl),. . . , p(b,)) since Tk(B) is a subalgebra of B. Moreover, we get p(sB(bl,.. . , h,,,)) = 'p(tB(bl, . . . ,b,)) since p is an endomorphism (condition (ii)). It follows the equality sB(bl, . . . , b,) = tB(bl, . . . , b,) since op(s), op(t) k and the restriction of cp to Tk(B)is the identity mapping on Tk(B)(condition (ii)). Therefore s FZ t t I d a .

>

>

>

>

>

We now have all the tools at our disposal t o present, for some varieties V ,a method t o construct a generating system of Nk(V), k 1, as Mel'nik did for k = 1 ([65]).We start with the following notation: For every i t I, let C;" be the set of all equations of the form:

>

.

e;

.

:= t ( z l , . . . , x,)

= r ( t ( z l , . . . , z,),

where m is the arity of t.

. . . , t(x1,. . . , z,)),

8.5 Generalization of Normalizations

293

Proposition 8.5.8 Let V := Mod{x = r ( x , . . . , x)} be a variety of 1, and let C := ( U C;,) U ( U {ei)). type 7 , where op(r) k

> >

t~Tk(X1

iEI

Then Nf (V) = ModCr Proof: c From zj = r ( z j , . . . , zj) E I d V for 1 < j we obtain in V the identities r(fi(x1, . . . , x n , ) , . .. , f i ( ~ 1 ,... ,xni)) M fi(z17...,zni)

< ni and i E I,

Thus, the variety N f ( V ) satisfies these identities. Therefore, we get U C; E I d N f ( v ) . By definition of r;, for t E Tk(X), we obtain it1

e; E I d N f ( V ) , since z

= r ( x , . . . , x ) E IdV.

Thus

U

{e;)

t€Tk(X)

c

I ~ N ~ ( v )Altogether, . we have Cr

c IdNf (V), that is,

Nf (V)

c ModCr.

> Let B E ModCr. To prove that

B E N f ( V ) we will show that condition (i) and condition (ii) of Theorem 8.5.7 are satisfied. Indeed, since B E ModC', B satisfies all identities of C', in particular all identities of U {e;}. That is, Tk(B) satisfies x = r ( x , . . . , x). ttTk

(X)

This means that Tk(B) E V. Therefore, we have proved condition (i). We consider the mapping cp : B B , b H r(b, . . . , b). Let bl, . . . , bnZ E B, we have

~ ( f ( ~ 7= ~rB(f:(bl,...,bn,),...,f:(bl,...,bn,)) ) ) B B . . h ) , . . . , rB(bn,,.. . bn,)), = f%(r since B satisfies fi(r(zl,. . . , X I ) , . . . , r(xni,. . . , zna))= 7

7

r(fi(x1, . . . , xn,), . . . , fi(x1, . . . , xni)). That is, cp is an endomorphism. The identities of U {e;} guarantee that (plTk(B)is the tETk (X)

identity mapping on T~(B). ~ e n c econdition , (ii) is satisfied.

Remark 8.5.9 Let F;i be the set of the following equations:

294

8 M-solid Varieties of Semirings

which is the generating system of N A( ~ o d {Mx r ( x ,. . . , x ))) found by Mel'nik in [65].

Proof: We have only t o show that for each t E T l ( B ) r, ( t ,. . . , t ) M t E E ( r r ) .Let t E T l ( B ) and assume that op(t) > 1. Then t = fi( 71,. . . , T,,), where yj are terms. Since r(f i ( x l ,. . . , x,,), . . . , f i ( x l , . . . , x , ~ ) ) M f i ( x l ,. . . , x , ~ ) E rr, replacing each variable xj, by yj, 1 j ni, we obtain r ( t ,. . . , t ) M t E E ( F r ) .

< <

Corollary 8.5.10 Let C

C1 U U { r j( x ,. . . , x )

=

M

x ) , where

1tJ

>

op(rj) k and N;(C1) Mod(C1U U Cr3).

=

C1. Let V := ModC. Then N ~ ( v = )

j tJ

Proof: We have the equalities IdNf(V) = Nf(IdV) =

N,E(E(& u u { r j( x ,. . . , X )

M

2)))

j tJ

=

E ( N f ( E ( C 1 )U )

U N , E ( E ( { r j ( x , . . . , x E) X ) ) ) ) .ltJ

by Lemma 8.5.1 (vii)

=

by Proposition 8.5.8 and since N,f(C1)= C1 E(C1U U (C'i)). jE J

Therefore, N , f ( V ) = Mod(& U U (C'i)). jE J

Now we are able t o determine generating systems of the knormalizations of concrete varieties. We start with the following lemma:

Lemma 8.5.11 Let V M Dbe the variety of all medial and distributive semirings. Let 2 ~Zf Y 2 M x + Y + Z , ~ XM x'), V2 := V ~ ~ D ( XX Y~Z , ~ X Y , 2 2 V1 := V ~ ~ D ( X ~ f~Y % x f Y , 22 % x 2 ) .

+

Then

8.5 Generalization of Normalizations 1. The following identities hold in V2: (i) xy2z M xyz2 M xyz. (ii) x3y M xy3 M ( ~ y ) ~ . (iii) x 3 + y = x + y 3 (iv) t3 M t

t

M

( ~ + y ) ~ .

3t for all t E T2(X)

2. The variety Vl satisfies the following identities: ~ (i) zY2M zy2M ( ~ y M) zy.

+

(ii) z2 y

M

x + y2 M ( x + Y ) M ~ X+Y.

Proof: I. The following identities hold in V2: ( i ) z Y 2 x M z y3x x(Y+Y+Y)~ M xyz xyz xyz (x+x+x)yz z3yx XYz (by the identities z2yz M zyz, 3z M z3 and by the distributive laws). xyz2 M zyz3 by the identity zy2z M zyx M zy(z z z) by the identity z% 3x M zyz zyx xyx by the distributive laws M xyz (similarly as we did before). (ii)x3y M ( x + x + x ) y M xy+xy+zy = X(Y + Y + Y ) zy3 (by the identity 3z M x3 E IdV2 and by the distributive laws). Moreover, we have z y zy zy M ( ~ y E) IdV2. ~ Altogether, we get x3y M xY3M ( ~ y E) IdV2. ~

= = --

+

+

7.2

+ + + +

7.2

+ +

(iii) Clearly, we have z" distributive laws.

y

M

(z + Y)"

z

+ Y Vyusing the

(iv) Let t E T2(X). Then there exist ti, i = 1 , 2 , 3 such that either t M tlt2t3 E IdV2 or t M t1 t2 t3 E IdV2 or t M (tl t2)t3 E IdV2 or t M t1(t2 t3) E IdV2 or t M t1t2 t3 E

+

+

+ +

+

8 M-solid Varieties of Semirings

296

+

+

IdV2 or t = t l t2t3 E IdV2. If t FZ (tl t2)t3 E IdV2 or t FZ tl(t2 t3) E IdV2 or t = tlt2 t3 E IdV2 or t = t l t2t3 E I d K , then by the distributivity there exist terms ti, i = 1 , 2 , 3 such that t = ti + t', + t', E Id&. Thus, we have t o consider only the following cases: t = tlt2t3 E IdV2 and t = tl t2 t3 E IdV2. Assume that t = tlt2t3 E IdV2. Then the following identities hold in V2 : t3 = tlt2t3tlt2t3tlt2t3 = = tlt2t3 = t by using the medial laws and the identities x2yz FZ xy2z = xyz2 = xyz E IdV2. For t = tl + t2+ t3 E IdV2, in a similar way as we did earlier, we obtain t% t E IdV2, using t3 = t t t E IdV2.

+

+

+

+ +

+ + 2.(i) By using the identity x + x = x2 and the distributive equa-

tions, we obtain in Vl the following identities: xy2 = x(y+y) = xy+xy = ( ~ y=) xy+xy ~ = (x+x)y

+

= x 2 y = xy.

(ii) Since x x FZ x2 E IdVl and since Vl satisfies the duality principle, we deduce from 2. (i) that x2 y = x y2 = x y E I d K . By the distributivity the variety Vl satisfies the equation x2 y = (x y) (x y) as an identity.

+

+

+

+

+

+

From now on, we will use the following abbreviations: 2Ass. if we mean both associative equations, 4Dist. if we mean the 4 distributive equations and 2Med. if we mean both medial equations. Then all tools are ready to show:

Theorem 8.5.12 The varieties IVk(VhfID),k = 1 , 2 are determined by: Nk(VMID) = V M D ( ~ 2=yxyz, ~ 2x y z = x + y + z, 3x = x3) =: V2,IVA(VMID)= VMD(x2y = X Y , ~ X y = x Y , ~ X= x2) =: Vl, where VhfID is the variety of all medial idempotent and distributive semzrzngs.

+ + +

+

Proof: Notice that VhfID = Mod{2Ass., 4Dist., 2Med., x2 = x = 2x). Since all of the equations of the generating system of V2 are satisfied as identities in VMrD and have the property that the number of operation symbols on both sides are greater or equal t o 2, ~ ~the ~ ~converse ) . inclusion, we conclude that IdV2 C I ~ I V ~ ( V For we look for a generating system of IdVMID which satisfies the con-

8.5 Generalization of Normalizations

297

ditions of Corollary 8.5.10. Let C1 := {2Ass., 4Dist., 2Med., x2yz M xyz, 22

= x} and C

+y + z

n

M

x

+ y + z},

(J Cz. Now we want t o 2=1 prove that ModC = VMID Clearly, C IdVMID, then ModC It is left t o show that x2 M x M 2x E E ( C ) . Indeed, VhfID. X ~ M X E C ==+ x 4 = x 2 ~ E ( C ) ==+ x3 M x2 E E ( C ) since x4 M x3 E E ( C ) + x = x2 E E ( C ) since x = x3 E C. Since C is hyperdualizable, we obtain x M x x E E (C). Therefore, VMID = ModC, so Nk(VhIID) = Nk(ModC). Finally, we obtain the following equalities: {x3 M x}, C3 := {3x

:=

>

c

+

6

=

N,E(E( Xi))

=

E ( C l U CX3U C3x) by Corollary 8.5.10.

i=l

The sets CX3and C3" (see Proposition 8.5.8) are given by the identities of Lemma 8.5.11,l. (i),(ii),(iii),(iv). But all of these identities are satisfied in V2 (Lemma 8.5.11,l.). Therefore IVF(IdVMID) C IdV2. Altogether, we have proved that IV:(VMID) = V2. We will now prove that IVA(VhfID) = Vl. Clearly, IdVl C IdNA(VMID).Let F1 := {2Ass., 2Med., 4Dist., x x M 2x}, r2:= {x2 M x}, r3:= {X M x}. Obviously, VMID = M o d ( r l U r2U r3),so in a similar way as we did before, we get IV?(VMIn) = M o d ( r l U rx2 U rX+").By Remark 8.5.9 the sets Fx2 and Fx+x are determined by: rx2 = {x2y = xy2 M M x + y} and rx+x = ( ~ y =) ~ xy, x2 y M x y M (x { ( x + x ) y = x ( y + y) = x y + x y M x y , x + x + y = x + y + y M x + y + x + y M x + y}. By using Lemma 8.5.11,2. and the duality principle we conclude that rx2 U rX+" C IdVl. Therefore, we have ~ ~ ~ ~ N ) .A ( V ~=~Vl. ~ ) IdVl I ~ N ~ ( V Altogether

+

+

+

+

+

>

In a similar way as we did for the variety VhfID we can also show

Theorem 8.5.13 The normalizations VENT and RA(2,21are determined by

of

the solid varieties

8 M-solid Varieties of Semirings and

Now that the problem of the normalization has been satisfactorily solved, the results will be applied in Section 8.6 t o obtain a complete description of the lattice of all pre-solid varieties of semirings.

8.6

All Pre-solid Varieties of Semirings

A pre-solid variety of semirings is a variety of semirings for which the set of all its identities is invariant under replacement of any occurrence of operation symbols by any binary term different from a variable. Such identities are called Pre-hyperidentities. More precisely, a variety of semirings is called pre-solid, if it is M-solid for the set of all Pre-hypersubstitutions of type r = (2,2). Projection hypersubstitutions of type (2,2) are hypersubstitutions a of type (2,2) such that a(f ) E {xl, x2} for all f E {F,G}. We denote by P ( 2 , 2 ) the set of all projection hypersubstitutions of type (2,2), i.e. P ( 2 , 2 ) := {a, , a, ,, a,,,,, , a,,,,,). A hypersubstitution a of type (2,2) is a Pre-hypersubstitution if a ( f ) is not a variable for all f E {F,G). Let Sr be the variety of all semirings and let C ( S r ) be the subvariety lattice of S r . Instead of SPTe(2,2)(S~) we will write P S ( S r ) . This chapter presents a proof which shows that the lattice of all pre-solid varieties of semirings contains exactly 13 elements. We will give some necessary conditions for pre-solid varieties of semirings. First, we will derive identities which are useful t o describe all pre-solid varieties of semirings. Since the hypersubstitution a ~ ( x , , x a ) , ~is( xa lPre-hypersubstitution ,xa) (i.e. belongs to P r e ( 2 , 2 ) ) , every pre-solid variety of semirings satisfies the duality principle. Moreover, we have:

Proposition 8.6.1 Let V be a pre-solid variety of semirings. Then (i) V is a variety of medial and distributive semirings.

8.6 All Pre-solid Varieties of Semirings

299

(ii) The following identities are satisfied in V : 32, = 2x1 = x,2 , x;, 2x1 2 2 2 3 = 21 x2 2 3 , x:x2x3 = 2122x3.

+ +

+ +

-

(iii) V is either normal or idempotent.

Proof: (i) In Proposition 8.1.4, we proved that every solid variety of semirings is medial and distributive. All hypersubstitutions which we needed in this proof, are Pre-hypersubstitutions. Therefore, every pre-solid variety of semirings is medial and distributive. (ii) Since every pre-solid variety satisfies the duality principle, we have only to show that 2x1 = xf = x?,xfxzx3 = x122x3 E IdV. Applying the Pre-hypersubstitution a1 (which maps F and G to G(xl, x2)) to the distributive identity

gives the identity

Applying the Pre-hypersubstitution o2 (which maps F and G t o G(xl, x l ) ) to the associative identity

we obtain G(xl, x l ) = G(G(xl, xl), G(xl, XI)). That is, x: = xf. Consequently, we have x: = xf = x; by (*). That is, xf = x:. It remains t o prove the identity 2x1 = xf. Applying the Pre-hypersubstitution a3 (which maps G to G(xl, x l ) and F t o F ( x l , x2)) to the distributive identity

we obtain x: = x: +x:. The duality principle and the distributivity give X I x1 = (xl xl)(xl xl) = 442:. It follows that XI X I =

+

+

+

+

300

8 M-solid Varieties of Semirings

x:, since x: + x f M x:. Hence, we obtain xf M 2x1. It is not difficult to show that x:x2x3 M ~ 1 x 2 E~ IdV. 3 (iii) We will prove that if V is not idempotent then V is normal. Indeed, assume that V is not idempotent and assume that the identity t = x is satisfied in V. Clearly, the variety V is not trivial because it is not idempotent. Applying the Pre-hypersubstitution 01 (which maps F and G t o G(x1,x2))t o the identity t M x E I d V and identifying all variables in the resulting identity with x, we obtain xn M x E IdV for some n 1. If n > 1, using x3 = x2 if necessary, we obtain x2 M xn M x. This contradicts the fact that V is not idempotent. Hence n = 1. This implies that t is the variable x since V is not trivial. Therefore, V is normal.

>

Proposition 8.6.1(iii) leads to the first description of the lattice PS(Sr) of all pre-solid varieties of semirings as follows:

Theorem 8.6.2 The complete lattice P S ( S r ) of all pre-solid varieties of semirings splits into two complete sublattices, the sublattice PSIdem(Sr) of all idempotent pre-solid varieties of semirings and the sublattice PSN(Sr)of all normal pre-solid varieties of semirzngs. Proof: The lattice CN(2,2) of all normal varieties of type (2,2) and the lattice Cldem(2,2) of all idempotent varieties of type (2,2) are complete sublattices of the lattice C(2,2) of all varieties of type (2,2). Therefore, since PSN(Sr)= P S ( S r ) n CN(2,2) (intersection of two complete sublattices) and since PSIdem(Sr) = PS(Sr)n CIdem(2,2) (intersection of two complete sublattices), it arises that PSIdem(Sr)and PSN(Sr)are complete sublattices. By Proposition 8.6.1 (iii) the lattices PSIdem (ST)and PSN( S r ) are disjoint and its union is P S ( S r ) . Now we characterize the idempotent part of the lattice P S ( S r ) .

Proposition 8.6.3 A pre-solid variety of semirings is idempotent if and only if it is solid. That is, the lattice PSIdem(Sr) of all idempotent pre-solid varieties of semirings is the &element chain 7 ( 2 , 2 ) C R42,2) C VBE C VMID. Proof: Clearly, every solid variety of semirings is pre-solid. Moreover, it is idempotent (Proposition 8.1.4). Conversely, let V be a

8.6 All Pre-solid Varieties of Semirings

301

pre-solid variety of semirings which is idempotent. We will show that V is solid. It is clear that Hyp(2,2) = P r e ( 2 , 2 ) U W P ( 2 , 2 ) . Each of the hypersubstitutions from the set W P ( 2 , 2 ) is equivalent (with respect to the relation w v )t o the Pre-hypersubstitutions ox:,,2 or i = 1 , 2 with t E W(2,2) (X2)because of the idempotency. Therefore, since V is pre-solid, Proposition 3.1.14(ii) shows that every identity in V is preserved by any hypersubstitution of type (2,2). That means, that V is solid. Furthermore, we have

Theorem 8.6.4 The varieties RA(2,2) and C(2,2) = Mod{xl +x2 = x3xq} are the only minimal elements of the lattice P S ( S r ) of all pre-solid varieties of semirings. Proof: First, we prove that the variety C(2,2)is a pre-solid variety of semirings. Clearly, C(2,2)is a variety of semirings. We show that C(2,2)is pre-solid. Obviously, all C(2,2)-normalforms of binary terms are X I , 2 2 and x1+x2 since the variety C(2,2) is normal. Consequently, we have to consider only the hypersubstitution which maps F and G to xl 22. But this hypersubstitution is --c(,,,, related to the identity hypersubstitution since xl x2 = ~ 1 x 2 Therefore . C(2,2) is pre-solid. Let V be a non-trivial pre-solid variety of semirings. Then Proposition 8.6.1 (iii) shows that either V is normal or V is idempotent. If V is normal, then V contains C(2,2) since it is wellknown that C(2,2)is the least normal variety of type (2,2). If not, then V is idempotent, so it is solid (Proposition 8.6.3) and we get RA(2,2) V (Theorem 8.2.2). rn

+

+

c

To offer more insight into the lattice of all pre-solid varieties of semirings, the notion of an outermost equation, which was introduced in 7.1 is needed. A variety V is called outermost, if all identities of I d V are outermost. Clearly, the equation xl x2 = x3xq is not outermost, so the variety C(2,2) is not outermost. Let V be a nontrivial pre-solid variety of semirings which is not outermost. Then V is not idempotent because all non-trivial idempotent pre-solid varieties of semirings (solid varieties of semirings by Proposition 8.6.3) are outermost. Therefore, V is normal (Proposition 8.6.1 ) and then C(2,2) C V (Theorem 8.6.4). Thus, it is clear that the variety C(2,2) is the least non-trivial pre-solid variety of semirings which is not outermost. This leads t o the following question: Are there non-trivial

+

302

8 M-solid Varieties of Semirings

pre-solid varieties of semirings which are not outermost? 1, we consider the following equations: For n

>

Theorem 8.6.5 Let V be a pre-solid variety of semirings. Then the following conditions are equivalent: (i) V is non-outermost. (ii) x:

= x;

E IdV.

(iii) For every n

> 3,

IV:

E I d V and

Nz E IdV.

Proof: (i)+(ii) Assume that V is non-outermost. Then there exists a non-outermost identity s = t E IdV. Without loss of the generality, we can assume that s starts with the variable xl and t with the variable x2. Let a be a Pre-hypersubstitution which maps F and G to G(xl, x l ) = xf. The application of a to the identity s = t E IdV, gives x y = x; E I d V with rn > 2, n > 2. Therefore, we get x,2 = x; E IdV, since x; = x: E I d V (Proposition 8.6.1 (ii)).

>

(ii)+ (iii) Let n 3. Then Proposition 8.6.1 (ii) and the presumption show that the following identities hold in V: = x:x2.. .x, 21x2.. .x, = x;x3.. .x, -" 2x2X3.. . Xn ... ... -

2

xn 2 Y, ... ... " YlY2.. . Y n E I d V follows from the duality principle. " " 7 2 .

-

IV:

(iii)+(i) This implication is obvious. Now we ask for greatest non-outermost pre-solid variety of semirings. Let V be a pre-solid variety of semirings such that there exists a least number m 4 and V satisfies the equations IVE and IV: as

>

8.6 All Pre-solid Varieties of Semirings

303

identities. Then V is not outermost and Theorem 8.6.5 shows that the identities IV; and NF belong t o IdV. Therefore, the variety v ( ~which ) is defined by

where VD is the variety of all distributive semirings, is a good candidate for the greatest non-outermost pre-solid variety of semirings. To show that v(" is pre-solid, we derive first some identities.

Lemma 8.6.6 For any term s E W(2,2)(X),with op(s) > 2, the equations xf M x i M s are satisfied as identities in ~ ( 3 ) . Proof: From the identities xf M x: and xlx2x3 M yly2y3 we obtain the identities x: M x,% x: M xi. If s is built up only by using of F or only by using of G, then there exist terms sl, s 2 and s3 such that the variety v ( ~satisfies ) the equations s M sl s2 s3 or s M ~ 1 ~ as identities since op(s) 2; w.1.g. we can assume that s M sls2s3. ~ yly2y3, x 3 we get the identities Then, using x2 M x3 and ~ ~ 2 M s M ~ 1 ~ M2 s;~ M3 x:. Therefore, we obtain the identity x: M s. If both operation symbols F and G occur in s, then there exist terms sl, s 2 and ss such that, either s M sl(s2+s3), or s M (s1+s2)s3,or s M sl s2s3,or s M s1s2 s3. Therefore, by the distributivity one can produce a sum with at least four summands and using the same idea as before, one obtains the identity s M x: E I ~ v ( " . Altogether x: M x2 M s E I ~ v ( ~ ) .

+ +

>

+

+

Theorem 8.6.7 The variety pre-solid variety of semirings.

v ( ~is) the

greatest non-outermost

>

In a similar way as we did in the proof of Theorem 8.3.3, we conclude that V is a variety of semirings. Moreover, it is presolid. Therefore, Proposition 8.6.1 and Theorem 8.6.5 show that I ~ v ( ~IdV, ) i.e., V ~ ( 3 ) .

c

c

C We show that

the variety

v ( ~satisfies ) the equations

2

~

3

304

8 M-solid Varieties of Semirings

and

as pre-hyperidentities. Let a be a Pre-hypersubstitution. Then the terms a(F) and a ( G ) have a number of operation symbols greater or equal to 1. Moreover, the equations (*) and (**) have the property that the number of operation symbols of both terms of each equation are greater or equal to 2. Then applying a to (*) and to (**) gives equations having the property that the number of operation symbols of terms are greater or equal to 2 or the terms are identically equal to x: in v(" because of the identity x: = x;. Lemma 8.6.6 guarantees that such equations are satisfied as identi. the inclusion has been proved. The equality ties in v ( ~ )Therefore, v ( ~=) V shows that v ( ~is)a non-outermost pre-solid variety of semirings. Furthermore, Proposition 8.6.1 ( (i) and (ii)) together with Theorem 8.6.5 ensure that ~ ( ( " 1is the greatest non-outermost pre-solid variety of semirings. Theorem 8.6.7 shows that every non-outermost pre-solid variety of semirings is a subvariety of the variety v ( ~ ) But . the variety v ( ~ ) is not commutative. Therefore, we consider the proper subvariety &(("I of ~ ( ( " which 1 is generated by the commutative equations. That is, K(" := v ( ~ ) ( $x X2 ~ M X2 21, X l X 2 = x2x1).

+

We obtain

Proposition 8.6.8 The variety ety of commutative semirings.

v,(~)

is the greatest pre-solid vari-

Proof: To show that &(("I is pre-solid, it is enough t o show that it satisfies the commutative equations xl + x2 = x2 + xl and xlx2 = ~ 2 x 1as Pre-hyperidentities since VJ3) = V ( ~ ) ( X ~ x2 = x2 xl,xlx2 M x2x1) and v ( ~is) pre-solid (Theorem 8.6.7). Using the fact that v,(~) satisfies the duality principle, we have only to show that the identity xlx2 = x2xl is pre-hypersatisfied in &(("I. Indeed, Lemma 8.6.6 and the inclusion &(("I c v(" show that x,2 = xi FZ t E 1 d ~ ( 0 , for t E W(2,2i(X) with op(t) 2. Thus, the application of the hypersubstitutions a such that a ( f ) E {x:, x1 2 2 , x1x2} for f E {F,G} are enough t o give our proof

+

+

>

+

8.6 All Pre-solid Varieties of Semirings

305

since VJ3) is normal. Notice that only the image of one of both operation symbols under a is interesting, since the commutative identity contains only one operation symbol. Hence, the hypersubstitutions a such that o(G) E {xf, x1 x2,x1x2) are enough. But all of these hypersubstitutions preserve the identity xlx2 = ~ 2 x 1E I~v,(("). Therefore VJ3) is a pre-solid variety of commutative semirings. Let V be a pre-solid variety of commutative semirings. Then V is nonoutermost. Moreover, Proposition 8.6.1 and Theorem 8.6.5 ensure V,(3). Altogether, v,(~) is the greatest pre-solid variety of that V rn commutative semirings.

+

c

The next problem in hand is the determination of more pre-solid varieties of commutative semirings. To solve this problem, more information about pre-solid varieties of semirings which are not outermost is needed.

>

Lemma 8.6.9 Let n 1 be a natural number. Let V be a nonoutermost and non-trivial pre-solid variety of semirings which is not commutative. Then the universe of the V-free algebra Fv(X,) freely generated by X, = {xl,. . . , x,), is { [ z ~I i ]E ~{ I ~ , . .~ . ,n ) ) ~ {[xixjlld~I 2 # j and 2 7 j E ( 1 , . . . 7n)) U {[xi + x j ] l d ~I 2 # j and i , j E (1,. . . , n)) U { [ x & ~ } .

Proof: Let T ( n ) := { [ x ~ I ] i ~ E~ {~I , . . . , n)) u { [ z ~ x I ~ ] ~ ~ ~ z # j and z,j E { I , . . . , n ) ) U {[xi xjlIdV I i # j and z,j E (1,. . . , n}} U { [ x : ] ~ ~ ~We ) . want to show that the elements of T ( n ) are pairwise different. Assume that V is not normal. Then V is idempotent (Proposition 8.6.l(iii)). Moreover, V is solid (Proposition 8.6.3). Since V is not outermost, then V has to be trivial, but in fact V is not trivial. It follows that V is nor[xi xjlIdV contains mal. Therefore, none of the classes = {xi}, with i, it,j E { I , . . . ,n}. Assume that Xi,, and xi x j = x: E IdV,i # j and i , j E { I , . . . , n ) . Then by using the identity x1 + x1 = xf E I d V and the duality principle, the following implications hold: xi x j = xf E I d V + xixj M XI X I E I d V + xixj M x: E IdV. Since V is not outermost, x: M x2 E I d V (Theorem 8.6.5). Hence xixj M x: gives 21x2 M x: and xi x j M x: gives x3 2 4 M x:. Therefore xlxz M x: M x3+x4 E IdV. This contradicts the fact that

+

+

+

+

+

+

+

306

8 M-solid Varieties of Semirings

V # C(2,2).Assume that xi+xj = xk+xl E IdV, i # j, k # 1 and (i # k or j # 1). W.1.g. assume that i # k . Then the substitutions xi H x l , xj H 2 2 and xk H 2 2 give xl +x2 = x2 + x l E IdV. If 1 = 1, this leads to the fact that the variety V is commutative but it is not commutative by hypothesis. If l # 1 then the substitution xl H x2 gives xl x2 = 2x2 = x: E IdV. But we have seen earlier that this equation cannot be satisfied as an identity in V. Now assume that xi x j = xkx1 E IdV, i # j, k # 1. Then applying the Pre-hypersubstitution defined by F H xl x2 and G H x:, we have xi xj = x; = x: E IdV,

+ +

+

+

that is, Xi

+ = Xj

X1

E IdV.

This leads also t o the contradiction V = C(2,2).Altogether, by the duality principle, we conclude that no two different classes of T ( n ) are collapsing. Finally, by Lemma 8.6.6, we conclude that for every t E W(2,2) (X,) there exists [t']ldV E T ( n ) such that t = t' E I d V since V is normal. Therefore, the lemma has been completely proved.

>

L e m m a 8.6.10 Let n 1 be a natural number. Let V be a nontrivial pre-solid variety of commutative semirings which is diflerent from C(2,2).Then the universe of the V-free algebra Fv(Xn)freely E { ~I ,~. . . , n)) u generated by Xn = { x l , . . . , x,), is { [ z ~I ]2 ~ {[xixj]IdV I 2 < j and 2 , j E { I , . . . ,n)) U {[xi xj]IdV I 2 < j and 2, j E (1,. . . , n)) U {[z:]I~v).

+

Proof: Since V is commutative, V is not outermost and the proof can be done in a similar way as for Lemma 8.6.9. T h e o r e m 8.6.11 There are exactly two non-trivial pre-solid varieties of commutative semirings, namely the varieties C(2,2)and ~ ( ~ 1 ) . Proof: Let V be a non-trivial pre-solid variety of commutative semirings different from C(2,2).Then V

= = =

HSP({FV(Xn) I n E ~ \ { 0 ) ) ) (section 1.4) H S P ( { FVc( 3 , (X,) I n E N\{O))) (Lemma 8.6.11) v,(~) (section 1.4).

8.6 All Pre-solid Varieties of Semirings

307

The last point t o consider in this section is the determination of all non-outermost pre-solid varieties of semirings. T h e o r e m 8.6.12 The lattice of all non-outermost pre-solid varieties of semirings is the 4-element chain 7 ( 2 , 2 ) c Ci2,2)c I/;j3) c v(3). Proof: Let V be a non-trivial pre-solid variety of semirings which is not outermost. If V is commutative, then V is either equal t o ~ 1 8.6.11). If V is not commutative, C(2,2)or equal t o ~ ( (Theorem then we obtain HSP({Fv(Xn) I n E N\{O}}) (section 1.4) HSP({FV(3,(Xn)I n E ~\{0}}) (Lemma 8.6.10) = v ( ~(section ) 1.4). ) the lattice Therefore, the chain 7 ( 2 , 2 ) c C(2,2jc ~ ( c ~v (1 ~forms of all non-outermost pre-solid varieties of semirings. V

= =

Up t o now, we have determined all idempotent pre-solid varieties of semirings and all non-outermost pre-solid varieties of semirings. The join VENTV C(2,2)of two elements of these classes is not outermost and also not idempotent. Therefore, there are more pre-solid varieties of semirings. Our aim now is t o determine all of them. We will show that the lattice of all pre-solid varieties of semirings is finite. Firstly, it is important to determine a generating system of the greatest pre-solid variety of semirings. Let Vgp be the class of all algebras of type (2,2) such that the associative and distributive laws are satisfied as pre-hyperidentities, i.e., Vgp = H ~ ~ e ( 2 , 2M0d{G(x7 ) G ( ~ 7 z)) G(G(x7 Y), z), G ( x 7 F ( ~ 7 z)) F ( G ( x , y), G(x, 2))). By the same method as we did in Theorem 8.3.3, we conclude that V, is a variety of semirings. Therefore, it is the greatest pre-solid variety of semirings. To get a generating ) . do this, we system of Vgp,we will prove that \:$p = VICIIDV ~ ( ~ 1To . need a technical lemma which describes all identities in v ( ~ ) Let F1 := NF(2,2). Let r2be the set of all equations s z t E W(2,2) (X)2 such that op(t) 2, op(s) = 1 and such that s contains only one 2, op(t) = 1 and t contains only one variable. variable or op(s) such that Let I73 be the set of all equations s z t E W(2,2)(X)2 op(t) = op(s) = 1 and each of both terms s and t contains only one

> >

308

8 M-solid Varieties of Semirings

variable. Then we obtain 3

Lemma 8.6.13 The set U ri is equal to the set I ~ v ( " . i=l

>

Proof: Clearly, Lemma 8.6.6 shows that rl C I ~ v ( " . Using the ~ )get, also that same lemma and the identity x x = x2 E I ~ v (we E2 U r3C I ~ v ( ~ ) .

>

+

Since the generating system of 3

the proof of the fact that U i=l

ri

I ~ v (is~a) subset

3

of U ri, i=l

is an equational theory leads 3

to the conclusion that

U) r i . I ~ v (C~i=l

Thus, we will show that

3

U Fi is closed under the five derivation rules. Indeed, the reflexi=l

ivity and the symmetry are clear. The application of the substitution rule to a given equation s = t gives an equation sf = t' such that op(tf) op(t) and op(sf) op(s), thus the substitution rule also holds. For the replacement rule, let sj FZ ttj t

>

3

U Fi, j

=

>

1,2. If there exists a number jo E {1,2) such that

i=l " -

>

>

sj0# tjo, then op(sjo) 1 and op(tjo) 1. Consequently, we obtain 2 for f E { F , G). That means op(f (tl,t2)) 2 and op(f (sl,s 2 ) ) that f (sl,s2) = f (tl,t2) E rl for f E {F,G). Otherwise, sj = t j for j = 1,2. Thus, for every f E {F,G), we have f (sl,s 2 ) = f (tl, t2) and f (sl,s2)= f (tl, t2) E r l . It is left to show the transi-

>

tivity. Let s

= t and t = t'

(i) s = t a n d t = t ' r~l .

>

3

E

U Fi. We consider the following cases:

i=l

Then s = t ' € F1.

(ii) s = t t El and t = t' E E2. If s = t , then obviously s = t = t' E E2. If not, then necessarily op(tf) = 1 with t' containing only one variable and op(s) 2. Therefore s = t' € F2.

>

This means that s (iii) s FZ t t El and t = t' E E3. t since op(t) = op(tf) = I , so s = t' E F3. (iv) s = t t E2 and t = t' E r l . This means that t' El and t = s t E2. By (ii) we have t' = s t E2.

=t

=

t

8.6 All Pre-solid Varieties of Semirings

309

>

(v) s = t E r2and t = t' E r 2 . If op(t) = 1 then op(s),op(tt) 2 and s = t' E rl.If op(t) 2, then op(s) = op(tt) = 1 with s and t' containing only one variable, hence s = t' E r3.

>

(vi) s = t t E2 and t E t' E r3. Then necessarily op(s) 2, op(tt) = 1 and t' contains only one variable, hence s = t' E E2.

>

and t = t' E El. For reasons of symmetry and by (vii) s = t t the use of (iii), we get s = t' E F3. (viii) s = t E r3and t = t' E F2. For reasons of symmetry and by the use of (vi), we get s = t' E r 2 . (ix) s = t E F3 and t s = t t E r3.

= t'

E r3.

Then obviously, we obtain

Corollary 8.6.14 Let s = t E W(2,2)(X)2. Then s and only if one of the following statements holds: (i) op(s) op(t)

= t @ I ~ v (if ~ )

> 2 and op(t) = 1 with t containing two variables or > 2 and op(s) = 1 with s containing two variables.

(ii) op(s) = op(t) s # t.

=

1 with s or t containing two variables and

>

(iii) (op(s) = 0 and op(t) 0 and op(s) 0 ands # t ) .

>

0 and

s

#

t) or ( ~ ( t )=

Proof: By the definitions of Fi, i E {1,2,3), we have s = t @ I ~ v ( if and only if one of the following statements holds: a) op(s) = 0 and op(t) 0 and s # t. b) op(s) = 1 and op(t) = 0. c) op(s) = 1 and op(t) = 1 and s # t and s or t contain two variables. d) op(s) = 1 and op(t) > 2 and s contains two variables. e) op(s) > 2 and op(t) = 1 and t contains two variables. f ) op(s) 2 and op(t) = 0. Case a) and case b) together with f ) lead to (iii). Case c) is (ii).

>

>

~)

310

8 M-solid Varieties of Semirings

Case d) and case e) lead to (i). Now all the tools are in hand t o prove

Theorem 8.6.15 The join VMID V v ( ~is) equal to VMD(x2yx= xyx, 2x y x = x y x,3x = 2x = x2 = x3) and it is the greatest gre-solid variety of semir.ings7 that is7 = VMIDV v ( ~ =) V M D ( ~ 2Myx~y x , 2 x + y + z M x + + + x 7 3 x = 2x M x2 M x3), where VMD is the variety of all medial and distributive semirings.

+ +

+ +

4,

+

Proof: First we show that VMID V v ( ~=)VnrD(x2yx= xyx, 2x y z = x y z, 32 = 2x M x2 M x3). Let W := V M D ( ~ 2=y ~ x y z , 2 x + y + z = x + y + x 7 3 x M 2x = x 2 = x 3 ) . Clearly, Id(VMID V ~ ( ~ 1 )IdW. For the converse inclusion, we start with the equalities:

+

+ +

>

3

=

U ( I ~ V M I Dn r i ) . i=2 I ~ N , A ( v ~ ~c~ )IdW. Let s = t

I~N,A(VMID) U

By Theorem 8.5.12, we have E IdVMID n r 2 . Then w.1.g. we can assume that t E {x x, x2). For t = x x, using that VMID is regular (Theorem 8.2.2 and Theorem 8.3.3), we conclude that the term s contains only the variable x. Furthermore, since op(s) 2 and 32 = 2x M x2 = x3 t I d W , we get s M x x t IdW. In a similar way, we also get s M t t I d W for t = x2. Hence, we have IdVMID n r2 IdW.

+

+

+

>

c

The set IdVMIDn F3 consists of the identities s M t t IdVhfID such that op(s) = op(t) = 1 and each of terms s and t is built up only by one variable. Thus, the regularity property of VMID and the identity x x = x2 E I d W guarantee that IdVMID n F3 IdW. Now that we have proved Id(VMIDV ~ ( ~ C 1 I) d W , we conclude that VMID V v(" = W. Thus, as join of two pre-solid varieties of semirings, W is a pre-solid variety of semirings. By Proposition 8.6.1 every presolid variety of semirings is a subvariety of W. Therefore, it is the

+

c

8.6 All Pre-solid Varieties of Semirings

311

greatest one. In addition, V, is also the greatest pre-solid variety of semirings (see the remarks after Theorem 8.6.12). Altogether, we obtain Vgp = VMID V Vi3) = W. Similarly as Theorem 8.6.15 we can also prove

Remark 8.6.16 VENT V xx yw).

+

vi3)= (VMIDV via)((x + y) (x + w)

M

Having determined the greatest pre-solid variety Vgp of semirings, we will now be interested in all subvarieties of V, which are presolid. We start with the following fact. It is well-known that in the case of semigroups, the set of all identities of the variety R B = RA2 (generated by all projection algebras of type (2)) is the set of all outermost equations of type (2). That is, the variety R B = RA2 is the least outermost variety of type (2) ([36]),but this is not true for varieties of semirings, because the equation xy z M x x is outermost but is not satisfied in RA(2,2).To prove a similar result in the case of varieties of semirings, we need a new concept.

+

+

Definition 8.6.17 A variety V of semirings is s-outermost, if for any identity of the form

si

+ ...+ < <

S, M

tl + . . . + t,,

E IdV,

< <

where si and t j (1 i n, 1 j n') are variables or products of variables, we get that the equations

s1 M tl and s,

M

t,t

M

t,t

are outermost, i.e., the identities

s1 M tl and s, belong t o IdRAi2,2). Now we can prove

Proposition 8.6.18 The variety RAi2,2) is the least s-outermost variety of semirings.

312

8 M-solid Varieties of Semirings

Proof: Since IdRA(2,2) is the set of all identities satisfied by all projection algebras, the variety RA(2,2)is s-outermost. We now show that every s-outermost variety of semirings contains RA(2,2).Let V be an s-outermost variety of semirings and let s M t E IdV. By the distributive laws, we can find variables or products of variables ti, sj, 1 i n , 1 j n', such that the identities

< <

< <

belong to IdRA(2,2)n IdV. Since s

M

t

E IdV, we get

Therefore, we will deduce that the equations sl M tl and s, M t , ~ are outermost because V is s-outermost. That is, the identities sl

M

tl and s,

M

t,~

+ + +

belong t o IdRA(2,2).Thus, we get in RA(2,2)the identities sl s, M tl t , ~ and sl . . . s, M t l . . . t,~, since x y z M x z E IdRA(2,2).Altogether, using the idempotency in the cases n = 1 or n' = I , we conclude that s M t E IdRA(2,2).

+ +

+

+

+

+

Corollary 8.6.19 Let V be a variety of semirings. Then the variety V V RA(2,2)is the least s-outermost variety containing V. Proof: From I d ( V V RA(2,2))= I d V n IdRA(2,2)and the fact that RA(2,2)is s-outermost, we get obviously that V V RA(2,2)is soutermost. Let W be a s-outermost variety of semirings containing V. Then Proposition 8.6.18 ensures that RA(2,2) C W. Therefore V V RA(2,2) C W . Similarly as it was done in the proof of Proposition 8.6.19, one can show that every s-outermost variety is outermost. The converse is not in general true. Nevertheless, we have

Lemma 8.6.20 An outermost pre-solid variety of semirings is soutermost. Proof: Let V be an outermost pre-solid variety of semirings and let tl . . . t, M s l + . . . S,/ E IdV,

+ +

+

8.6 All Pre-solid Varieties of Semirings

< <

313

< <

where ti and sj ( 1 i n, 1 j n') are variables or products of variables. Using the hypersubstitutions which map F t o the binary term 25, i = 1,2 and G to G ( x l ,x 2 ) ,we obtain

and

si

3

ti, E IdV

Since V is outermost and the terms t l ,s l , t, and s , ~are variables or products of variables, the identities tl 3 sl and t , 3 s , ~are also outermost. Therefore V is s-outermost. The following facts are helpful in giving a complete description of the lattice P S ( S r ) .

v(" is determined by: R A ( 2 , 2 j V ~= ( 3v )~ ( x + ? J + x3 3 x 3 2 x 3 x 2 3 x 3 ~ X X X x, 2 y z z x y z , x + x + y + z 3 x + y + z ) , where VD is the variety of all distributive semirings.

Proposition 8.6.21 The gre-solid variety RA(2,2)V

x2 3 x3 3 x z x , x 2 yz 3 x y z , x + x + y + z 3 x + y + z ) . Clearly IdV' c I d ( R A ( 2 , 2V) ~ ( ~ 1 ) . In a similar way as we did in the proof of Theorem 8.6.15, we have Proof: Let V' := VD(x+y+x

3

3x

3

2x

3

With the aim of having a generating system of I d N k ( R A ( 2 , 2 1we ), determine first a generating system of IdRA(2,2)which satisfies the conditions of Corollary 8.5.10. Let 0 := {2Ass., 4Dist., x2yz 3

XyZ, X + X + y + x

3

X + y + z , XyX

3

X+z+X,

X3 3 X , X 3

32).

We will prove that E ( R ) = IdRA(2,2).Clearly, all identities of R hold in R A ( 2 , 2 )so , E ( R ) IdRA(2,2).The converse inclusion will be true by proving that the identities x 3 x 2 ,xyz 3 x z belong t o E ( a ) , since the duality principle holds in the variety generated by a . Indeed,

c

314

8 M-solid Varieties of Semirings

and

(a)

Then we get x4 FZ x3 3 E ( O ) . Hence x2 FZ x t E (by using ( 1 ) ) . We are going to prove the identity xyx = xx. The following identities belong to E ( R ) : xx = ( x y x x ) x M x y ( x x x ) M xyx by and C" from Corollary 8.5.10. We have (2). Now we form = {X3y M X Y 3 M ( x ~ X) 3 ~ f , = X+y3 = ( X + ? J ) ~ } U { = ~ ~t I t t W(2,2) ( X ) and op(t) 2 ) and C3" = { ( 3 x ) yFZ x ( 3 y ) FZ 3 ( x y ) ,3x y = x 3y = 3 ( x y ) ) U {3t = t 1 t t W ( 2 , 2 ) ( Xand ) op(t) 2). Corollary 8.5.10 shows that a generating system of I d N e ( R A ( 2 , 2 1 ) is given by the union of the 2-normal part of R and U C3x. Similarly as we did in Lemma 8.5.11 and Theorem 8.5.12, we get I ~ N & ( R A ( ~ C , ~ IdV'. )) The set r3 n IdRA(2,2jconsists of the identities s FZ t t IdRA(2,2)such that op(s) = op(t) = 1 and the terms s and t are built up only by one variable. Thus, the outermost property of RA(2,2)guarantees that the equation s = t is built up by the same variable x . Then the equation s M t is one of the following equations:

zx3

zx3

+

+

>

>

+

zX3

But all of the aforementioned identities belong t o IdV'. Therefore r3n IdRA(,,,) IdV'.

c

+ + + > >

Consider r2n I d R A p 2 ) . Let s = x x E F 2 n IdRA(,,,), with op(s) 2. Then by distributivity, there exist variables or products . . . s, E IdRA(2,2)n IdV' with of variables such that s FZ sl n 1 and op(sl . . . s,) 2. If n = I , then s = sl M x x M x2 E IdRA(2,2).Moreover, we have s = sl = xxi, . . . ximx M x2 E IdRA(2,2),with m I , since op(sl) 2 and RA(2,2)is outermost. Thus, we get s = sl = xxi, . . . ximx = x2 M x x E IdV' because xyx FZ x x t IdV'. If n 2 then from sl . . . s, = x x t I d R A ( 2 , 2 )it, follows that both terms sl and s, start and end with x. Moreover, using the identities xxx M 2x = x2 M x y x E IdV' , we get sl s, M x x E IdV' and sl . . . s, = x x E IdV'. Hence s = sl . . . s, FZ x x t IdV'. Therefore, we get the inclusion F 2 n IdRA(2,2)C IdV' since x x FZ x2 E IdV'. This finishes the proof of the fact that IdV' I d ( R A ( 2 , 2V) ~ ( ~ 1 ) .

>

>

+

+

>

>

+

+

+ + +

+

+ + + + + + + + + >

+

+

8.6 All Pre-solid Varieties of Semirings

315

Lemma 8.6.22 The variety RA(2,2)V v ( ~is) the only outermost ) which is not pre-solid variety of semirings which contains v ( ~and regular. Proof: Let V be an outermost pre-solid variety of semirings which ) which is not regular. We shall show that V = contains v ( ~and RA(2,2jV ~ ( ~ Clearly, 1 . the variety V is normal since Vi3) j V. From the outermost property of V, it follows that V is s-outermost (Lemma 8.6.20). Moreover, the variety RA(2,2)V v ( ~is)the least soutermost variety containing v ( ~(Corollary ) 8.6.19). It follows that V V ( ~C) V. For the converse inclusion, by Proposition 8.6.21 and by the duality principle and also by the fact that V is pre-solid, it is enough t o prove that zyz = x2 E I d V (see Proposition 8.6.1). Indeed, since V is not regular, there exists an identity s = t E I d V such a variable y occurs in s but not in t . Identifying in s and t all variables different from y by x, we obtain the identity st FZ tt E IdV. Applying the hypersubstitution a,,,,, to st FZ tt E IdV, we get x n ym z = zp, where n , m > I , p > 2, by the outermost and the medial properties of V. Since V is pre-solid, using the identities of Proposition 8.6.1, we get xyx = x2 E IdV. Lemma 8.6.23 Let V be a regular and outermost pre-solid variety of semirings such that v(" ) V. T h e n V = (V n VMID) V v ( ~ ) .

Proof: Clearly, we have V = V V v ( ~2)(V n VMID)V v ( ~because ) vMID) v ~ ( 3 )I ~ ( Vv v("), of v ( ~ )V. ~t is left to prove I ~ ( (nv i.e., we have to show IdNk(VnVMID)~(Id(vnvMID)nr2)~(Id(vn VMID) n r3)C I d I V ~ ( V U ) (IdV n F2) u (IdV n r 3 ) . Indeed, as a pre-solid variety of semirings, V is a subvariety of V,, C IVf(VMID) (see the proof of Theorem 8.6.15). It follows that I ~ N ~ ( v ~ ~ ~ IdV. This implies that IdNk(VMID) I d N k ( V ) i.e. Nk(V) N k (VhIID). Moreover, N k ( V n VMID) = Nk (V) n Nk (VhIID) = N&(v) by Remark 8.5.5. Therefore I d ( ( V n VhfID) V v ( ~ )=) I ~ N ~ (uv( )I ~ ( v ~ v nr2) M ~nD ( I)~ ( v ~ v m M3 )~. Thus, D ) we have t o prove that I ~ N ~ ( v ) u ( I ~ ( v ~ v ~ I I D ) ~ ~ ~C )u(I~ I d N . ( V ) u (IdV n r2)U (IdV n F,). The set I d ( V n VMID) n r2 is the set of all identities from I d ( V n VMID) having the forms s = x + x or s = x2, where o p ( s ) 2. Consider s = x x E I d ( V n VMID). From the regularity of V and VMID (see Theorem 8.2.2, Theorem 8.3.3 and the s resumption), it follows that the vari-

c

c

c c

c

>

+

316

8 M-solid Varieties of Semirings

is regular. Thus, the term s contains only the variable ety V n V M I D x. Furthermore, since 3x = 2x = x2 = x3 E IdV (V is pre-solid) and op(s) 2, we get s = x x E IdV n F2. The same goes for s = x2. Therefore, we get ( I d ( V n V M I Dn) ) IdV n r 2 . Using again the outermost and the regularity properties of V and of VMID we see that I d ( V n V M I Dn)F3 is the set of all identities of the form x + x ~ x + x , x + x = x 2 , x 2 = x + x o r x 2 2~ x .

>

+

r2)

Corollary 8.6.24 Let V be an outermost pre-solid variety of semirings such that v(" ) V . Then there exists a solid variety S of semirings such that V = S V ~ ( 3 ) Moreover, . the variety V belongs to the chain RA(2,2)V ~ ( c ~VENT 1 Vv(~ c VMID ) V~ ( ~ 1 . Proof: If V is not regular then Lemma 8.6.22 shows that V = RA(2,2) V ~ ( ~ 1If ) V. is regular then by Lemma 8.6.23, we have V = (V n VhIID) V ~ ( ~But 1 . the variety V n VMIDis an idempotent pre-solid variety of semirings, so it is solid (Proposition 8.6.3). Moreover, the variety V n VMIDis one of the varieties RA(2,2),VENTor V j I Dbecause V is outermost. It is left t o show 1 , V ~ ( and ~ VMID 1 V v(~) that the varieties RA(2,2) V ~ ( ~ VENT are pairwise different. Indeed, the variety RA(2,2)V ~ ( is ~ dif-1 V v(~ since ) RA(2,2)V Vi3)is ferent from VENTV v(" and VMID not regular but VENTV v ( ~ and ) VMIDV v ( ~are) regular. Finally, VENTV v ( ~ and ) VMIDV v(" are different since the entropic equation ( x y) (z u) = xz yu holds in VENT Vv ( ~ but) not in V M I D .

+

+

+

From the previous corollary, the following problem arises: Having an outermost pre-solid variety V of semirings, under which condition does one obtain ) V ? An answer is given as follows:

v(~)

Lemma 8.6.25 Let V be an outermost pre-solid variety of semirings such that the equation x 2 y = xy does not hold in V as an identity. Then we obtain v ( ~ )V ). Proof: We shall show that IdV C I ~ v ( i.e., ~ ) s, = t # I ~ v (implies " s = t $ I d V . We provide the proof by contradiction. Assume that s = t # I ~ v and ( ~s )= t E I d V . As s = t # I ~ v ( ~ ) , by Corollary 8.6.14 and for reasons of symmetry, we have t o consider the following cases:

8.6 All Pre-solid Varieties of Semirings

>

317

(i) op(s) 2 and op(t) = 1 with t containing two variables. W.1.g. we can assume that t E {x y, xy). Then since V is pre-solid, we get 6[s]= 6[t] E IdV, where a is the hypersubstitution which maps F and G to zy. Furthermore, using the outermost property of V 2, there together with the medial laws and the fact that op(s) 2 and m I ) or (n exist natural numbers n and rn with ( n 1 and m 2) such that xmyn = xy E I d V by identifying all variables in 6[s] different from y by x. The identity x2yz == XYz, from Proposition 8.6.1, ensures that x2y = zmyn (since V is presolid). This leads t o the contradiction that z2y = zy E I d V because z2y = zy $ IdV.

+

>

>

>

>

>

(ii) op(s) = op(t) = 1 with s # t and t or s contains two variables. Then since V is outermost, w.1.g. we have s, t E {z y, zy). Furthermore, we get in V, z y = zy. Thus V = C(2,2)(see the proof of Lemma 8.6.9). This contradicts that V is outermost.

+

+

>

(iii) op(s) = 0 and op(t) 0 and s # t. From x2y = xy # IdV, it follows that V is not idempotent. Then V is normal (Proposition 8.6.1). This contradicts the fact that s = t E I d V since V is nor~ ) s = t $ IdV. m mal. Altogether, we obtain s = t $ I ~ v (implies We now have all the tools at our disposal to prove the main result of this chapter.

Theorem 8.6.26 Let V be a variety of semirings. Then V is presolid if and only if (i) V is solid or V is a normalization of a solid variety of semirings, or (ii) V

=

~ ( ~or1 ,

(iii) there exists a solid variety S of semirings such that V v(3).

=

SV

Proof: + Let V be a pre-solid variety of semirings. We consider the following cases: (i) x2y M xy E IdV. As a pre-solid variety of semirings, V satisfies the duality principle. Then we get x x y FZ x y E IdV.

+ +

+

318

8 M-solid Varieties of Semirings

Therefore, using the identities of Proposition 8.6.1, we conclude that I ~ ( I V ~ ( V ~ ~IdV, ~ ~ ) )i.e., V C IVn(vhIID) (see Theorem 8.5.12). If V is idempotent. Then V is solid (Proposition 8.6.3). If not, then V is normal (Proposition 8.6.1) and we have the equalities: IVA(Vn VhfID) = IVA(V)n IVA(VMID)(Remark 8.5.5) = V n IVA(VMID)since V is normal = v since V N~(VM~~). Moreover, as an idempotent pre-solid variety of semirings, V nVhfID is solid. Therefore, V = IVA(Vn VMID) is a normalization of a solid variety V n VMID.

c

(ii) x2y s x y @ IdV. If V is not outermost then V = 7 ( 2 , 2 ) or V = C(2,2) or V = vL3)or V = v ( ~(Theorem ) 8.6.13). But the identity i~~~ M xy holds in 7 ( 2 , 2 ) and C and does not hold in ~ (and 3~ ( (Corollary 8.6.14). Therefore V = 7/,(3)or V = V(li). If V is outermost then Lemma 8.6.25 shows that v ( ~C)V. Furthermore, by Corollary 8.6.24 there exists a solid variety S of semirings such that v = Sv ~ ( 3 ) . +== This is clear by Proposition 8.6.3 and Theorem 8.6.12 and also

by the use of the fact that the join of two pre-solid varieties is also a pre-solid. We remember that VENT= VBE. In summary, the lattice of all presolid varieties of semirings is the lattice represented by the picture below:

~

1

8.6 All Pre-solid Varieties of Semirings We will use the following notation: ass^ := G(G(x,y), 2) FZ G(x7G(y7z)), IdemG := G ( x , x ) FZ x, := G(x, F ( y , z)) FZ F(G(x, y), G(x, z)), lDist E N T : = G ( F ( x ,y), F ( z , u)) = F(G(x7 4,G(Y,u)), CanG := G(G(x,y), z) FZ G(x, 2). a pre-hyperidentity baFor each of the varieties v(" and vMIDvv(" sis was given (see the proof of Theorem 8.6.7 and Theorem 8.6.15). For the other non-trivial pre-solid varieties of semirings, we have

Theorem 8.6.27 The following pre-solid varieties are determined by: (i) VhIID = HP,e(2,2) Mod{AssG, 1Dist, IdemG}. (ii) VENT= HPre(2,2) Mo~{AssG,1Dist, id em^, E N T ) . (iii) RA(2,2)= Hp,e(2,2) Mod{AssG, lDist, IdemG,CanG)

vJ3)= Hp,e(2,2)Mod{AssG,N,6, lDist, G(x, y) G(y, 2)). IDbst, E N T ) . (ix) VENTV v ( ~=)Hpre(2,2jMod{AssG,

(viii)

FZ

Proof: (i), (ii) and (iii) can be proved in a similar way as we did in Lemma 8.3.2 and Theorem 8.3.3. In (iv), there is nothing to prove. (v) Using Theorem 8.5.12 and Proposition 8.6.1, it is clear that ,s G : x), Y) FZ Id(vh11~V C(2,2))C I ~ ( H P T ~ ( ~ , ~ ) M o l~D{zA~ ~tG(G(x7

320

8 M-solid Varieties of Semirings

c

G(x, y)}). Since {AssG,l D i s t , G(G(x, x), y) RZ G(x, y)) Id(VMIDvC) and since VhfIDVC(2,2)is pre-solid, we have Id(VhfIDV C(2,2)) 2 Id(H~re(2,2)Mod{AssG, lDist,G(G(x, X ) Y) RZ G(x, Y)}) (vi) and (vii) can be done in a similar way. (viii) can be proved in a similar way as we did for 8.6.8).

v ( ~(Theorem )

(ix) and (x) can be proved in a similar way as we did for VMrDV V ( ~ ) (Theorem 8.6.15). Using Theorem 8.6.27 and Lemma 8.4.7 , one will be able to determine the pre-hyperidentity basis for every subvariety of VhfIDV v ( ~ ) as we did for D in the case of hyperidentity basis.

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Glossary O ( A ). . . . . . . . . . . . . . . . . . . . . . . 1 o n ( A ). . . . . . . . . . . . . . . . . . . . . . 1 fAIB . . . . . . . . . . . . . . . . . . . . . . . 1 A0 . . . . . . . . . . . . . . . . . . . . . . . . . 1 ( A ;F A ) . . . . . . . . . . . . . . . . . . . . 1 ( A ;(f,A ) i E I ) . . . . . . . . . . . . . . . . 2 ( f , A)iE1 . . . . . . . . . . . . . . . . . . . . . 2 Alg(r) . . . . . . . . . . . . . . . . . . . . . 2 r := (ni)iEI . . . . . . . . . . . . . . . . .2 A{., y } . . . . . . . . . . . . . . . . . . . . .4 V { x ,y } . . . . . . . . . . . . . . . . . . . . .4 A B ........................4 V B ........................4 ..................... 5 ( X ) , . . . . . . . . . . . . . . . . . . . . . . .5 Sub(B). . . . . . . . . . . . . . . . . . . . . 6 Aut (A). . . . . . . . . . . . . . . . . . . . . 8 E n d ( A ). . . . . . . . . . . . . . . . . . . . 8 i d A . . . . . . . . . . . . . . . . . . . . . . . . .8 Con (A). . . . . . . . . . . . . . . . . . . . 8 AA . . . . . . . . . . . . . . . . . . . . . . . . . 8 V A. . . . . . . . . . . . . . . . . . . . . . . . . 8 A18 . . . . . . . . . . . . . . . . . . . . . . . 10 nat ( 8 ). . . . . . . . . . . . . . . . . . . . . 10 f;ilB . . . . . . . . . . . . . . . . . . . . . . 1 0 ker h . . . . . . . . . . . . . . . . . . . . . 10 nat(ker h ) . . . . . . . . . . . . . . . . 10 Coni,, (A). . . . . . . . . . . . . . . . 11 A~. . . . . . . . . . . . . . . . . . . . . 11

~ c a

n

j EJ

AJ . . . . . . . . . . . . . . . . . . . . . . . . 12 . . . . . . . . . . . . . . . . . . . . . . . . . 12 Al x . . . x A,. . . . . . . . . . . . . 12 Q1 o Q2 . . . . . . . . . . . . . . . . . . . . . 12 X , . . . . . . . . . . . . . . . . . . . . . . . . 16

pk

W,"" ( X n) . . . . . . . . . . . . . . . . . 19 W T ( X ). . . . . . . . . . . . . . . . . . . . 19 & ( X ) . . . . . . . . . . . . . . . . . . . . 19 W T ( X n .) . . . . . . . . . . . . . . . . . . 19 F T ( X n ). . . . . . . . . . . . . . . . . . . 19 S n ( t 7 t 1.7. . . t, ) . . . . . . . . . . . . 19 S n . . . . . . . . . . . . . . . . . . . . . . . . 19 . . . . . . . . . . . . . . . . . . . . . 19 n - clone r . . . . . . . . . . . . . . . 20 S"lA . . . . . . . . . . . . . . . . . . . . . . . 21 S2A . . . . . . . . . . . . . . . . . . . . . . . 21 eni lA . . . . . . . . . . . . . . . . . . . . . . . 21 N+ . . . . . . . . . . . . . . . . . . . . . . . . 21 N . . . . . . . . . . . . . . . . . . . . . . . . . 21 ( C ) . . . . . . . . . . . . . . . . . . . . . . . 21 t A. . . . . . . . . . . . . . . . . . . . . . . . . 21 wT(xJA . . . . . . . . . . . . . . . . . . 21 w, ( x ). . .~. . . . . . . . . . . . . . . 21 T ( A ). . . . . . . . . . . . . . . . . . . . . . 21 T ( ,) (A). . . . . . . . . . . . . . . . . . . 21 A + s = t . . . . . . . . . . . . . . . . . 22 K + s = t . . . . . . . . . . . . . . . . 22 I d A . . . . . . . . . . . . . . . . . . . . . . . 22 I d K . . . . . . . . . . . . . . . . . . . . . . 22 F K ( X ). . . . . . . . . . . . . . . . . . . . 23 ModC . . . . . . . . . . . . . . . . . . . . .24

Glossary

S ( K ). . . . . . . . . . . . . . . . . . . . . .25 H ( K ) . . . . . . . . . . . . . . . . . . . . . 25 P ( K ) . . . . . . . . . . . . . . . . . . . . . 25 I ( K ) . . . . . . . . . . . . . . . . . . . . . . 25 P s ( K ) . . . . . . . . . . . . . . . . . . . . 25 V ( K ). . . . . . . . . . . . . . . . . . . . . 26 C ( r ). . . . . . . . . . . . . . . . . . . . . . 26 I ( r ) . . . . . . . . . . . . . . . . . . . . . . 26 C ( V ). . . . . . . . . . . . . . . . . . . . . . 27 C F S = t . . . . . . . . . . . . . . . . . .27 C k s = t . . . . . . . . . . . . . . . . . 28 P ( A ). . . . . . . . . . . . . . . . . . . . . . 29 LC . . . . . . . . . . . . . . . . . . . . . . . . 30 C L . . . . . . . . . . . . . . . . . . . . . . . . 30 K L ( Y ). . . . . . . . . . . . . . . . . . . . 31 S , . . . . . . . . . . . . . . . . . . . . . . . . 32 Fix(cp) . . . . . . . . . . . . . . . . . . . . 32 cps . . . . . . . . . . . . . . . . . . . . . . . . 32 L

V . . . . . . . . . . . . . . . . . . . . . . . . . 32

clone r . . . . . . . . . . . . . . . . . . . . 51 H y p ( r ) . . . . . . . . . . . . . . . . . . . 53 6[t]. . . . . . . . . . . . . . . . . . . . . . . 53 7, . . . . . . . . . . . . . . . . . . . . . . . . . 53 (a, . . . . . . . . . . . . . . . . . . 54 O h . . . . . . . . . . . . . . . . . . . . . . . . . 54 aid . . . . . . . . . . . . . . . . . . . . . . . . 54 'Flyp(7). . . . . . . . . . . . . . . . . . . .54 kero . . . . . . . . . . . . . . . . . . . . . . 55 A u = u . . . . . . . . . . . 57 M-~YP

-v . . . . . . . . . . . . . . . . . . . . . . . . 57 o ( A ). . . . . . . . . . . . . . . . . . . . . . 57 P ( V ) . . . . . . . . . . . . . . . . . . . . . 58 -v I M . . . . . . . . . . . . . . . . . . . . 60 NICf( V ) . . . . . . . . . . . . . . . . . . . 60 $ [ S = t] . . . . . . . . . . . . . . . . . 62 xICf xfr [C]. . . . . . . . . . . . . . . . . . . . . 62 x&[A]. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . 62 k . . . . . . . . . . . . . . . . . . . . . . . . . 63 Hhf Id . . . . . . . . . . . . . . . . . . . . . 64 Hhf Mod . . . . . . . . . . . . . . . . . . 64 HId . . . . . . . . . . . . . . . . . . . . . . 64 HMod . . . . . . . . . . . . . . . . . . . . 63 A I= s = t . . . . . . . . . . . . 65 XA

M-~YP

S M( 7 ). . . . . . . . . . . . . . . . . . . . . 66 S M( V ). . . . . . . . . . . . . . . . . . . . 67 L ( V ). . . . . . . . . . . . . . . . . . . . . . 67 a ( W ) . . . . . . . . . . . . . . . . . . . . . 67 q ( M ) . . . . . . . . . . . . . . . . . . . . . 67 Q ( L ). . . . . . . . . . . . . . . . . . . . . . 67 S ( H y p ( 7 ) ). . . . . . . . . . . . . . . . 69 C ( C ( V ) ). . . . . . . . . . . . . . . . . . 69 L (C( V )) . . . . . . . . . . . . . . . . . . 69 S- ( H y p ( 7 ) ). . . . . . . . . . . . . . . . 69 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 70 fl . . . . . . . . . . . . . . . . . . . . . . . . . . 70 a ( M ) . . . . . . . . . . . . . . . . . . . . . 69 P ( M ) . . . . . . . . . . . . . . . . . . . . . 69 CM . . . . . . . . . . . . . . . . . . . . . . . 70 ML . . . . . . . . . . . . . . . . . . . . . . . 71 Ms, . . . . . . . . . . . . . . . . . . . . . . . 73 t . . . . . . . . . . . . . . . . . . . . 74 M-~YP

C M-~YP

tl = t2 . . . . . . . . . . 75

Glossary

v,, . . . . . . . . . . . . . . . . . . . . . . . . 79 FTn. . . . . . . . . . . . . . . . . . . . . . . . 79 clonesFr, . . . . . . . . . . . . . . . . . 80 Substc . . . . . . . . . . . . . . . . . . . . 80 0 . . . . . . . . . . . . . . . . . . . . . . . . . 80 i d . . . . . . . . . . . . . . . . . . . . . . . . . 80 Id,V . . . . . . . . . . . . . . . . . . . . . .81 n - cloneV . . . . . . . . . . . . . . . . 83 IdsFV . . . . . . . . . . . . . . . . . . . . 83 HypsF (r,) . . . . . . . . . . . . . . . . 83 cloner . . . . . . . . . . . . . . . . . . . . 84 Vhf - . . . . . . . . . . . . . . . . . . . . . . . 84 S$ . . . . . . . . . . . . . . . . . . . . . . . . 84 F: . . . . . . . . . . . . . . . . . . . . . . . . 84 S ( T ). . . . . . . . . . . . . . . . . . . . . . 87 PT . . . . . . . . . . . . . . . . . . . . . . . . 87 RAT . . . . . . . . . . . . . . . . . . . . . . 87 R A 2 . . . . . . . . . . . . . . . . . . . . . . 87 Bi . . . . . . . . . . . . . . . . . . . . . . . . 87 ( I Df ) . . . . . . . . . . . . . . . . . . . . . 89 ( A B ; ) . . . . . . . . . . . . . . . . . . . . .89 ( M f . g.). . . . . . . . . . . . . . . . . . . . 89 ( A B f ). . . . . . . . . . . . . . . . . . . . .89 EST . . . . . . . . . . . . . . . . . . . . . . . 89 P f i , . . . . . . . . . . . . . . . . . . . . . . 91 I P f i , . . . . . . . . . . . . . . . . . . . . . 91 N F ( t ) . . . . . . . . . . . . . . . . . . . . 92 N, F . . . . . . . . . . . . . . . . . . . . . . 95 op, ( t ). . . . . . . . . . . . . . . . . . . . . 97 vb, ( t ). . . . . . . . . . . . . . . . . . . . . 97 Ka . . . . . . . . . . . . . . . . . . . . . . . 104 ( a ). . . . . . . . . . . . . . . . . . . . . . . 103 E ( S ). . . . . . . . . . . . . . . . . . . . . 104 E ( M ). . . . . . . . . . . . . . . . . . . . 104 R e g ( M ). . . . . . . . . . . . . . . . . . 104 Reg (S) . . . . . . . . . . . . . . . . . . 104 PA. . . . . . . . . . . . . . . . . . . . . . . 104 'FIA . . . . . . . . . . . . . . . . . . . . . . 104 I ( a ). . . . . . . . . . . . . . . . . . . . . . 104

aLb . . . . . . . . . . . . . . . . . . . . . . 105 ZJ . . . . . . . . . . . . . . . . . . . . . . . . 105 aRb . . . . . . . . . . . . . . . . . . . . . . 105 3 . . . . . . . . . . . . . . . . . . . . . . . . 105 'FI . . . . . . . . . . . . . . . . . . . . . . . . 105 I m a . . . . . . . . . . . . . . . . . . . . . 105 RH,p(T, ) . . . . . . . . . . . . . . . . . 106 L H ,p(Tn) . . . . . . . . . . . . . . . . . . 106 leftmost(t) . . . . . . . . . . . . . . 107 rightmost ( t ). . . . . . . . . . . . . 107 R, ( t ). . . . . . . . . . . . . . . . . . . . 107 L, ( t ). . . . . . . . . . . . . . . . . . . . . 107 o p ( R p ( t ) ). . . . . . . . . . . . . . . . 108 op(Lp(t)). . . . . . . . . . . . . . . . 108 war ( t ). . . . . . . . . . . . . . . . . . . . 109 vb(t) . . . . . . . . . . . . . . . . . . . . . 109 vbj ( t ). . . . . . . . . . . . . . . . . . . . 109 f irstops(t). . . . . . . . . . . . . . . 109 op ( t ). . . . . . . . . . . . . . . . . . . . 109 le f tdepth(t). . . . . . . . . . . . . . 109 rightdepth(t) . . . . . . . . . . . . 109 Reg ( T ) . . . . . . . . . . . . . . . . . . . 109 Reg ( n ). . . . . . . . . . . . . . . . . . . 109 B i j ( r ) . . . . . . . . . . . . . . . . . . . 111 Hypreg(r) . . . . . . . . . . . . . . . I l l s, . . . . . . . . . . . . . . . . . . . . . . . 111 W ({x}) . . . . . . . . . . . . . . . . . . 114 W + ( { x ) ). . . . . . . . . . . . . . . . . 114 M x ( L ). . . . . . . . . . . . . . . . . . . 114 MA ( L ) . . . . . . . . . . . . . . . . . . . 114 My ( J ) . . . . . . . . . . . . . . . . . . . 114 M b ( J ) . . . . . . . . . . . . . . . . . . . 114 F, . . . . . . . . . . . . . . . . . . . . . . . 114 F, . . . . . . . . . . . . . . . . . . . . . . . 114 M I . . . . . . . . . . . . . . . . . . . . . . . 114 M2 . . . . . . . . . . . . . . . . . . . . . . . 114 M3 . . . . . . . . . . . . . . . . . . . . . . . 114 M4 . . . . . . . . . . . . . . . . . . . . . . . 114 M5 . . . . . . . . . . . . . . . . . . . . . . . 114

Glossary

MT . . . . . . . . . . . . . . . . . . . . . . 114 M D . . . . . . . . . . . . . . . . . . . . . . 114 S f i n ( H y p ( r ). . . . . . . . . . . . . 115 L x . . . . . . . . . . . . . . . . . . . . . . . 116 L. . . . . . . . . . . . . . . . . . . . . . . . 116 L1 . . . . . . . . . . . . . . . . . . . . . . . 116 at . . . . . . . . . . . . . . . . . . . . . . . . 119 F(2)( ( 2 .y ) ) . . . . . . . . . . . . . . 119 E , . . . . . . . . . . . . . . . . . . . . . . . 121 E x . . . . . . . . . . . . . . . . . . . . . . . 121 S1 . . . . . . . . . . . . . . . . . . . . . . . 131 S 2 . . . . . . . . . . . . . . . . . . . . . . . 131 S3 . . . . . . . . . . . . . . . . . . . . . . . 131 D ( V ) . . . . . . . . . . . . . . . . . . . . 131 Jt . . . . . . . . . . . . . . . . . . . . . . . . 137 W P ( 2 .2 ) . . . . . . . . . . . . . . . . 138 P ( 2 . 2 ) . . . . . . . . . . . . . . . . . . . 138 P r e ( 2 . 2 ) . . . . . . . . . . . . . . . . . 138 tn . . . . . . . . . . . . . . . . . . . . . . . . 145 t:i . . . . . . . . . . . . . . . . . . . . . . . 145 H2Op . . . . . . . . . . . . . . . . . . . . . . 153 ~ , . t. . . . . . . . . . . . . . . . . . . . . . . 171 St . . . . . . . . . . . . . . . . . . . . . . . . 171 pl.t . . . . . . . . . . . . . . . . . . . . . . . 171 Ct . . . . . . . . . . . . . . . . . . . . . . . 174 C s . . . . . . . . . . . . . . . . . . . . . . . 174 S,.,. . . . . . . . . . . . . . . . . . . . . . 175 P ( n ) . . . . . . . . . . . . . . . . . . . . . 182 D ( n ). . . . . . . . . . . . . . . . . . . . . 182 F ( n ) . . . . . . . . . . . . . . . . . . . . . 182 Ex%. . . . . . . . . . . . . . . . . . . . . . 182 a1 + a2. . . . . . . . . . . . . . . . . . . 184 H ypPF ( r ). . . . . . . . . . . . . . . . 185 H y p S F ( r ). . . . . . . . . . . . . . . . 185 H y f ( r ). . . . . . . . . . . . . . . . . 186 ( O C ) . . . . . . . . . . . . . . . . . . . . 187 d e p t h ( a ) . . . . . . . . . . . . . . . . . 187 v ( a ) . . . . . . . . . . . . . . . . . . . . . 187 . . . . . . . . . . . . . . . . . . . . . . 188

sR

dc. . . . . . . . . . . . . . . . . . . . . . . 188 P ( t ) . . . . . . . . . . . . . . . . . . . . . 193 S E M . . . . . . . . . . . . . . . . . . . . 199 CSg . . . . . . . . . . . . . . . . . . . . . . 199 C . . . . . . . . . . . . . . . . . . . . . . . . 200 LZ . . . . . . . . . . . . . . . . . . . . . . 200 N B . . . . . . . . . . . . . . . . . . . . . . 200 RB . . . . . . . . . . . . . . . . . . . . . . 200 R e g B . . . . . . . . . . . . . . . . . . . . 200 R Z . . . . . . . . . . . . . . . . . . . . . . 200 Z . . . . . . . . . . . . . . . . . . . . . . . . 200 S L . . . . . . . . . . . . . . . . . . . . . . . 200 P r e . . . . . . . . . . . . . . . . . . . . . . 203 Reg . . . . . . . . . . . . . . . . . . . . . . 204 (x,) . . . . . . . . . . . . . . . . 204 Ci: . . . . . . . . . . . . . . . . . . . . . . 206 VRC. . . . . . . . . . . . . . . . . . . . . . 206 V ( C ) . . . . . . . . . . . . . . . . . . . . 210 VRS. . . . . . . . . . . . . . . . . . . . . . 216 Vzyc. . . . . . . . . . . . . . . . . . . . .237 VHS. . . . . . . . . . . . . . . . . . . . . . 252 VPS. . . . . . . . . . . . . . . . . . . . . . 255 VPC. . . . . . . . . . . . . . . . . . . . . . 259 M g ( V ) . . . . . . . . . . . . . . . . . . 261 Z , . . . . . . . . . . . . . . . . . . . . . . . 262 VhIID. . . . . . . . . . . . . . . . . . . . 268 shd. . . . . . . . . . . . . . . . . . . . . . . 268 Vgs. . . . . . . . . . . . . . . . . . . . . . . 272 V M r D ( s t ). . . . . . . . . . . . . . 276 VBE. . . . . . . . . . . . . . . . . . . . . . 278 VENT. . . . . . . . . . . . . . . . . . . . 278 N E ( r ). . . . . . . . . . . . . . . . . . . 284 N E . . . . . . . . . . . . . . . . . . . . . . 284 N: . . . . . . . . . . . . . . . . . . . . . . 284 E ( C ) . . . . . . . . . . . . . . . . . . . . 284 TI,( X ). . . . . . . . . . . . . . . . . . . . 290 Tk( B ). . . . . . . . . . . . . . . . . . . . 290 TI,( B ). . . . . . . . . . . . . . . . . . . . 290 ei . . . . . . . . . . . . . . . . . . . . . . . . 292

we,

Glossary

Xk . . . . . . . . . . . . . . . . . . . . . . . 292

r;%. . . . . . . . . . . . . . . . . . . . . . . 293 r' . . . . . . . . . . . . . . . . . . . . . . . 294 VMD. . . . . . . . . . . . . . . . . . . . . 294 2Ass . . . . . . . . . . . . . . . . . . . . . 296 4Dist . . . . . . . . . . . . . . . . . . . . 296 2Med . . . . . . . . . . . . . . . . . . . . 296 rl . . . . . . . . . . . . . . . . . . . . . . . 297 I'z . . . . . . . . . . . . . . . . . . . . . . . 297 I?3 . . . . . . . . . . . . . . . . . . . . . . . 297 Sr . . . . . . . . . . . . . . . . . . . . . . . 298 PS(ST). . . . . . . . . . . . . . . . . . 298 PSId,, (ST). . . . . . . . . . . . . . 300 PSN( S r ) . . . . . . . . . . . . . . . . .300 C(2.2) . . . . . . . . . . . . . . . . . . . . . 301 N z . . . . . . . . . . . . . . . . . . . . . . 302 N z . . . . . . . . . . . . . . . . . . . . . . 302 v ( ~. . ). . . . . . . . . . . . . . . . . . . . 303 ~ ( . . .~. . . 1. . . . . . . . . . . . . . . . 304 VD . . . . . . . . . . . . . . . . . . . . . . . 313 V, . . . . . . . . . . . . . . . . . . . . . . . 307

Index 0-simple semigroup, 105 ID-semiring, 268 M-hyperequational classes, 64 M-hyperequational logic, 76 M-hyperequational theories, 64 M-hyperidentity, 57 M-solid variety, 57 NF-term, 93 Pre-hypersubstitution, 298 V-proper hypersubstitutions, 222 n-ary operation, 1 n-solidity, 82 O P ( ~ ) ,17 absolutely free algebra, 19 absorption law, 3, 89 Aczd, J., 51 algebra, 1 absolutely free, 19, 24 derived, 57 directly indecomposable, 13 indexed, 1 non-indexed, 1 relatively free, 23, 24 simple, 8 subdirectly irreducible, 15 term, 19 alphabet, 16 anarchic algebra, 19 arity of an operation, 2 Arworn, S., 37, 47

associative law, 2 automorphism, 7 automorphism group, 8 bands, 50 base set, 2 Belousov, V. D., 51 bijective hypersubstitution, 103, 109 binary clone, 77 Birkhoff, G., v, 16, 74 Boolean algebra, 5 Boolean lattice, 272 Brandt-semigroup, 260 class equational, 25 clone, 21 generated by a set of operations, 21 of operations, 21 of terms, 20 clone of a variety, 85 clone substitution, 80 closed set, 25, 30 closure operator, 6, 29, 32, 35 additive, 41 closure properties, 9 commutative semigroup, 51, 200 compatibility, 8 complete lattice, 25, 32, 37 complete sublattice, 37 completeness, 28 completeness theorem

Index of

M-hyperequational logic, 76 complex product, 105 complexity of a term, 17 composition, 21 congruence Ms,-totally invariant, 76 fully invariant, 23, 71, 76 generated by a set, 9 totally invariant, 71 congruence relation, 8 conjugate pair of additive closure operators, 49 conjugate pair of closure operators, 41 conjunction, 5 consistency, 28 Consistency and Completeness Theorem, 28 deduction rules, 27 Denecke, K., 37 depth of a term, 17 derivation rules, 27, 285 diagonal algebra, 88 Dikranjan, D., 41 direct product, 11, 12, 25 directly indecomposable algebra, 13 disjunction, 5 distributive lattice, 50 dual-solid, 206 duality principle, 268 embedding, 7 endomorphism, 7, 54 endomorphism monoid, 8 entropic law, 270

equation, 22 equational class, 25 equational logic, 28 equational theory, 25, 71, 285 Main Theorem, 26 equationally complete variety, equationally definable class of algebras, 25 essentially binary, 124 extensivity, 6, 9, 29 factor congruence, 13 finite hyperidentity basis, 261 finite identity basis, 261 finitely based variety, 260 fixed point, 25 formal deduction, 74 formal language, 16 full clone, 21 full clone of terms, 52 full term, 19 full term clone, 84 fully invariant congruence, 23 fully invariant congruence relation, 71 fundamental operation, 2, 51 fundamental semigroup, 132 Galois closed subrelation, 37 Galois connection, 25, 34, 35, 49, 64 Ganter, B., 37 general NF-term, 93 generalized metabelian law, 89 generating system, 6 graph of a relation, 1 graph of an operation, 1, 2 Green's relation 23, 105

Index Green's relation 'H, 105 Green's relation 3, 105 Green's relation C,105 Green's relation R,105 Green's relations, 105 group, 2 groupoid, 2 Guili, E., 41 heterogeneous algebra, 52, 53, 84 homogeneous algebra, 53 homomorphic image, 25 Homomorphic Image Theorem, 11 homomorphism, 7 hyperamssocia8tjive, 252 hyperdualizable identity, 268 hj~perequationalclass, 96 hyperequational theory, 71 hyperidentity, 50, 57, 64 hypersatisfaction, 50, 57 hypersubstitution, 53 V-equivalent , 57 normal form, 61 hypersubstitution rule, 74 hypersubstitution rule of inference, 74 hypervariety, 51 idenipotency, 6, 9, 30 idempotent law, 3, 50, 89 idempotent semigroups, 50 identity, 22 inductive definition, 16 infimum, 4 injective hypersubstitution, 103, 109 intensivity, 30 invertibility, 2

isomorphism, 7 Jbnsson, B., v join operation, 3, 5 K-free algebra, 23 kernel of a mapping, 10 kernel operator, 32 lattice, 3 bounded, 4 complete, 4 distributive, 3 algebraic, 26 lattices dually isomorphic, 49 left ideal, 105 left zero bands, 201 left-seminearring, 103 local subsemigroup, 106, 131 locally finite variety, 26, 261 Main Theorem for Conjugate Pairs of Closure Operators, 43 Main Theorem of Equational Theory, 26 maxdepth of a term, 17 medial law, 50, 198 meet operation, 3, 5 Menger algebra of rank n, 77 Adenger system, 84, 86 mindept h of a term, 17 minimal variety, 27 monoid, 2 monotonicity, 6, 9, 29 Movsisjan, Yu. M., 51 multi-based algebra, 52, 53, 84

Index muk-sorted algebra', 84 natural homomorphism, 10 negation, 5 Neumann, W. D., 51 nil-semigroup, 200, 263 normal bands, 201 normal form term, 51 normal identity, 201, 284 normal variety, 284 normalization of a variety, 284 one-based algebra, 53 operat ion nary, 2 finitary, 2 fundamental, 2 join, 3 meet, 3 nullary, 1 term, 21 operation symbol, 51 operation symbol count, 17 operators H, S, P, 25 outermost equation, 301 outermost identities, 201 overcommutative varieties of semigroups, 200 Poschel, R., 37 Plonka sum, 274 pair of factor congruences, 13 partial order, 3 periodic semigroup, 200, 263 permutable relations, 12 permutationally full term, 19, 186 power set, 5 pre-hypersubstitution, 138 pre-solid variety, 255

principal ideal, 105 product direct, 11 subdirect, 13 projection, 12, 272 projection algebra, 87 projection hypersubstitution, 138, 298 projections of a direct product, 12 rectangular algebra, 87 rectangular band, 88, 89, 201 regular bands, 201 regular element, 103 regular hypersubstitution, 109 regular identities, 201 regular-solid variety, 205 regularity, 183 Reichel, A4., 41 relation trivial, 8 relatively free algebra, 23 replacement rule, 27 restriction, 1 right ideal, 105 right zero bands, 201 root of a tree, 17 rules of consequence, 27 rules of derivation, 74 satisfaction, 22 self-dual, 206 semantic tree, 17 semi-weak endomorphic image, 71 semigroup, 2 semigroup with zero, 104 semilattice, 5

Index semilattice ordered system of algebras, 274 semilattices, 201 semiring, 268 distributive, 268 idempotent, 268 medial, 268 simple algebra, 8 solid congruence, 71 solid vasietJy,57, 87, 96 solid variety of semigroups, 252 strongly full term, 19, 79, 186 subalgebra, 5, 25 Subalgebra Criterion, 5 subalgebra lattice, 6 subclone, 21 subdirect product, 13 subdirectly irreducible algebra, 15, 26 substitution rule, 28, 74 s~~bvasiet~y, 27 subvariety lattice, 27 superposition of terms, 20, 52 supremum, 4 Tarski, A., v, 32 Taylor, W., 51 term, 16 n-ary, 16 nullary, 16 term algebra, 19 term operation, 21, 22 tot ally invariant congruence relation, 71 tree, 17 trivial relation, 8 trivial variety, 26, 201

two-sided ideal, 105 type, 16 of an algebra, 2 of terms, 16 unitary Menger algebra of rank 2, 77 unit,ary Menger algebra, of rank n , 20, 53, 54, 77 universe of an algebra, 2 variable, 16 variety, 26 M-solid, 57 minimal, 27 solid, 57 vertex. 17 weak projection hypersubstitution, 138 Wille, R., 37 words, 198 yields, 27 zero, 105 zero element, 104, 105 zero-semigroups, 20 1 Zimin word, 262

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