Semigroups (Translations of Mathematical Monographs 3)

n. We write lB~8)

=

{XI">' Xt+!, X;+2, ... } E ~('l(i) (s

=

1,2,3, ... ; i

= 1,2).

In view of (e) we have:


=

lB~S)

=

(s

1,2, ... , n),


lB}2)

=

=

•.• ]

= lB~.

2. The equation

[Xf] 0 [X1]

=

[Xr, Xf)

(i

= 1,2)

is clearly true only when p and q are the integers 2 and 3. Since
= [X&J,


=

[Xf].

2 0 [Xl5] = [Xl> 2 Xl' 4 Xl' 5 Xu1"" ] contams . aII umts . 0 f -VI m(2) ,w h'IC h are But [Xd monogenic semigroups in 'l(l' except for one: [Xn since [Xl] 0 [XJ] = [Xl, XJ] does not contain [Xi] and [X;] for arbitrary 1 > 4 and does not contain [Xf] and [X~] for I = 4. Therefore, we cannot have n = 2. Suppose that n is even and greater than two. Since [Xi] ::l [X~] and rp([Xm = [X~], we have [X~] ::l [X~'], that is, m is even. for the elements YI = Xi and Y2 = X~ we have

which contradicts the assumption that n is the smallest number with the stated property. It remains to assume that n is odd.

134

o

MULTIPLICATION

Since [Xf] c lBi+\ we have [X2'] n < n. In l:(m2) we have

<m-

[X2'-n] 0

But [x1m-n] 0

m(n+l) -

:.(.)1

-

c

OF

SUBSETS

lB~.

lB~ :::>

[x1m- n'

[CHAP.

This implies that m

III

< 2n, that is,

[X2'].

X 1n+l , X 1n+2 '

• . .

] 3

xn1

Xr

is possible only if = (Xr,-n)". This means that m - n is a divisor of n. Therefore m - n is odd. Since n is odd, m must be even. But then, since n is a minimum and n > 2, we obtain the contradiction:

ep[Xf]

=

[X2'] = [(X242)2]

= ep[(Xi"J2)2]

= ep[Xi"],

[Xf]:;= [Xr'].

7.11. As follows from 7.7, the class of all groups does not correspond to the class of semigroups defined by subsemigroup characteristics. It is natural to ask, therefore, whether there are groups, and if so which groups they may be, that have subsemigroup characteristics isomorphic to the subsemigroup characteristic of some semigroup which is not a group. The converse question also arises. Both these problems have been elaborated in detail by R. V. Petropavlovskaja [4; 5; 6]. She has determined the structure of groups with this property and moreover has discovered several classes of groups that are defined by subsemigroup characteristics. We shall examine a few cases of this sort. If all nonunit elements of a group are of infinite type, the group is said to be torsion-free. Such groups are important in group theory. We shall show that they are defined by subsemigroup characteristics. 7.12. We shall show first that an infinite cyclic groupis defined by its subsemigroup characteristic. THEOREM. Ijml is an infinite cyclic group and the semigroup m 2 is such that l:cml ) and l:(m2) are isomorphic, then m2 is an infinite cyclic group. PROOF.

ml =

[AI' All, Ell, where EI is the unit of

mI'

According to 7.10,

m2 has a single idempotent E2 and ep(El ) = E2• ml admits a decomposition into three subsemigroups:

The three components of this decomposition have no unit in common in l:(ml ), and every infinite monogenic subsemigroup in ml is a unit in l:(ml ) either of[AI]' or of [All]. In virtue of7.10, ()I), this implies that m2 admits the decomposition: m2 = [A 2 ] U [A;] U E2 ,

ep[A I ]

= [A2]'

where [A 2] and [A 2] are infinite.

ep[AII] = [A~],

Sec. 7]

THE SUBSEMIGROUP CHARACTERISTIC OF

A

SEMIGROUP

135

Since [Ad 0 [All] ~ E10 we have [A 2 ] 0 [A~] :;) E2 • It follows that A 2' Ai" A22 At" ... A?l = E2, IXI 1 (if A 2' is the empty symbol, the argument is the same). This implies that A 2X = E 2, where X:;:: E2 • Since [AI] 5 E I, we have [A 2] 5 E2 and therefore X EO [A 2]. Accordingly, X = A;s. If it were true that A2A2 :;:: E2 we would have A2A2 = A~ or A"A~ = A~q and from A.,A.;" = E.2 we would have = E2 or A'm = E2• But neither is admissible. Therefore, A2A2 = E2 •

>

Ar

OJ

..........

........

Since

(A2A2)3 = A~ . A2A~ . A2A2 . A2

= A2 . A2A~ . A2 = (A~A2)2,

we have A~A2 = E 2• The element A2E2 cannot belong to [A 2] since otherwise A2E2 = A;"', on multiplication by A~, would imply E2 = A;'" +1. Therefore, either A2E2 = E2, or A2E2 = A 2· Suppose A2E2 = E2 • Then E2A2 . E2A2 = E2A2, that is, E2A2 = E2 • From A2E2 = E2 and E2A2 = E2 we obtain, on multiplication by A~, A~E2 = E2, E2A; = E 2· It follows that [A~] 0 [A~] = [A~, A~] consists of elements of the form A~k, A;k, E2 (k = 1,2, ... ) and therefore does not contain A z, which contradicts the assumption that [Ail 0 [All] ~ [All. Therefore, AzEz cannot be equal to E2• From the equation A2E2 = A2 we obtain

Since [A 2 ] is infinite, this equation can hold only if r = 1. In the same way that we proved that A2E2 = A2 we may show that E2A2 = A2 , A2E2 = A 2, E2A2 = A~. These equations, together with those derived earlierA2A~ = E 2 , A2A2 = E2-imply that the infinite semigroup m2 = [A 2, A2, E 2 ] is an infinite cyclic group. 7.13. THEOREM. If ml is a torsion-free group (7.11) and m2 is a semigroup such that ~(ml) and ~(m2) are isomorphic, then m2 is a torsionjree group. PROOF. Since ml has a single idempotent E'1(" we find in accordance with 7.10, that m2 has a single idempotent E2• Except for E'1(" every subsemigroup of ml is infinite. Therefore, by 7.9 and 7.10, (IX), every subsemigroup of m2 distinct from £2 is infinite. Suppose that cp is an isomorphism of ~(ml) on ~(m2); A2 is an arbitrary element ofm2 , distinct from E 2 • According to 7.10, (y), to each [A 2] corresponds an Al E m1 such that CP([AID = [A 2]. In ~(ml) there exists an infinite cyclic group l ~ [A J ]. In virtue of7.10, (IX) and 7.12, CP(~l) is an infinite cyclic I , such that group, and cp(m 1) ~ [A2]' Therefore, an arbitrary element A2 of the semigroup m2 , distinct from E 2 , has E2 as a unit and possesses an inverse with respect to E 2 • It follows that m2 is a group and, from what we have said earlier, is a torsion-free group.

m

m

136

MULTIPLICATION OF SUBSETS

[CHAP. III

7.14. The furthest advances in this direction have been based on the theory of commutative semigroups. R. V. Petropavlovskaja [3] has shown that every aperiodic commutative group is defined by a subsemigroup characteristic. 7.15. With the aid of the semigroup ~ (m) of all subsemigroups of a given semigroup m, one may pose similar problems as to the definability of'll itself with respect to other semigroups of subsemigroups. For example, when studying groups it is wholly natural to consider the ensemble of subgroups ~* ('ll). This is a subsemigroup of the semigroup ~ ('ll). Historically, the first question along the line developed in this section concerned those groups for which the isomorphism of ~* ('lll) and ~* ('ll2) always implies the isomorphism of 'lll and 1ll2. The essential results were first obtained by Baerl and by L. E. Sadovski.2 This line has been even further developed. 3 It is of some interest to note that Baer's fundamental result was subsequently obtained by R. V. Petropavlovskaja [3] in connection with work on the semigroup of all subsemigroups of a group. 7.16. The following theorem plays a significant part in the derivation of these results. THEOREM. Suppose that for the groups ffi l and ffi2 the subsemigroup characters effi l ) and~ (ffi2) are isomorphic. Then the semigroups of the subgroups ~* (ffi l ) and ~* (ffi 2) are isomorphic.

~

PROOF.

Let cp be the isomorphism of ~ (ffi l ) on

~

(ffi 2) and

We suppose that lB2 E ~* (ffi2)' We choose from lBI an arbitrary element Xl' If Xl is of finite type, then [Xl] contains an element inverse to Xl' Therefore X1- 1 E lB1 . If Xl is of infinite type, then by 7.9 and 7.10, (IX), cp([XI ]) is an infinite monogenic semigroup: cp([XI ]) = [X2]. Since [Xl] c lBI' we have [X2] c lB 2 • But lB2 is a group, and therefore [X2 , X2- 1] c lB 2 • But then we have for arbitrary E ~ (ffi l ),

.s

1 The significance of the system of subgroups for the structure ofgroups, Amer. J. Math. 61 (1939), 1-44. 2 On structural isomorphisms offree groups, Dokl. Akad. Nauk SSSR 32 (1951), 171-174. (Russian) Structural isomorphisms offreegroups and of free products, Mat. Sb. (N.S.) 14 (56) (1944), 155-173. (Russian) On structural isomorphisms of free products of groups, Mat. Sb. (N.S.) 21 (63) (1947), 63-82. (Russian) 3 See, for example, M. Suzuki, Structure of a group and the structure of its lattice of subgroups, Springer, Berlin, 1956. B. 1. Plotkin, Generalized solvable and generalized nilpotent groups, Uspehi Mat. Nauk 13 (1958), no. 4 (82),89-172. (Russian)

Sec. 7]

THE SUBSEMIGROUP CHARACTERISTIC OF A SEMIGROUP

137

According to 7.9 and 7.10, (IX), ~ is an infinite cyclic group. Therefore XI-I E ~ C ~1' Since ~1 contains for everyone of its elements Xl the inverse XI-I, ~1 E 2:* «(\)1)' In an altogether similar way we may show that ~1 E 2:* «£)1) implies ~2 E 2:* «(\)2)' We have found, then, that the isomorphism cp of the semigroup 2: «£)1) on 2: «(\)2) is a one-to-one mapping of 2:* «£)1) on 2:* «£)2), and is therefore an isomorphism of 2:* «(\)1) on 2:* «(\)2)' 7.17.

These results mean that for groups every isomorphism of 2: «£)1) to

Z «£)2) can be obtained as an extension of some isomorphism of 2:* «£)1) to 2:* «(\)2)' R. V. Petropavlovskaja has shown [3] thatthe converse is notin general true. There are groups (£)1 and (£)2 for which the isomorphisms of 2:* «£)1) to 2:* «(\)2) exist but cannot be extended to isomorphisms of 2: «£)1) to 2: «(\)2)' Moreover, there are groups (\)1 and (£)2 (and even commutative groups) such that 2:* «(\)1) and 2:* «£)2) are isomorphic whereas 2: «£)1) and 2: «£)2) are not. Thus, the requirement that 2: «(\)1) and 2: «£)2) be isomorphic is stronger, in the case that (\)] and (\)2 are groups, than the requirement that 2:* «£)1) and 2:* «£)2) be isomorphic.

CHAPTER

IV

IDEALS 1. The Concept of Ideals and Their Simplest Properties

1.1. We have had the opportunity to convince ourselves of the important role of subsemigroups, i.e., of those subsets of semigroups that are closed under multiplication. A natural strengthening of this condition is the requirement that the subset be closed under multiplication by any element of the semigroup. Subsets of semigroups with such properties play an important role in the theory of semigroups. DEFINITION.

afm

A nonempty subset X of a semigroup 'll is said to be a LEFT IDEAL

if

X is said to be a RIGHT

IDEAL

IllX eX. if XIll eX.

;:r is said to be a TWO-SIDED IDEAL ifi! is simultaneously a left and a right ideal, if it is nonempty and IllX c X and XIll c X.

i.e.,

X is said to be an

IDEAL

if it is either a left or a right ideal of Ill.

In commutative semigroups the concepts of ideal, left ideal, right ideal and two-sided ideal clearly coincide. Since XX c IllX, XX c XW, an ideal is always a subsemigroup. One should note that the word "ideal" is often used to signify only two-sided ideals. Left and right ideals are sometimes spoken of as one-sided ideals. The term "ideal" appeared first in the theory of algebraic numbers where its use was motivated by well-known reasons. Later it passed into other branches of algebra where it became so strongly rooted that it would be impossible to change it in spite of its being quite unsuitable. The properties of right, left and, in particular, two-sided ideals of a semigroup are not only interesting in themselves but are closely connected with various other properties of the semigroup. For example, their role in divisibility is immediately clear. In many cases the structure of a semigroup is determined 138

Sec. 1]

CONCEPT OF IDEALS AND THEIR SIMPLEST PROPERTIES

139

in varying degrees by the existence and interrelations of its ideals. A significant part of the work done on semigroups utilizes the concept and properties of ideals. The present chapter and the following one will be completely concerned with the investigation of various properties of ideals. Moreover, we will often employ ideals in later discussions. 1.2. In the multiplicative semigroup of the natural numbers the set of all the even numbers is an ideal. In the semigroup of all real functions defined on the whole real axis considered with respect to the operation of composition the set of all the constants is a two-sided ideal. The set of all the periodic functions is a left ideal. The set of all the functions differing from zero for all values of the independent variable is a right ideal. In the multiplicative semigroup 9n n of all complex square matrices of order n the set of matrices for which all the elements of a given fixed column are equal to zero is a left ideal. Moreover, it is not a right ideal if n > 1. The set of matrices for which all the elements of a given row are equal to zero is a right ideal but is not a left ideal if n > 1. The set of all singular matrices is a two-sided ideal. In an arbitrary semigroup m for any nonempty subset 91 c m the product mm is clearly a left ideal, the product mm a right ideal and the product m91m a two-sided ideal. The set of all factorable elements, i.e., ~m, is obviously a twosided ideal of \!L If an element X is irreducible, then the set m\X is a two-sided idealofm.

1.3.

In a finite monogenic semigroup m = [Xl of type (h, d) (III, 3.7) the set

Xk = {Xk, Xk+l, ... , XMd-l} clearly forms an ideal of m for any k = 1,2, ... ,h. There are no other ideals in m. In fact, let Xk be the least power of X belonging to some ideal :r of the semigroup m. Then X also contains Xk+l, Xk+2, ... , XMd-l. The number k cannot be larger than h, for it follows from XMr E :r (0 < r < d) that Incidentally, it follows from this that the first component of the pair (h, d) may be defined as the number of ideals in the semigroup m. If m = [X] is an infinite monogenic semigroup, then clearly all its ideals are sets of the form (k = 1,2,3, ... ). 1.4. The important role of ideals in the theory of rings and algebras is weIlknown. Clearly, any left, right or two-sided ideal of a ring will be respectively a left, right or two-sided ideal of the multiplicative semigroup of this ring. The converse is in general not valid. The ring of all the integers will serve as an example of

140

IDEALS

[CHAP.

IV

this. Its subset consisting of zero and all the integers greater in absolute value than some arbitrary fixed natural number is obviously a two-sided ideal of the multiplicative semigroup of all the integers but is not an ideal of the ring. For a ring with an identity it is easy to give all the cases when all the ideals of the multiplicative semigroup are also ideals of the ring. (Aubert [1] formulated this result for the commutative case.) THEOREM. In a ring with an identity every left ideal of the multiplicative semigroup of the ring will be a left ideal of the ring itself if and only if for any two elements one is the right divisor of the other. PROOF. (1) Let every left ideal of the multiplicative semigroup of the ring Ill: be a left ideal of the ring itself. For any X, Y E Ill: the set Ill:X U Ill: Y is clearly a left ideal of the semigroup. Since it must also be a left ideal it must also contain the element X + Y. Let (X + Y) E Ill:X. Then for some Z E Ill: we have

X+ Y=ZX, Y= (Z- Em)X.

(2) Assume that for any two elements of Ill: one is always the right divisor of the other and that £ is an arbitrary left ideal of the multiplicative semigroup of the ring Ill:. Let X, Y E £ and A E Ill:. We have AX En. Iffurthermore Y = ZX, then X - Y= EmX - ZX= (Em - Z)XE£, (Y-X)=(-Em)(X- Y)E£

and, consequently, £ is a left ideal of the ring. 1.5. Although it is not true even for multiplicative semigroups of rings that the theory of ideals of rings and the theory of ideals of semigroups coincide, there is nevertheless a definite connection between them. Since the contents of a whole sequence of properties of rings are connected only with the operation of multiplication it is natural to ask whether it is possible to carryover these properties directly or, in a generalized sense, into the theory of semigroups. Among such properties the characteristics of distinct factorizations occupy an important position. They are related to the arithmetic characteristics and are related in an essential way to ideals. As examples of investigations with the indicated tendency, see the works of V. I. Arnol'd [1], Asano and Murata [1], Aubert [1], Weaver [2], Dubreil-lacotin [2], Kawada and Kondo [1], Clifford [2], Lesieur [3], Mackenzie [1], L. M. Rybakoff [1], and Skolem [2; 3; 5]. 1.6. Let us point out some of the simplest properties of ideals. Let Ill: be an arbitrary semigroup. (a) Ill: is a two-sided ideal of itself. (P) If Ill: has a zero Om, then OUl is a two-sided ideal of Ill:. (I') The union of any collection of left ideals is itself a left ideal.

Sec. 1]

CONCEPT OF IDEALS AND THEIR SIMPLEST PROPERTIES

In fact, if lB;. (A

E

141

r) are left ideals of Ill, then

Ill· (U lB;.) ;'er

= U(IlllB;.) c

UlBk

;'er

;'er

(0) The intersection of any collection of left ideals is itself a left ideal ifit is not empty. In fact, using the notation of (y) we get that for any f1 E r

Ill· (n lB;.)

c

IlllB I'

lB I"

c

;'Er

Hence Ill,cn lBJ c n lBi" ;.€r

I'€r

(e) IflB is a subsemigroup of Ill, :!: a left ideal of III and the intersection oflB and:!: is nonempty, then lB !l :!: is a left ideal of lB. In fact, since lB !l :!: c :!:, it fallows that

lB . (lB !l :!:) c lB:!: c :!:,

and since lB !l :!: c lB we have lB . (lB !l :!:) c lB . lB c lB.

Hence, lB . (lB !l:!:) c lB !l :!:.

(S) If an element X is included in some left ideal:!: of the semigroup III while an element Y is not included in :!:, then X does not divide Y on the right.

For if X divided Yon the right, i.e.,

y=ZX,

Z

Y = ZX c

mx c

then

E

Ill,

X.

(1]) If X is a semigroup without nonfactorable elements (i.e., X:!: = X), III is a supersemigroup of X such that:!: is a left ideal ofm and Ill' is a supersemigroup ofm such that mis a left ideal ofm', then X is a left ideal of the semigroup m'.

In fact, 1.7. In connection with the property (1]) of 1.6, it is necessary to note that in general the relationship of "being a left ideal" is nontransitive. If X is a left ideal of the semigroup mand mis a left ideal of the semigroup Ill', then it is not necessary that :!: be a left ideal of Ill'. The same is true of right and two-sided ideals. As an example we consider the semi group consisting of the four elements Ill'

= {A, E,

C, O}

142

[CHAP.

IDEALS

IV

in which the product of any two elements is equal to 0, with the exception of the one product AB = C. The associativity of the operation is verifiable without difficulty inasmuch as, for any three elements X, Y, and Z from Ilt', (XY)Z = 0,

X(YZ)

= o.

Ilt = {B, C, O} is clearly a two-sided ideal of Ilt'. Its subset X = {B, O} is a two-sided ideal of m:. However, X is not even a left ideal of Ilt' since Ilt'X:3 AB

=

C.

1.S. Semigroups are usually very rich in ideals. Moreover, the further away (in some sense) the semi group is from a group the more ideals it has. Groups are the limiting case in this respect. A semigroup is a group if and only if it has no proper ideals. In fact, ifllt is a group then each of its elements divides any other element ofllt on both the right and the left. Thus, by 1.6 (0, no element of m: may be in any proper left or right ideal. If Ilt contains no proper ideals, then for any A E Ilt IltA

= Ilt,

(since IltA and Allt are proper left and right ideals of m:). By III, 1.2 it follows from this that Ilt is a group. 1.9. We will give some of the simplest properties of two-sided ideals. Here, considering the especially important role of two-sided ideals, we also include some properties whose validity follows directly from the properties of left and right ideals considered in 1.6. Let Ilt be an arbitrary semigroup. (IX) The union of any collection of two-sided ideals of a semigroup Ilt is itself a two-sided ideal ofm:. CP) The product of two two-sided ideals of Ilt is a two-sided ideal of Ilt. (y) The intersection of any collection of two-sided ideals of m: is a two-sided ideal ofllt if it is nonempty. (b) The intersection of two two-sided ideals ofm: is a two-sided ideal ofm:. In fact, if Xl and X2 are two-sided ideals of Ilt, then their intersection is nonempty since it clearly contains their product Xl . X 2 • But then, according to (y), this intersection is a two-sided ideal of Ilt. (e) A subset of Ilt consisting of one element X is a two-sided ideal of Ilt if and only if X is the zero of the semigroup m:. (0 Ifm: has a zero Om, then 02( is included in every two-sided ideal ofm:. Cry) If 'is is a subsemigroup of Ilt and X is a two-sided ideal of m:, then the intersection 'is n X, if it is not empty, is a two-sided ideal of the semigroup 'is.

Sec. 1]

CONCEPT OF IDEALS AND THEIR SIMPLEST PROPERTIES

143

(6) Jf'X is a two-sided ideal 0/91, then the set U consisting 0/ all the elements U E 91 such that U91 c ::t is a two-sided ideal 0/91.

In fact, the set U is nonempty since obviously U :::l::t. Further, for any U E U and A E 91 we have (A U)91 c A::t c ::t, (UA)91 c U91 c 'X,

i.e., AU, UA

E

U.

1.10. It follows from 1.9 that we may introduce naturally in the set of all two-sided ideals r of the semigroup 91 several operations with respect to each of which it will be a semigroup. It follows from 1.9 (ct.) that r is a commutative semigroup with respect to the operation of set union. It follows from 1.9 «(3) that r is a semigroup with respect to the operation of multiplication of subsets of semigroups. It follows from 1.9 (0) that r is a commutative semigroup with respect to the operation of set intersection. 1.11. The properties indicated in 1.10 of the semigroups of all the two-sided ideals of the semigroup 91 are connected with the properties of 91 itself. Without going into these connections in detail let us consider as an illustration the question of identities and zeros in these semigroups. (ct.) We consider r with respect to the operation of set union. 91 itself, being an element of r, will clearly be the zero of the semigroup r. A two-sided ideal 91 will be the identity of r if its union with any two-sided ideal ::t is again 'X. But this occurs only when 91 c :t. Thus the identity of r is a two-sided ideal 91 which is included in every two-sided ideal of 91. It follows from this that the semigroup r possesses an identity if and only if the intersection of the set of all the two-sided ideals of 91 is nonempty. «(3) We consider r with respect to the operation of multiplication of subsets of semigroups of 91. The zero of the semigroup r will be a two-sided ideal 91 such that 91::t = ::t91 = 91 for any two-sided ideal::t. But since 91::t c ::t, consequently, 91 must be included in all the two-sided ideals of the semigroup 91. If such a two-sided ideal 91 exists, i.e., the intersection of the set of all twosided ideals of 91 is nonempty, then it must be the zero of the semigroup r. In fact, for such an 91 and any two-sided ideal 'X of the semigroup 91, the products 91'X and 'X91 are two-sided ideals of 91 belonging to 91. Consequently,

91::t = 'X91 = 91. H a two-sided ideal 9J1 is the identity of the semigroup r, then, in particular, it must satisfy 919J1 = 91. But 919J1 c 9J1 and thus 9J1 = 91.

144

[CHAP.

IDEALS

IV

Thus the semigroup 'l( itself is the only possible identity in r. If ~r is a semigroup with an identity, then 'l( must be the identity ofthe semigroup r. In fact, for any X i.e., X'l(

E

r,

= X.

X'l(

C

X'l( ;:, XE21 = X,

X,

Analogously, 'l(X

= X.

If ~ possesses no identities, then in some cases 'l( does not have to be an

r

r

identity of the semigroup and, consequently, in this case has no identities. An example of this is the multiplicative semigroup of all the even natural numbers for which 'l( . 'l( ~ ~ and hence 'l( is not an identity of r. (y) We consider r with respect to the operation of set intersection. As in the preceding case r will possess a zero if and only if the intersection of the set of all the two-sided ideals of the semigroup 'l( is nonempty. This intersection is the zero of r. As may be clearly seen, the semigroup 'l( itself will always be the identity of r. 1.12. If r' is the collection of all the two-sided ideals of some ring 'l(, then we consider in r' the following operation (usually called the multiplication of idea Is). If Xl' X z E r', then X3 = Xl 0 X 2 is the collection of all the elements of 'l( representable in the form Til)T~I)

+ TiZ)T~Z) + ... + Tin)T~n)

(Ti i ) E XIT~i)

C

X2 ;

i

= 1,2, ... ,n).

It is not difficult to see that r' is a semigroup with respect to this operation. The investigation of this semigroup of ideals plays an important role in the theory of rings. l 1.13. Many properties of semigroups may be characterized with the aid of various properties of systems of ideals of semigroups. As Iseki [11] showed, the important property of regularity of a semigroup (II, 6.1) may be thus characterized. Here there is a well-known similarity with the situation in the theory of rings. 2 THEOREM. In order for a semigroup 'l( to be regular, it is necessary and sufficient that for each of its left ideals £ and for each of its right ideals 9\ we have

9\£ = 9\ n £. (1) Since 9\£ C 9\ and 9\£ C £, we see that 9\£ C 9\ n £. Let ~ be regular. We choose in 9\ n £ an arbitrary element A. We find for it an X E ~ such that A = AXA. But since A E £ it follows that XA E £. Thus A = AXA = A . XA E 9\£. PROOF.

Consequently, 9\ n £

C

9\£ and thus 9\ n £ = 9\£.

1 See, for example, N. Jacobson, The theory of rings, American Mathematical Society Mathematical Surveys, Vo!. I, Amer. Math. Soc., New York, 1943. 2 L. Kovacs, A note on regular rings, Pub!. Math. Debrecen 4 (1956), 465-468.

Sec. 2]

CHAINS OF SUBSETS OF AN ARBITRARY SET

(2) Let the given property hold for the ideals in~. For an A a right ideal 9t = A'l.t u 'l.t. By assumption, we then get

145 E~

we choose

9t = 9t (") ~ = 9t~ = (A~ u A)~ c A~. Consequently, A E A'l.t. We can prove similarly that A C ~A. But A~ is a right ideal of~, while ~A is a left ideal of~. Thus A E A'l.t (") ~A

= A~~A

from which it follows that for some X

E

C

A~A,

'l.t it is true that A = AXA.

1.14. COROLLARY. In order for a commutative semigroup ~ to be regular, it is necessary and sufficient that for each of its ideals ~ we have ~~=~.

PROOF.

(1) If ~ is regular, then by 1.13 we have ~~

= ~ (") ~ = ~.

(2) Let the indicated property of ideals hold in~. For an arbitrary pair of its ideals, ~1 and ~2' their intersection ~1 (") ~2 will itself be an ideal. Thus ~1 (") ~2

= (~1

n

~2)· (~1 (") ~2) C ~~2'

and since always ~1 (") ~2 :::l ~~2' we obtain ~l of'l.t follows from this by 1.13.

(") ~2

=

~1~2.

The regularity

2. Chains of Subsets of an Arbitrary Set 2.1. We consider some general concepts and properties of subsets of an arbitrary set. Their choice is determined by our desire to introduce with their aid in the following sections a series of properties of ideals of semigroups which were obtained earlier by means of direct constructions by N. N. Vorob'ev [3; 6] and Green [1]. Let r be some nonempty collection of nonempty subsets of some set ill1. DEFINITION. A set MEr is said to be MINIMAL in r if none of its proper subsets belong to r. A set MEr is said to be UNIVERSALLY MINIMAL in r ifit is a subset of every set belonging to r. MAXIMAL and UNIVERSALLY MAXIMAL sets in r are d~fined analogous~y. 2.2. In the remainder of this section, unless we indicate to the contrary, we will understand by r an arbitrary nonempty collection of nonempty subsets of a set ill1 which satisfies the conditions: (ex) the set ill1 itself belongs to r; (p) the union of any nonempty collection of sets from r belongs to r; (y) the intersection of any collection of sets of r belongs to r if it is nonempty.

146

[CHAP. IV

IDEALS

2.3. If the intersection Mo of all the sets of r is nonempty, then Mo is clearly a universally minimal set in r. In this case r has no minimal sets other than Mo. If the indicated intersection Mo is empty, it is easy to see that there does not exist a universally minimal set in r. 2.4.

Clearly, the following relation in 9J1 is an equivalence.

DEFINITION. Elements x and y of9J1 are said to be r-EQUIVALENT of r that contains one of them must contain the other. A class of r-equivalent elements is said to be a r-LAYER.

if any set

2.5. 9J1 is the nonintersecting union of classes ofr-equivalent elements, i.e., the nonintersecting union of all of its r -layers.

2.6. For a nonempty subset N of the set 9J1, its r-envelope is the intersection of all the sets of r containing N. Of course there are such sets, for 9J1 itself belongs to r. Obviously a r-envelope is the universally minimal set in the collection of those sets of r which contain N. 2.7. THEOREM. Two elements x andy of9J1 belong to the same r-Iayer only if their r-envelopes coincide.

if and

PROOF. (1) Let the r -envelopes of the elements x and y coincide. If some set MEr contains x, then it contains the r-envelope of x, and hence the renvelope of y and thus y itself. The converse is analogous. Thus x and yare r -equivalent. (2) Let x and y be r -equivalent. The r -envelope of x, inasmuch as it belongs to r and contains x, must also contain y. The converse is analogous. Hence the r-envelopes of x and y coincide. 2.8. There is another approach to the concept of a r-layer other than those which were employed in 2.4 and 2.7. DEFINITION. Two distinct sets from r are said to be ADJACENT in r if one of them Ml contains the other M2 and there is no set M' in other than Ml or M2 such that M 1 :::> M':::> M 2 •

r

2.9. THEOREM. The set N c 9J1 will be a r-layer if and only following two conditions is satiified: (1) N belongs to r and is a minimal set in r; (2) there are in r two adjacent sets Ml :::> M2 such that

if one

of the

N= M 1\M2 .

PROOF. (1) Let N be some r-Iayer. We denote by Ml the r-envelope of N and by M2 the union of all the sets in r that are in Ml but have no common elements with N (M 2 may be empty).

Sec. 2]

CHAINS OF SUBSETS OF AN ARBITRARY SET

147

Clearly, MI \M2 ::::> N. Let x EN and y E MI \M2 • If x is contained in some set M' E r, then, by the definition ofa r-Iayer, M' ::::> N, i.e., M' ::::> M I , and thus M' =:1 y. If Y is included in some set Mil E r, then Ml n Mil =:1 y. Since y E M 2 , MI n Mil is not included in M 2. Thus MI n Mil possesses an element z belonging to N. But then Ml n M":3 x, i.e., x E Mil. From what has been said it follows that x and yare r-equivalent. It follows from this that M I \M2 c N. Hence, N = M 1 \M2 . It follows from the definition of a r-Iayer that there does not exist a set M3 E r such that Ml ::::> M3 ::::> M 2 , where Mg ':;6 Ml> Ms ':;6 M 2 . If M2 is empty, then N = M is a minimal set in r. (2) If M is a minimal set in r, then any set M' from r either contains M or has no elements in common with M. In fact, in the contrary case M n M' would be a set from r contained in M and distinct from M. Thus any two elements of Mare r-equivalent. Elements that do not belong to M cannot be r-equivaJent with elements from M. Let the sets MI and M2 be adjacent in r, where MI :::J M 2• Clearly, no element of M 1\M2 can be r-equivalent with any element that is not included in M I \M2 • If we can show that any two elements of MI\M2 are r-equivalent, this will mean that MI \M2 is a r-Iayer. Let us assume the contrary. Let two elements x and y of MI \M2 not be r-equivalent. This means that one of them, say x, is included in some set M' E r which does not contain the

element y. But then the set from r, Mil = (M'

n

M 1 ) U M 2,

is included in Ml and contains M 2. Moreover, it is distinct from Ml since it does not containy, and it is distinct from M2 since it contains x. The existence of such a set contradicts the fact that Ml and M2 are adjacent. 2.10. There is another approach to the concept of a r-layer that is associated with the concept of a r -chain. DEFINITION. A nonempty collection 2: c r is said to bea r-CHAIN if, for any two sets in 2:, one must necessarily be a subset of the other. A r-chain L: is said to be a PRINCIPAL r-chain if there does not exist a r-chain L:' distinct from L: such that 2: c 2:'. 2.11. THEOREM. (f3) and (y). PROOF.

Any principal r-chain 2: itself possesses properties 2.2 (cx:),

Adjoining the set 9J1 to 2:, we clearly obtain a r-chain containing 2:.

It must coincide with L:. Consequently, 9J1 E 2:.

Let M be the union of some sets from 2:: M=UM;.. ;.

By the definition of r, M belongs to r.

148

IDEALS

[CHAP.

IV

We take an arbitrary set B from:l:. If B is contained in some component M~ of our union, that is, if B c: M~, then B c: M. If B is not included in any component M A, then, by the definition of a r -chain, B must contain all the MA and hence B :::> M. From this it follows that the collection consisting of.:l: and the set Mwill bear-chaincontaining.:l:. Itmustcoincidewith.:l:. ConsequentIy,M E.:l:. Analogously, if M' is a nonempty intersection of sets from .:l:, then it either contains or is contained in every set from.:l:. Thus, adjoining the set M' to .:l:, we 0 btain a r -chain containing:l:. It must coincide with .:l:, from which it follows that M' E.:l:. 2.12. The role of principal r -chains is determined by the fact that any rchain can always be extended to a principal r-chain. THEOREM.

For any r-chain:l:, there is always a principal r-chain.:l:' such that

.:l: c: .:l:'. PROOF. Let \P be the collection of all the r-chains containing a given r-chain.:l:. If.Q c: \p, where.Q is such that, for any two r-chains belonging to .0, one must always be contained in the other, then .:l:.Q, the union of all the rchains in.Q, will itself be a r-chain. In fact, assume M I , M2 E .:l:.Q. Then there are r-chains:l: 1 and:l: 2 in.Q such that Ml E.:l:l and M2 E .:l:2' If.:l:l c: .:l:2' then the sets Ml and M2 both belong to the chain .:l:2 and thus one of them is a subset of the other. This means that:l:.Q is a r-chain.

By what has been said we may apply Lemma III, 4.6 to \p. By this lemma there exists in \p a r-chain :l:' which is not included in any other r-chain of \p. Clearly, :l:' will be the principal r-chain that we are looking for. 2.13. By Theorem 2.11 all the definitions and results above for the collection r are applicable to a principal r-chain.:l:. Clearly,.:l: is a .:l:-chain for the collection .:l: itself. Here it is obviously a principal.:l:-chain and there can be no other principal .:l:-chains different from .:l: itself. Consideration of the principal r-chain.:l: is helpful because the concepts of the r-layer and the :l:-layer turn out to be equivalent. THEOREM. Jf.:l: is a principal r -chain, then each r -layer is a .:l:-layer and each .:l:-layer is a r-layer.

PROOF. (1) Let us assume that in some r-Iayer N there are two elements belonging to two distinct .:l:-layers. This means that one ofthese elements, say x, is included in some set N' from .:l: which does not contain y, the second of the elements. But this is impossible, for x and yare r -equivalent and N' E.:l: c: r. (2) Now let us assume that there are in some .:l:-layer P two elements belonging to two distinct r-layers. This means that one of these elements, say u, is included in some set M from r which does not contain D, the second of the elements.

Sec. 2]

CHAINS OF SUBSETS OF AN ARBITRARY SET

149

P cannot be a minimal set in~. Otherwise, adjoining to ~ the set P n M, which is nonempty since it contains u and is distinct from P since it does not contain v, we would obtain a new r-chain containing ~ and distinct from~. But this would contradict the fact that ~ is a principal r -chain. Since P is not a minimal set in~, by 2.9 there are in ~ two sets Ml and M2 adjacent in ~ such that

The set from r, is included in M 1 and contains M 2· It is distinct from M 2 since it contains u, and distinct from M 1 since it does not contain v. Consequently Mil does not belong to~. But then, adjoining the set Mil to ~, we would clearly obtain a new rchain containing ~ and distinct from ~, which is impossible. (3) By 2.5 the set 9J1 may be represented as the union of all the r -layers, and by 2.11 as the union of all the ~-layers. If we take one of the components of the first factorization N, and one of the components of the second factorization P, it follows from the discussions in the first two parts of the proof that Nand P either coincide or have no common elements. It directly follows from this that the two unions considered consist of the same elements. Thus, each r -layer is included in the second union, i.e. is some ~-layer, and conversely. 2.14. It follows from 2.13 that all the r-layers may be obtained, starting from any principal r -chain. If ~ is a principal r -chain, then the set N will be a r -layer if and two conditions is sati.ified: (1) N belongs to ~ and is a minimal (and thus a universally minimal) set in~. (2) There are in ~ two adjacent sets Ml ::;, M2 such that COROLLARY.

only

if one of the following

N= M 1\M2 •

2.15.

We note also the following corollary of 2.13.

COROLLARY. If ~l and ~2 are two principal r -chains, then the collection of all ~l-layers coincides with the collection of all ~2-layers.

In fact, by 2.13 the collection of all ~l-layers and the collection of all ~-layers both coincide with the collection of all r-layers. 2.16. In the study of the collection r the conditions of minimality and maximality often play an important role. r is said to satisfy the condition of minimality if each nonempty subset r t ofthe set r contains a set minimal in r'. It follows directly from the definition that if r satisfies the property of minimality, then any r-chain forming a descending sequence

150

[CHAP.

IDEALS

IV

possesses the property that, starting with some n, all its members coincide. It is not difficult to convince oneself of the converse. Let the indicated property about decreasing sequences be satisfied in r. In an arbitrary r' c r we take an arbitrary setMI E r'. If MI is not minimal in r', then we takesomeM2 E r'suchthat M2 c M I, M2 ;of MI' Then, starting from M 2 , we choose Ms c M 2 , Ms ;of M2 and so on. By assumption, the construction of such a sequence must break off at some member Mn. Obviously, Mn E r' will be minimal in r'. 2.17. The condition of maximality is defined and its equivalence with the conditions of stability of any increasing sequence is established analogously. Another necessary and sufficient condition for the property of maximality is the requirement that in each MEr there exists a finite subset M' c M such that the r' -envelope of M' is equal to M. In fact, let the condition of maximality be satisfied. We choose in the set MEr an arbitrary element Xl and consider its r-envelope Xl' If Xl ~ M, we choose in M\X1 an element X 2 and consider the r-envelope X 2 of the set {Xl' x 2}. We then construct X s, and so on. We obtain an increasing sequence of sets in r, Xl

C

X2

C

.•. C

Xn

C

.•.•

Since Xi - 1 ;of Xi' the sequence must break off at some member X m , which means that M = Xm, i.e., M is the r-envelope of the finite set {Xl' X 2, •.• , x m }. Now let each set of r be the r-envelope of some finite set. We consider an arbitrary increasing sequence of sets from r: Ml c M2 C

Msc .... The set N

= Un M n belongs to

r and is the r -envelope of some finite set For some k all the Xi are included in M k • Consequently, Mk = N::> Mi (i = 1,2, ... , m), i.e., Mk = Mk+1 = ....

{Xl' X2' ... ,Xm }.

2.18. The collection r satisfies simultaneously the condition of minimality and the condition of maximality only when it is finite. In fact, because of the condition ofminimality there must exist in r a minimal set MI' In the collection of all sets in r containing MI but distinct from MI there must exist a minimal set M 2 • Repeating the argument, we obtain an increasing sequence which must break off:

MI c M2

C

.•. C

M n- 1

C

Mn.

It follows from the construction that we have obtained a principal r-chain. The number of its layers is finite. By 2.13 it follows from this that the number of all the r -layers is also finite. Since each set of r is the union of certain r -layers, the number of all the sets in r is also finite.

3. Principal Ideals and Ideal Layers 3.1. As was already mentioned, the study of properties of collections of subsets of an arbitrary set that was made in the preceding section will now be

Sec. 3]

PRINCIPAL IDEALS AND IDEAL LAYERS

151

applied to collections of ideals of semigroups. Corresponding properties of ideals were obtained in part by Green [1], and in part by N. N. Vorob'ev [3; 6]. We take as me the set of all the elements of the semigroup ~, and as r a collection of ideals of ~ such that ~ itself belongs to r and the union and intersection of any collection of ideals from r belong to r if they are nonempty. Then all the results obtained in § 2 are automatically satisfied for such a collection of ideals. 3.2. We will be interested in the following three collections: (C(:) the collection of all the left ideals of a semigroup; (f3) the collection of all the right ideals of a semigroup; (y) the collection of all the two-sided ideals of a semigroup. It follows from 1.6 (C(:), (y) and (0), that each ofthese possesses the above property. All the results obtained in § 2 for the collection r are valid for each of these collections. Each of these results, when formulated for the collection (C(:), (f3) or (y), is an important property of ideals of semigroups. Clearly, there is no need to reformulate in detail the results of § 2 for the indicated collections of ideals since this may be done without difficulty and in a completely automatic way. 3.3. When used in connection with ideals, the terms introduced in the preceding section are correspondingly changed. Let r be the collection of all the left ideals of a semigroup ~. An ideal minimal in r is said to be a minimal left ideal of ~. r-equivalence becomes left ideal equivalence. A r-Iayer is said to be a left ideal layer. A r-envelope becomes a left ideal envelope. The corresponding concepts are analogously formed when r is the collection of all the right ideals of ~ or the collection of all the two-sided ideals.

3.4.

THEOREM.

If 91 is a nonempty subset of the semigroup ~91

~,

then

u 9c

is the left ideal envelope of 91; is the right ideal envelope of 91; ~91~

u

~91

u

91~

u 91

is the two-sided ideal envelope of 91. PROOF.

Since

91) = ~~91 u ~91 c ~91 c ~91 u 91, it follows that ~91 U 91 is a left ideal of~ containing 91. But every left ideal of~ containing 91 must also contain ~91 and 91. Thus ~91 u 91 is the intersection of the set of all the left ideals of ~ that contain 91., ~(2I91 U

152

[CHAP.

IDEALS

IV

The argument is analogous for right ideals. Since 1ll(1ll911ll u 11191 u mill u 91) = 1ll1ll911ll u 1ll1ll91 u 1ll911ll u 11191 c 1ll911ll u 11191 u 91~l u 91,

we see that 1ll911ll u 11191 u 91~ u 91 is a left ideal of III containing 91. Similarly, we can show that it is also a right ideal. It is easy to see that any two-sided ideal ~ containing 91 must also contain 1ll911ll u 11191 u 911ll u 91. Thus, this twosided ideal is the intersection of the set of all the two-sided ideals of ~ containing 91. 3.5. When III possesses an identity, the expression for ideal envelopes is simplified, since 11191 :::> 91,

91~:::>

91, 1ll911ll:::> III 91,

~91Ill:::>

mill,

1ll91~{:::>

91.

COROLLARY. Ifill is a semigroup with an identity, then 11191 is the left ideal envelope of 91 c Ill, 911ll is the right ideal envelope and 1ll9l~ is the two-sided ideal envelope.

3.6. Especially important is the case when the subset 91 consists of one element. DEFINITION. A left ideal of the semigroup ~ which is the left ideal envelope of one of its elements is said to be a PRINCIPAL LEFT IDEAL.

Principal right ideals and principal two-sided ideals are defined analogously.

It follows directly from 3.4 and 3.5 that a principal left ideal of a semigroup III has the form IllX u X, where X is some element

of~.

If ~ possesses an identity, then IllXuX= 1llX.

Analogously, a principal right ideal has the form XIll U X, or XIll in the presence of an identity. A principal two-sided ideal has the form IllXIll

U

IllX

u

XIll

u

X,

or IllXIll if III has an identity. 3.7. The relation of inclusion between principal left ideals of elements of a semigroup determines the important relation of divisibility on the right (and analogously of divisibility on the left). THEOREM. Let A and B be two distinct elements of a semigroup Ill. The element A is divisible on the right by the element B if and only if the left ideal envelope of the element A is included in the left ideal envelope of the element B.

Sec. 3] PROOF.

PRINCIPAL IDEALS AND IDEAL LAYERS

153

If, for some X E m:,

A= XB,

then

m:A

u A = m:XB u XB c \lIB u B.

On the other hand, if

m:A then, taking into account that A i.e., for some X

E

m:

uA c ;6

m:B u

B,

B, we have

A c m:B, it must be true that

A=XB. 3.8. By 2.5 a semigroup may be represented as the nonintersecting union of all of its left ideal layers, as the nonintersecting union of all of its right ideal layers and also as the nonintersecting union of all of its two-sided ideal layers. As was shown in the preceding section, an ideal layer (left, right, or two-sided) may be determined in various different ways. By 2.7 two elements A and B of a semigroup m: belong to the same left ideal layer if their left ideal envelopes coincide. Obviously, for this it is necessary and sufficient for A to be contained in the left ideal envelope of B while B is contained in the left ideal envelope of A. It is also possible to use Theorem 3.7. In order for two distinct elements of a semigroup to be included in the same left ideal layer it is necessary and sufficient for each of them to be divisible on the right by the other. It is clear that an analogous assertion is valid for right ideal layers.

3.9. If a left ideal B of a semigroup m: contains some element X, then by definition of left ideal equivalence, B will also contain all the elements leftideally equivalent with X. It follows from this that the left ideal B is the nonintersecting union of certain left ideal bands of the semigroup. Of course, in the general case not every union of left ideal bands is a left ideal. The left ideal B can also be obtained from principal left ideals. For each X E B the left ideal envelope of X is contained in B and contains X. Thus B is the union of the left ideal envelopes of all the elements X contained in B. The converse is clear: any union of principal left ideals is a left ideal. If there occur in such a union the left ideal envelope of an element X and the left ideal envelope of an element Y, where X is divisible on the right by Y, then by 3.7 it is possible to exclude from the union the left ideal envelope of the element X. Thus in certain cases (but of course not always), for example, when m: is finite, each left ideal may be represented as the union of principal left ideals such that no two of them are left ideal envelopes of elements where one is divisible by the other on the right. Clearly, it is impossible to discard from such an ideal any of the principal left ideals included in it. The representation of a left ideal B in the form of the

154

IDEALS

[CHAP.

IV

indicated union .\} = U. X. has the following important property. If.\} is represented in the form of some union of principal left ideals .\} = U; U", then all the X. must be included in its components Ui. In fact, ifX. is a left ideal envelope of the element T., then T. is included in some U. which is the left ideal envelope of some element Ap" But AI' must be included in some X;.- By the definition of left ideal envelopes we obtain X. c Up' Up C XA, i.e., X. c XA- By assumption this is possible only for X. = X;., and then X. = Up" From w:q.at has been proven it follows in particular that every left ideal can have no more than one representation in the form of the union of principal ideals such that no two of them are left ideal envelopes of elements where one element is divisible by the other on the right. The situation is analogous for right ideals. Two-sided ideals are also all the possible unions of principal two-sided ideals.

3.10. As was indicated in the work of Munn and Penrose [1] (see also Miller and Clifford [1]), the existence and properties of idempotents in left ideal and right ideal layers is connected with certain important properties of semigroups. We first consider the relationship between idempotents included in the same left ideal layer. THEOREM. If the idempotents 11 and 12 are both included in the same left ideal layer, then each of them is a left zero for the other. If they are both included in the same right ideal layer, then each of them is a right zero for the other. PROOF. 12 is included in the left ideal IltI2• Inasmuch as 11 belongs to the same left ideal layer as 12 , it must also belong to this left ideal, i.e., for some Xem:, But then XI2 = 11• 3.11. COROLLARY. No two distinct idempotents included in the same left ideal layer can be commutative with each other. 1112

=

X12 · 12

=

In fact, by the preceding discussion, we have for the indicated idempotents 11 and 12 1112 = 11> 1211 = 12, and then for 1112 = 1211, we obtain 11 = 12.

3.12. THEOREM. In order for an element A of a semigroup m: to be regular (II, 6.1) it is necessary and suffiCient for the left ideal layer containing A to have an idempotent. PROOF.

(1) If A is regular, then, for some B em:, ABA = A,

(BA)2 = BA.

Sec. 3]

PRINCIPAL IDEALS AND IDEAL LAYERS

155

Since A = ABA is divisible on the right by BA and BA is divisible on the right by A, it follows by 3.8 that A and the idempotent BA lie in the same left ideal layer. (2) If the idempotent I lies in the same left ideal layer as A, then, for some B and C E 'll,

I=BA, From this we obtain

ABA

A

=

CI.

= Al = CI· 1 = C1 = A,

i.e., A is regular. 3.13. It follows directly from the theorem that in any left ideal (and analogously in any right ideal) layer either all the elements are regular or all of them are nonregular. If all the left ideal layers contain idempotents, then all the right ideal layers also contain idempotents, and conversely. The existence of idempo tents in all the left ideal layers is a necessary and sufficient condition for a semigroup to be regular. 3.14. THEOREM. In order for a semigroup to be inverse (II, 7.2) it is necessary and sufficient that there be in each of its left ideal layers and in each of its right ideal layers a unique idempotent. PROOF. (1) If'll is an inverse semigroup, then it is regular and by 3.12 each of its left ideal layers and each of its right ideal layers contains idempotents. Since all the idempotents of an inverse semigroup are commutative (II, 7.4) it follows by 3.11 that no layer contains more than one idempotent. (2) Let there be in each left ideal layer and in each right ideal layer of a semigroup III exactly one idempotent. By 3.12, III is regular. Let us assume that the two elements B1 and B2 are both regularly associated with some element A E 'll. Since BiA and A (equal to ABiA) are divisible by each other on the right, by 3.8 the idempotent BiA is included in the same left ideal layer as A. Consequently, BIA = B2A. Considering right ideal layers analogously, we obtain ABI = AB2 • Because of this B1 = B1 ABI = B2ABl = B2AB2 = B2· Thus an arbitrary element A has only one element regularly associated with it. 3.15. Using the theorem just proven, it is also possible to weaken the condition that singles out the inverse semigroups in the class of all regular semigroups. COROLLARY. Ifin a regular semigroup none of the idempotents have elements regularly associated with them other than themselves, then the semigroup is inverse. PROOF. By 3.12 there are idempotents in each left ideal layer. Let us assume that some of these contain two idempotents II and 12 • By 3.10,

1112

=

II'

1211

= [2·

156

IDEALS

[CHAP.

IV

From this we get that 11/2/1

= 1111 =

II'

12//2 = 1212 = 12,

i.e., 11 and 12 are regularly associated. By assumption this is possible only when 11 =/2, The argument is analogous for right ideal layers. By Theorem 3.14 the semigroup is inverse.

3.16. equality

COROLLARY.

If, in a regular semigroup ~,for any idempotent I the

(XE~) lXl= I is satisfied only for X = 1, then ~ is a group.

PROOF. By 3.15, ~ is an inverse semigroup. It follows from this that all its idempotents are commutative (II, 7.4). LetJ1 and 12 be idempotents of~. Since they commute, 1112 is an idempotent. Obviously,

Thus 11 = 1112 and 12 = 1112 , i.e., II = 12, Consequently, there is only one idempotent in III and, by II, 6.10 ({J), ~ is a group. 3.17. Theorem 3.14 allows us to clarify the structure of systems of principal left ideals and of principal right ideals of an inverse semigroup (II, 7.2). Let ~ be an inverse semigroup and ~ the commutative subsemigroup of its idempotents (II, 7.4). Let £ be an arbitrary principal left ideal which is the left ideal envelope of an element A. By 3.14 there must exist some idempotent I in the same left ideal band that contains A. Since the left ideal envelopes of A and I must coincide, £=~luI=~I.

If the idempotents 11 and 12 are distinct, the equality WI = ~12 is impossible, for it would mean that II and 12 would lie in the same left ideal layer, which would contradict 3.14. Thus the idempotents of~ and its principal left ideals may be put into one-toone correspondence, 1,....,~I.

Similarly, the idempotents and the principal left ideals may be put into one-toone correspondence, 1,....,/~.

These correspondences preserve the partial orderings in the sets of principal ideals and in ~ where in the sets of principal ideals the ordering is defined as the relationship of inclusion, while in ~ it is defined in the following manner. We assume that II ;;., 12 if 1112 = 1211 = 12 , In fact, if

Sec. 4]

157

TWO-SIDED IDEAL CHAINS

then, for some X

'lr,

E

from which it follows that

1211 Conversely, for 1211

=

'lr12 Since, for II, 12 E.£),

3.18.

=

XII . II

XII

=

12,

= 12 we have

Is = 1112

E

.£),

= 'lr12 II IlIa

C

'lrl1.

= 13 ,

it follows from what was proven in 3.17 that

On the other hand, if X

E

'lrll n 'lr12' then, for some AI' A2

E

'lr,

X = AlII = A 212 • Also, since II and 12 commute (II, 7.4), we see that

X

=

A212 = A2I212

=

XIz = All112 = All11112 = XIi2

= XI3•

This means that X E 'lrla' Thus we have shown that ~Ul

n'lrI2

= ~Us'

The intersection of the principal left ideals of an inverse semigroup is itself a principal left ideal. It follows from this that the collection of all the principal left ideals forms a commutative semigroup of idempotents with respect to the operation of intersection. From the proven equality

'lrIl n 'lrI2

= 'lr(1112)

it follows that the correspondence

'lrl

f""oo.I

I

is an isomorphism between this semigroup and the subsemigroup of all the idempotents .£) of the semi group 'lr.

4. Two-Sided Ideal Chains 4.1. The properties of ideal layers of a semigroup are closely connected with the properties of chains of ideals. If is the collection of all the two-sided ideals of the semigroup ~r, then a -chain is said to be a two-sided ideal chain. Left ideal chains and right ideal chains are defined analogously. It follows from 2.12 that every two-sided ideal chain can be extended to a principal two-sided ideal chain. The situation is the same for left ideal and right ideal chains.

r

r

158

IDEALS

[CHAP.

IV

Properties concerned with the satisfaction of the conditions of minimality and maximality in the collection of all the left, right or two-sided ideals are formulated directly from the properties considered in 2.16, 2.17 and 2.l8. 4.2. The study of adjacent ideals (2.8) is also related to that of ideal chains. For the fact that two two-sided ideals are adjacent is equivalent to the fact that they form a two-sided ideal chain that has the property that there is no other two-sided ideal chain for which the given ideals serve as ends. By 2.9 a subset ~ of a semigroup m: is a two-sided ideal layer if and only if~ is a minimal two-sided ideal ofm: or if there exist two adjacent two-sided ideals::t1 and :t2 in m: such that ~ =

:t1\:t2·

The analogous statement is true for left ideal and right ideal layers. 4.3. If~ is a principal two-sided ideal chain, then by 2.13 and 2.14 all the two-sided ideal layers may be obtained as ~-layers, i.e., all the two-sided ideal layers consist of a minimal two-sided ideal belonging to ~ (if such a one exists) and of sets of the form ::tl\:t2' where :tl and :t2 are adjacent members in the chain~. The analogous proposition is true for left ideal and right ideal layers and chains. 4.4. Further discussions in this section will be concerned only with twosided ideal chains. This stems from the great importance of two-sided ideals in comparison with one-sided ones, and also from the fact that the construction given here for ideal factors is practicable only for two-sided ideal layers. We first note the important role of minimal two-sided ideals. 3 A semigroup cannot possess more than one minimal two-sided ideal. The validity of this follows from 2.2 and from the following property. A minimal two-sided ideal of a semigroup is always a universally minimal twosided ideal.

In fact, by 1.9 (15) the intersection of a minimal two-sided ideal ::t with an arbitrary two-sided ideal ::t is a two-sided ideal of the semigroup contained in :t. Consequently, it must coincide with ::t, from which we have that :t c :t/. The existence in a semigroup of a minimal two-sided ideal is always an important property. Its role is essential for the consideration of various properties of the semigroup. In the literature the minimal two-sided ideal is often called the kernel of the semigroup, or the ideal kernel, or the kernel of Suskevic. This latter term is explained by the fact that A. K. Suskevic [3] first drew attention to the role of such ideals and investigated their various properties. The numerous subsequent works in this direction are to a great degree connected with the development and generalization of the first results of A. K. Suskevic. l

3 It must be remembered that in the literature the term "minimal two-sided ideal" is sometimes given a wider sense, which includes among minimal two-sided ideals the minimal two-sided nonzero ideals discussed in § 3 and § 4 of the next chapter.

Sec. 4]

TWO-SIDED IDEAL CHAINS

4.5. DEFINITION. Let 91 be IDEAL FACTOR corresponding to

159

a two-sided ideal layer of a semigroup Ill. By the

91 we will mean the following semigroup 91*: (1) If 91 is a semigroup, then 91* = 91. (2) If 91 is not a semigroup, then 91* consists of all the elements of 91 and of a new element 0*. If for x, Y, Z E 91 it is true in III that XY = Z, then we assume that XY = Z also in 91*. In all the remaining cases we assume that NIN2 = 0* (N!, N2 E 91*). The associativity of the operation defined in 91* may be verified without difficulty. We note that in the literature the ideal factor is sometimes defined even in the first case by the external adjunction of a zero to the semigroup 91. It is easy to see that the distinction here is completely immaterial. In view of 4.2 the notation sometimes used for the ideal factor, 91* = :Il - :I2, is quite natural. 4.6. We will show by example that the construction of factors cannot in general be effected for left ideal layers. Let D be the denumerable set Q

=

{(Xl' (X2' ... , PI'

P2' ... , Yv Y2' ...}.

We consider in the semigroup 6 0 the following three transformations, given by the permutations:

A= B=

c=

((Xl (Xl ((Xl (X2 ((Xl (Xl

(X2

(Xa

PI

P2 Pa

YI

Y2 Ya

(X2

(Xa

Y2 Ys Y4

YI

YI

(X2

(Xg

PI

P2 Pa

YI

Y2 Ys

(X4

(Xs

(Xl

(Xg

C(5

YI

Yl Y1

(X2

(Xg

PI

P2 Pa

YI

Y2 yg

(X2

(Xs

YI Yl YI

YI

Yl YI Yl

) ...) ... ...). ... ... ...

;

;

It follows from II, 3.1 that A is divisible on the right by B, and B is divisible on the right by A. Neither A nor B is divisible on the right by C. We denote by 91 the left ideal layer of the semigroup 6 0 that contains A. By 3.8, B E 91, but C E 91. By direct multiplication we verify that AB = Band AA = C. Since AA E 91, 91 is not a semigroup. We consider the set 91* consisting of all the elements of 91 and of a new element 0*. We define an operation in 91* as was done in 4.5. As a result of this 91* turns out to be a multiplicative set. However, 91* is not a semigroup, for the property of associativity is not satisfied in 91*. In fact, by the operation in ~, AB = Band AA E 91, so that in 91*,

A(AB) = AB = B,

(AA)B = O*B = 0*.

4.7. THEOREM. Let 91* be the ideal factor of a semigroup III corresponding to a two-sided ideal layer 91. If 91* = 91, then 91* is a semigroup that contains no proper two-sided ideals. If 91* = 91 U 0*, then either 91* is a semigroup having

160

IDEALS

[CHAP.

IV

no proper two-sided ideals other than the zero ideal 0*, or 91* is a semigroup in which the product of any two elements is equal to the zero element 0*. PROOF.

(1) Let 91* = 91. By 4.2, 91

=

~1\~2'

where ~l and ~2 are adjacent two-sided ideals (or ~2 is the empty set while ~l is a minimal two-sided ideal). Let ~ be an arbitrary two-sided ideal of 91. We consider the set

~' = ~l~~l U ~2'

Since ~l and ~2 are two-sided ideals of Ill, it clearly follows that ~ is also a two-sided ideal of Ill, where Also, ~' :;6 ~2' since

Consequently, ~'= ~l = 91 U X2 • But XIS Xl C ~ U X2 • Thus ~' will contain 91 only when 91 c~. Hence ~ cannot be a proper ideal of 91. (2) Let 91* = 91 u 0 *, and, as in the first part, 91

= ~1\~2'

Let the semigroup 91* have elements whose product yields an element other than the zero 0*, i.e., X1 X2 C ~2' Let S* be an arbitrary two-sided ideal of the semigroup ~* possessing elements other than the zero 0*. We denote by ~ the set of all the elements of III that are included in S*. The set ~ is nonempty, and the set is a two-sided ideal of III included in Xl and containing X2 • We denote by iJ the collection of those elements Fin Xl for which XIF c X2" iJ is a two-sided ideal of Ill, since

Here Xl ~ iJ ~ X2 • But Xl~ C X2• Thus iJ = X 2 • It follows from this that for each T E Xl\X2 there exists a T' E Xl such that T'T E X 2 • We show similarly that for each T E ~\X2 there exists a T" such that IT" E ~. Let HE S. For some H' E ~l' H'HE~2

and for some H" E Xl'

(H'H)H" E X2•

It follows from this that the two-sided ideal~' of the semigroup III is distinct from X2• Consequently,~' Xl' But, from the fact that ~* is a two-sided

=

Sec. 4]

TWO-SIDED IDEAL CHAINS

161

ideal of 91*, it follows from the rule of multiplication in 91* that ~' c ~ U

:.r

2•

This is possible only when ~ = 91. But then ~* = 91*, i.e., the two-sided ideal ~* of the semigroup 91* cannot be a proper ideal of the semigroup. 4.8. If Sl is the minimal two-sided ideal of a semigroup Ill, then by 4.2, Sl is a two-sided ideal layer. Here, by 4.5, Sl* = R By 4.7 we obtain from this the following important property (which may also be developed independently). COROLLARY. IfSl is the minimal two-sided ideal of a semigroup Ill, then.R is a semigroup having no proper two-sided ideals.

4.9. A more detailed investigation of ideal factors under certain limitations on the semigroup was carried out by N. N. Vorob'ev [6]. He also considered some multiplicative properties of elements in left ideal layers. 4.10. We know that every semigroup is the union of all its two-sided ideal layers which pairwise have no common elements. By 4.5, to each two-sided ideal layer there corresponds in a natural way a semigroup closely associated with it, the ideal factor. This ideal factor consists of the same elements as the layer itself, with perhaps the addition of one new zero element 0*. Thus the semigroup-Ill may in some sense be considered as consisting of sets of semigroups, namely all its ideal factors. The latter, as we saw in 4.7 is either the very simply constructed semigroupin which the product of any two elements is equal to zero, or is a semigroup having no proper two-sided ideals or having only the one proper two-sided zero ideal. This situation explains the interest in semigroups having no proper nonzero two-sided ideals. Such semigroups are often said to be simple, a term which we will not employ in this book. We will devote a considerable part of the next chapter to the investigation of these semigroups.

4.11. Let 1:: be an arbitrary principal two-sided ideal chain of a semigroup Ill. All the ideal factors of III are obtained from its two-sided ideal layers. But the collection of the latter coincides with the collection of all the 1::-layers. Thus, all the ideal factors of the semigroup III can be obtained from the 1::-layers. Let 1::1 and 1::2 be two principal two-sided ideal chains of the semigroup Ill. As we have already remarked (2.15), the collection of all the 1::1-layers and the collection of all the 1::2-layers are identical. Hence the collection of the ideal factors obtained from the 1::1-layers coincides with the collection of ideal factors obtained from the 1::2-layers. This proposition is a strengthened parallel of the results of a number of group theorems of the type of Jordan-Holder. 4 The first 4

See, for example, A. G. Kuros, Theory of groups, 2nd ed., GITTL, Moscow, 1953;

§§ 16 and 56. (Russian)

162

[CHAP.

IDEALS

IV

corresponding result was obtained by Rees [1] and was similar in form and content to group-theoretic discussions. Later N. N. Vorob'ev [6] strengthened this result to the degree in which it is presented here. 4.12. Some semigroups have a unique principal two-sided ideal chain. Since any two-sided ideal can be included in some principal two-sided ideal chain (4.1), this situation occurs for those and only those semigroups in which the set of all the two-sided ideals is linearly ordered with respect to inclusion. THEOREM. Let the collection of two-sided ideals of a semigroup III satisfy the condition of maximality (2.17). In order for III to possess a unique principal twosided ideal chain, it is necessary and suffiCient that all the two-sided ideals oflll be principal two-sided ideals. PROOF. (1) Let III possess a unique principal two-sided ideal chain. Let X be an arbitrary two-sided ideal of Ill. By 2.17, it is the two-sided ideal envelope of one of its finite subsets {TI , T2 , ••• , Tn}. We consider the two-sided ideal envelopes of the elements of this set: Xl' X2, ••• , X n . Since they are all included in the same principal two-sided ideal chain, we have, for an appropriate numeration,

It follows from this that Xn is the principal two-sided ideal containing the elements TI , T2 , ••• , Tn. Since Xn C X, it follows that Xn = X. (2) Let all the two-sided ideals of III be principal. We take two arbitrary two-sided ideals Xl and X2 which are the two-sided ideal envelopes of the elements Al and A 2 • The two-sided ideal ::ra = Xl U X2 is also the two-sided ideal envelope of some element A3 . This element is included in Xl or in X 2 • Let A3 E Xl' This means that A3 = XA I Y, where X and Yare elements of III or are empty symbols. In turn, A2 E X3 and thus A2 = UA3V, where U and V are elements oflll or are empty symbols. Thus A2 = UXA I YV, from which it follows that A2 E Xl and hence X2 C Xl' We have shown that of two arbitrary twosided ideals of III one must necessarily be contained in the other. It follows from this that III possesses a unique principal two-sided ideal chain.

4.13. We will show that the multiplicative semigroup 9J1 n of all the complex square matrices of order n belongs to the type of semigroup considered above. We denote by Xk (0 k n) the set of matrices from ffi1n whose rank does not exceed k. As is well-known, the rank of a product of matrices does not exceed the rank of the factors. It follows from this that ::rk is a two-sided ideal of 9J1 n • It tutns out that ffi1n has no other two-sided ideals. Let M be a matrix of greatest rank r belonging to the two-sided ideal U of the semigroup 9J1.., and let Nbe an arbitrary matrix of rank p r. As is known from the theory of matrices, by means of so-called elementary transformations the

< <

<

Sec. 4]

TWO-SIDED CHAINS

163

matrices M and N may be brought to the form 1

M'=

1

o o

}p N'=

o o (elements not written are equal to zero). But the elementary transformations, as is well-known, may be realized by multiplication of the matrix on the left and on the right by nonsingular matrices. Thus, M' = P1 MP2 , N' = Q1 NQ2' where PI' P2, QI' and Q2 are nonsingular and thus possess inverse matrices. Since clearly N' = M' N', N= Q1 1N'Q;;1 = Q1 1M'N'Q;;1 = Q1 IP1MP2 N'Q;;1 and thus N E U. It follows from the above argument that U = ::tr• Thus the semigroup 9J1:n possesses only the following two-sided ideals, which form its unique principal two-sided ideal chain: 9J1:n = Xn :::; Xn - 1 :::; . . . :::; Xl :::; Xo = o. We note that, by 4.12, each of these ideals is a principal two-sided ideal.

4.14. For an arbitrary nonempty set n, the cardinality of which we denote by m, we consider the semigroup 6 0 of all of its transformations. For. any cardinality n > 1 not exceeding m, we denote by Xn the collection of all the transformations X E 6 0 such that the cardinality of xn is less than n.

164

IDEALS

[CHAP.

IV

For any S E 50 and X 5 :l:n (XS)Q = X(SQ) c XQ,

and thus XS E :l:n. Since the cardinality of XQ is less than n, the cardinality of (SX)Q = S(XQ) will also be less than n and hence SX E :l:n. Thus:l: n turns out to be a two-sided ideal. As A. I. Mal' cev [4] has shown, 6 0 has no other proper two-sided ideals. Let U be an arbitrary two-sided ideal of the semigroup 50 that is distinct from 50. We choose an arbitrary transformation U E U and denote the cardinality of UQ by 1. Let V be an arbitrary transformation for which the cardinality of VQ does not exceed 1. The set Q may be represented in the form of the nonintersecting union Q =

U Q~U), ~

where each component Q~U) consists of all the elements of Q that the transformation U takes into the same element. Analogously, for the transformation V, Q

= U Q~V). 'ry

Inasmuch as the cardinality of the set of components of the first union is r, while the second does not exceed r, there exists a one-to-one mapping rp of the collection of components of the second union into the first. Choosing an arbitrary element in each Q~U), we define a transformation Y E 6 0 such that, for ~ E Q~V), it is true that Yoc = a
a.

V= XUY. Since U E U, it follows from this that also V E U. It follows from the above argument that to a two-sided ideal U there corresponds a class of cardinalities having the property that along with a particular cardinality all the smaller ones belong to the given class. These are those cardinalities for which there exists a U E U such that UQ has the given cardinality. We may assume that for a given class there exists such a cardinality n and that this class is the class of all the cardinalities less than the cardinality n. By the preceding argument, U will be identical with :l:nSince, given any two unequal cardinalities n1 and n2, one of them must be smaller than the other, and since it clearly follows from n l > n 2 that:l: n1 ::>:tn2 the set of all the two-sided ideals of the semigroup 6 0 turns out to be linearly ordered by inclusion, i.e., forms a unique principal two-sided ideal chain of the semigroup 50.

Sec. 5]

THE INTERRELATIONS OF IDEAL EQUIVALENCES

165

5. The Interrelations of Ideal Equivalences 5.1. Along with the interest presented by the independent study of each of the ideal equivalences, i.e., left ideal, right ideal, and two-sided, there is a substantial connection between these equivalences. We first consider an important relation between left ideal equivalence and right ideal equivalence that was obtained by Green [1]. Let an element Al of a semigroup m: belong to a left ideal layer ,21 and to a right ideal layer 911 , Similarly, let an element A2 belong to the layers,22 and 91 2 , ljthe intersection oj£2 and 912 is nonempty, the intersection oj£2 and 912 is also nonempty. In fact, let ,21 n 91 2 3 Z.

If Z ¥= Al and Z ¥= A 2 , then by 3.8 there exist in the proper semigroup elements T, U, V and W such that

TAl

=

Z,

We take the element Since

Al

=

UZ

=

UA 2 W= XW,

X= A1V,

it follows by 3.8 that X and Al are in the same right ideal layer, i.e., X E Similarly, it follows from

mI'

that X E ,22' Hence, in the case under consideration 911 n £23 X. When Z = AI' we have Al E 91 2, i.e., 912 = 911, and hence Similarly, for Z

£2

n 9t1 = £2 n

91 2 3 A 2 •

£2

n 9t1 =

911 3 AI'

= A2,

£1

n

5.2. If we employ the concept of the product of relations (I, 5.2), then the fact that for some elements Al and A2 the intersection ,21 and 9t2 is nonempty means that Al and A2 are in the relation I . r, where I designates left ideal equivalence and r right ideal equivalence. Analogously, £2 n 9t ¥= 0 signifies that Al and A2 are in the relation r . L Thus, the property proven in 5.1 is simply the commutativity of the relations I and r (I, 5.3): I' r = r' 1. 5.3. The relation I· r (5.2) was introduced and considered similarly by Green [1]. Later it was investigated in the work of Clifford [9J and of Miller and Clifford [1]. As Green showed, in some cases this relation coincides with twosided ideal equivalence. For example, this is true for periodic semigroups. In fact, let m: be a periodic semigroup.

166

[CHAP.

IDEALS

IV

If two of its distinct elements A and B are associated by the relation (5.2), then there exists an element X such that A and X divide each other on the right and B and X divide each other on the left. It follows from this that X belongs to . the two-sided ideal envelope of A, while A in turn belongs to the two-sided ideal envelope of X. Consequently, X and A are two-sided-ideally equivalent. X and B are also two-sided-ideally equivalent. Hence A and Bare two-sided-ideally equivalent. Now let A and B be two-sided-ideally equivalent. Then by 3.6, for some Xl' X 2 , X3 and X4 , which are elements of U or empty symbols,

From this we obtain and thus also

B = (XI X a)l'B(X4X2)k for any natural k. We choose k so that (X4 X 2)k is an idempotent, which is possible since U is periodic. Then it follows from the indicated equality that

B(X4X2)k For Z

=

= B.

BX4 we have Z· X 2(X4X 2)k-1

= BX4 X2(X4 X 2)k-1 = B(X4X 2)k = B

(or ZX2 = B if k = 1). It follows from this that Band Z are right-ideally equivalent. We show similarly that the elements Z' = XaB and Bare leftideally equivalent. This means that Z' and B divide each other on the right. But then Z' X4 and BX4 will also divide each other on the right, i.e., will be leftideally equivalent. But Z' X4 = A, while BX4 = Z. Thus A and Z are leftideally equivalent. From what was proved above we thus obtain the result that A and Bare connected by the relation I . t (5.2). 5.4. As was shown by Green [1] the intersection of arbitrary left ideal layers and right ideal layers of a semigroup often turns out to be a group. He obtained the following condition for this. THEOREM. Let B be a left ideal layer and 91 a right idea/layer of a semigroup U; let ~ = B n 91. If then

~

is a group.

PROOF. Let X and Y be elements of ~ such that XY E R By 3.8 it is necessary to find a U and V which are elements ofU or empty symbols such that

U·XY= Y,

XY'

v= x.

Sec. 5]

THE INTERRELATIONS OF IDEAL EQUIVALENCES

For the element

[=

167

UXYV we clearly have [=

YV,

X[= XYV= X,

[=

UX,

IY= UXY= Y,

12 = I· [= UX· YV = I. It follows from this that X and 1 are Jeft-ideally equivalent, i.e., 1 E Q, while Yand 1 are right-ideally equivalent, i.e., [E ~. Thus [E st Let ZEst Since Z and J are left-ideally and right-ideally equivalent we see that, for some Zl' Z2' Z3' 24 E ~,

It follows from this that

[Z=Z,

ZI=2.

Thus [is a two-sided identity in the set 5t Since [Z3/· 2 = 1ZaZ = II = [,

[Z31 = [Za[ . [

= IZa[· Z . IZ4[ = [ . 1241 = [Z4['

it follows that the element [Z31 is left-ideally equivalent with [ and thus belongs to Q. Similarly, 1231 E~. Hence 1Z31 belongs to St and is a two-sided inverse to 2 with respect to I. Let T, ZESt and 2Z' = I. Inasmuch as TZ is divisible by T on the left and T is divisible by T2 on the left, i.e.,

T= T1= T·ZZ'= TZ·Z', both the elements T and TZ must be included in the same right ideal layer, i.e., TZ E~. We show similarly that TZ E Q. Consequently, TZ E St, which completes the proof that St is a group. 5.5. In particular, if St contains an idempotent, the intersection M will necessarily be nonempty.

n St

COROLLARY. IfQ is a left ideal layer and ~ a right ideal layer of a semigroup, where Q n !R contains an idempotent, then Q n ~ is a group (and, consequently, such an idempotent is unique).

5.6. Let 1 be an arbitrary idempotent of a semigroup ~ and let (})z be the set of all the completely regular elements for which 1 is a two-sided regular identity. As we saw in III, 1.14 and III, 1.16, (})z is the maximal subgroup of~ having 1 as its identity. Since all the elements of a group divide each other on the right and on the left, all the elements of ()) z will be both left-ideally and rightideally equivalent with I.

168

[CHAP.

IDEALS

IV

Considering 5.5, we conclude that

PROOF. (1) We denote by 2: the set of all the left ideal layers 9l of the semigroup ~ such that each of them is included in 'Xl \'X2 and such that 'X2 u 9l is a left ideal of~. 2: is nonempty since £ E 2:. We consider the set ~

= (mEL: U 9l)

U

'X2

= U (9l mE:!

u 'X~.

Since 9l U 'X2 for 9l E 2: is a left ideal of~, ~ is also a left ideal. ~ is contained in 'Xl' contains 'X 2 , and is distinct from 'X2 (since £ c 'X 2 ). If we succeed in showing that ~ is a right and thus a two-sided ideal of~, this will mean that ~ = 'Xl' The validity of the theorem follows directly from this, for £', as one of the left ideal layers belonging to 'Xl \::r2 = ~\'X2' will necessarily coincide with some 9l E 2:. (2) Let A E ~ and X E~. For some 9l E 2: the element X belongs to the left ideal 9l U 'X 2 • We assume that there exist no fewer than two left ideal layers contained in the left ideal (9l u ::r2)A but not contained in 'X 2 • This means that in (9l U 'X2 )A\['X2 n (9l U 'X 2)A]

there are no fewer than two left ideal layers. By 2.9 it follows from this that the corresponding left ideals are not adjacent, i.e., there lies between them some left ideal £" different from both of them: (9l u 'X 2 )A

::::>

£"

::::>

::r2

n (9l u 'X2 )A.

We denote by £1 the collection of all the elements Y E ~ such that YA E £", and by £2 the intersection of (9l U 'X2) and '£1' If T E ~, then T A E (9l u ::r2)A and T A E 'X 2 , and thus T A E ~ (\ (9l u 'X 2 )A C £". Consequently, T E £1' Thus 'X2 = £1' and £1 and £2 are left ideals.

Sec. 5]

THE INTERRELATIONS OF IDEAL EQUIVALENCES

169

Since £2 C £1 it follows by the definition of £1 that £2A C £". If L" c £", then L" = ZA, where Z EmU ::r2. By the definition of £1 the element Z must belong to £1· Consequently, Z E £2· If it were true that £2 c ::r2 , then Z would also belong to ::r2 , and thusan arbitrary element L" from £" would also belong to ::r2, which is impossible since £" is included in u ::r2 )A but contains ::r (91 u ::r2)A and is distinct from it. Thus £2 cannot in fact be included in "::r2. Since £2 is contained in 91 u ::r2 it follows that £2 has elements in common with 91 and thus, being a left ideal, must necessarily contain the whole left ideal layer m. We showed above that ::r2 c £1. Since £2 = u ::r2) () £1' we have ::r2 c £2' and since 91 = £2' we have 91 u ::r2 c £2. Thus by the definition of £2'

em

Q

()

em

Since £2 c £1 it follows that

em u ::r )A c 2

£",

which contradicts the original assumption concerning £/1. The contradiction so obtained means that in fact (91 u ::r2)A cannot contain more than one left ideal layer not included in ::r2 • (3) If (91 u ::r2 )A c ::r2 , then

XA

E

(91 u ::r2 )A

E

::r

2

c

'P.

em

If some of the elements of u ::r2 )A do not belong to ::r2 , then all of them, by what was proven in the second part, form one left ideal layer 91'. Since 91 E ~, therefore 91 u::r2 is a left ideal of and thus (91 u ::r2)A is also a left ideal. Thus 91' u ~ = u ::r2)A u ::r2

m

em

is a left ideal, i.e., 91'

E~.

XA

It follows that E

(91 u

~)A c

Inasmuch as in both possible cases XA

E

91' u

::r2 c 'P.

\.p, the set

'P is a right ideal.

5.9. COROLLARY. In a semigroup mlet the collection of principal left ideals satisfy the condition of minimality. If~ and::r2 are two adjacent two-sided ideals of'll, and 91 is a left ideal layer included in ::r1\::r2, then::r2 U 91 is a left ideal of'll:. PROOF. In the set of all principal left ideals included in ::r1 but not included in ::r2, there must exist an ideal £ that is minimal in the set. The left ideals ::r2 and ::r2 U £ are adjacent. In fact, if £' is a left ideal such that ::r2 c £' c ::r2 U £, £' =;t6 ::r2,

then we choose in £'\~ some element X and consider its left ideal envelope £x. Since X E ::r2, we have X E£ and thus also £ x c £. Thanks to the minimality of £ it follows that £x = £. But £x c £'. Consequently, £' ::l £ and thus

£'

= ::r2 U

£.

170

[C.!{AP.

IDEALS

IV

It follows from what has been proven that C:t2 u ,2)\:r2 is a left ideal layer. Its union with :r2 yields a left ideal 'U. By 5.8 it follows from this that the union of the left ideal layer in with :r2 must also yield a left ideal. 5.10. Any two-sided ideal chain of a semigroup is a left ideal chain of the semigroup. Of course, a principal two-sided ideal chain will not in general be a principal left ideal chain, but by 2.12 it may be extended to a principal left ideal chain. Usually this may be done in a number of ways. Using the corollary of 5.9 it is possible to describe how all the possible extensions may be realized under the corresponding assumptions. In a semigroup QI let the collection of principal left ideals satisfy the condition of minimality. Let ~ be an arbitrary principal two-sided ideal chain of 'U. For arbitrary adjacent two-sided ideals :r1 and :r2 of the chain ~ we consider the collection
U in) (ffie<1)'

U

:r2

will be a left ideal of QI containing :r2 and contained in Xl' Clearly, all such left ideals can be obtained in this way since each of them contains a whole layer in E
(U

ffie<1)'

left ideal of'U containing X 2 and contained in :rl • The set of all such left ideals constructed for any pair of adjacent two-sided ideals Xl and :r2 from the chain ~ forms a left ideal chain ~' which is an extension of the two-sided ideal chain ~. It is a principal left ideal chain of QI. In fact, let the addition of the left ideal ,2 to the chain ~' give a left ideal chain of SU. Let :r2 c ,2 c :rl , where :rl and :r2 are adjacent two-sided ideals belonging to~. The ideal ,2, like any left ideal, is the union of certain left ideal layers. Consequently, ,2

= ( 9le'Y U in)

U

:r2•

Let in E 'F and let 91' precede 91 in the ordering relation indicated above. We consider the left ideal from ~': ,2' = (

U in)

9!e <1) ,

U

:r2 ,

where 91', i.e. 91' E 'F. We have shown that 'F is such that ,2 itself turns out to be an ideal belonging to ~'.

Sec. 5]

THE INTERRELATIONS OF IDEAL EQUIVALENCES

171

It is obvious that every principal left ideal chain which is the extension of a given principal two-sided ideal chain ~ can be obtained by the method described. 5.11. The results obtained, and also various other investigations of N. N. Vorob'ev [6] and Green [1], show the importance of the condition ofminimality for principal ideals. In this connection a relation between such conditions established by Green also turns out to be significant. THEOREM. If in a semigroup the collection of all the principal left ideals and the collection of all the principal right ideals both satisfy the condition ofminimality, then the collection of all the principal two-sided ideals also satisfies the condition oj minimality. PROOF. Let r' be some nonempty set of principal two-sided ideals of a semigroup Ill. We denote by ~ the set of all the elements of III the two-sided ideal envelopes of which belong to r'. We denote by
n

~

¥:- 0.

Since
mo

Since ZESt, the left ideal envelope £' of the element AXo must, by the definition of
mo

172

[CHAP.

IDEALS

IV

If'¥ is empty, then Xo E.R and analogously to the preceding (considering Yo as an empty symbol), we show that the two-sided ideal envelope of the element Xo is the minimal set in r'o 6. Isolated Ideals 6.1. The role of a subsemigroup mof some semigroup mdepends to a great extent on the relationship between mand the elements of m\m. Essentially, the question is: Can the products of various elements of m\m be equal to an element of mand, in particular, can some power of an element of m\m be equal to some element of m?

m

m

DEFINITION. A subsemigroup of a semigroup is said to be ISOLATED if, for any X E mand any natural n, it always follows from xn E m that X E m. mis said to be COMPLETELY ISOLATED if, for any X, Y E m, it always follows from XY E mthat X and Y belong to m.

The investigation of the basic characteristics of isolation and complete isolation has been conducted up to now only for ideals. Completely isolated ideals are also called in the literature simple ideals, while isolated ideals (this term is due to P. G. Kontorovic) are sometimes said to be semisimple (for example, Croisot uses this term). 6.2. We note several properties of isolated subsemigroups of an arbitrary semigroup m. (0:) A completely isolated subsemigroup of a semigroup mis isolated. (fJ) mitself is a completely isolated subsemigroup of itself (y) A subsemigroup mwill be completely isolated if and only ifm\m is either a subsemigroup or is empty. (0) The intersection of any set of isolated subsemigroups is itself an isolated subsemigroup if it is nonempty. (8) The union of any set of isolated left ideals is an isolated left ideal. The union of any set of completely isolated left ideals is a completely isolated left ideal. (1) In order for an ideal Q3 to be isolated it is sufficient thatJor any X Em, it always follows from X2 E m that X E m. In fact, let the ideal mnot be isolated. This means that for some element X in I!l\m it is true that xn E Q3. Let the element X from m\m and the natural number n be so chosen that n is the least of all the possible numbers with this property (for any X E m\m). The number n cannot be even and different from two, for otherwise (xn/2)2 E m and both the possible cases

m

(1) xn/2 E Q3;

contradict the minimality of n.

(2) xn/2

=

Y Em,

Sec. 6]

ISOLATED IDEALS

173

The number n cannot be odd (of course n ~ 1), for otherwise xn+1 E lB

(which follows from the fact that xn+1 = X· xn = xn . X, X ll left or a right ideal), and again both the possible cases (1) x(n+1)12

E

lB;

(2) X(n+1)!2 = Y ElB,

E

lB, while lB is a

y2 E lB

contradict the minimality of n. Consequently, n = 2, i.e., for a nonisolated ideal there necessarily exists an element X E lll\lB such that

X 2 E lB. (8) IflB is an isolated two-sided ideal of a semigroup III that is a two-sided ideal of one of its supersemigroups Ill', then lB is a two-sided ideal oj the semigroup Ill'. In fact, for any X E Ill' and B E lB the product XB belongs to III since lB c Ill, and '2.( is a two-sided ideal for Ill'. Let us assume that XB E lB. Since lB is isolated in '2.(, we also have (XB)2 E lB. However, this is impossible, for

(XB)2 = XBX· BE (XBX)lB

E

lB.

We show analogously that BX E lB is also impossible. Consequently, lB is a two-sided ideal of '2.('. 6.3. We note that the property analogous to (0) does not apply for a completely isolated subsemigroup. An example is the semigroup composed of three elements A, B, and 0 for which A2 = A, B2 = B, and XY = 0 in all the remaining cases. Both {A, O} and {B, O} are completely isolated two-sided ideals. At the same time their intersection 0 is not a completely isolated subsemigroup since AB = O. We remark also that the property analogous to (rj) does not apply for subsemigroups that are not ideals. An example is the subsemigroup lB of the multiplicative semigroup of all the natural numbers that consists of numbers of the form 8k (k = 1, 2, 3, ... ). Clearly N2 = 87<: is possible only for N E lB. However, lB is not an isolated subsemigroup, for 2 3 E lB, but 2 E lB. 6.4. As Croisot (5] noted, it is convenient to distinguish the semigroups that are characterized by various ideals by use of the concept of classes of regularity (II, 6.11). (0:) In order for all the left ideals of a semigroup III to be isolated it is necessary and suffiCient that III = [[(0, 2). To see this, let all the left ideals of '2.( be isolated. For an arbitrary A E '2.( the left ideal '2.(A 2 is isolated. But A 3 E '2.(A 2• Thus A E IllA 2, i.e., for some X E III A

which also means that A

E [~{(O,

2).

= XA2,

174

[CHAP.

IDEALS

On the other hand, if III

=

IV

(£:m(O, 2), il is some left ideal of III and A2 E il

(A E Ill), then for some X E III

A

= XA2.

Since A 2 E il it follows from this that also A E il. By 7.2 (Yj) this yields the isolation of il. (P) In order for all the right ideals ofa semigroup III to be isolated it is necessary and suffiCient that III = (£:~1(2, 0). The proof is analogous to the proof of (ex). (y) In order for all the ideals (both the left and the right) of a semigroup III to be isolated it is necessary and sufficient that

III = (£:m(O, 2)

=

(£:m(2, 0).

This is a direct corollary of (ex) and (P)· (a) In order for all the two-sided ideals of a semigroup III to be isolated it is necessary and sufficient that for any A E III A E IllA 21ll.

In fact, if any two-sided ideal is isolated, then, in particular, the two-sided ideallllA 21ll will also be isolated. But A4 E IllA 21ll. Thus A E IllA 21ll. Now let any element A be included in IllA21ll and let ::r be an arbitrary twosided ideal of m:. If A2 E::r (A E m:), then A E IllA 2 m: c Ill::rlll c ::r and ::r is an isolated ideal (6.2, (1]».

6.5. Let.ft be some nonempty subset of a semigroup m:. We consider the collection of all the isolated subsemigroups of III that contain.ft. By 6.2 ({3) it is nonempty. A minimal subsemigroup in this collection (2.1) is said to be an isolated subsemigroup ofm: minimal over R Similarly, a minimal semigroup in the collection of all the completely isolated subsemigroups of m: containing .ft is said to be a completely isolated subsemigroup of m: minimal over R By what was said in 2.3 it follows from the properties 6.2 (ex), ({3), and (a) that there exists an isolated subsemigroup ofm: universally minimal over .ft which is the unique isolated subsemigroup of m: minimal over.ft. It is often called the isolator of .ft. However, there may be a number of completely isolated subsemigroups minimal over R Clearly, it is for isolated subsemigroups and only for them that the isolated subsemigroup minimal over them coincides with the subsemigroup itself. For completely isolated subsemigroups and only for them, a completely isolated subsemigroup minimal over them coincides with the subsemigroup itself. 6.6.

The property of isolation was studied in detail by P. G. Kontorovic [1;

2; 3] for a class of semigroups imbedded in groups (III, 1.10). We will consider below the description obtained by him for isolated and completely isolated

Sec. 6]

ISOLA TED IDEALS

175

subsemigroups minimal over a given ideal. We relate these investigations to a class of semigroups that contains the class of semigroups imbedded in groups. 6.7. We will say that a semigroup m satisfies the commutator condition if for any A, B there always exist in elements LA,B and RA,B such that

Em

m

AB

= LA.,BA,

AB = BRA . B.

Clearly, the commutator condition signifies the coincidence of the relations of left and right divisibility. This condition is important, in particular because it is satisfied by all commutative semigroups and also by every semigroup for which there exists a supergroup (J) such that G-1mG em

for each G E (J). In fact, for such a semigroup we may obviously take for LA..B

where A-I and B-1 are elements inverse to A and Bin (J), respectively (because of the definition of (J) both LA •B and RA •B belong to m). 6.8. LEMMA. Let a semigroup m satisfy the commutator condition (6.7); then any ideal of is two-sided.

m

PROOF.

Let £ be a left ideal of m. If X

XA

E

£ and A

E

m, then

= LA.xXE£.

Consequently, £ is at the same time a right ideal of m. We show similarly that each right ideal of m must necessarily be two-sided. 6.9. THEOREM. Let m be a semigroup satisfying the commutator condition and let X be one of its ideals. The isolated subsemigroups of minimal over X will be those and only those subsemigroups \l3 for which m\ \l3 is a subsemigroup of m maximal in the set of subsemigroups ofm contained in m\x. If this set is empty, then mitselfis a completely isolated subsemigroup ofm minimal over X. The subsemigroup \l3 is an ideal of m.

m

PROOF. (1) Let.Q be a subsemigroup of m belonging to m\X and not being contained in any subsemigroup belonging to m\X other than n itself (ifm\X does not contain any subsemigroups of m, then n is the empty set). We denote by 9l the ideal which is the union of all the ideals of m lying in \l3 = m\n. Since .Q = m\\l3 c m\X, we have Xc \l3 and thus Xc 91. We assume that the product of some elements Al and A2 of m\9C is included in 91. Both sets

176

[CHAP.

IDEALS

IV

are obviously left ideals. Since they are not contained in ~, they must have elements in common with .Q:

(the case when Fi

= Ai is completely analogous). But F1F2 = SlA 1S 2A2 = S1L.4.1,ssA1A2 E ~ c:

~,

which contradicts the fact that F1F2 is the product of elements belonging to the semigroup .Q and thus must be included in.Q. It follows from this that ~ is a completely isolated ideal. Since ~ :::J ~ it follows that m:\~ is a subsemigroup of m: belonging to m:\~. Because m:\~

:::J

m:\~ =

.Q

this is possible only for m:\ ~ = m:\~, i.e., when ~ = ~. Thus ~ turns out to be a completely isolated subsemigroup and an ideal. Let~' be a completely isolated subsemigroup of I!! such that ~c~' c~.

Then m:\~' is a subsemigroup of m: belonging to m:\~ and containing .Q = m:\~. By the original assumption about .0 it follows that m:\~' = I!!\~, i.e., ~' = ~. This proves that ~ is a completely isolated subsemigroup of m: minimal over ~. (2) Let ~ be a completely isolated subsemigroup minimal over~. .Q = m:\ ~ is a subsemigroup of I!! that belongs to m:\X (6.2 (y». By III, 4.7 there exists in I!!\X a subsemigroup .Q' containing .0 that is not included in any subsemigroup belonging to m:\X other than .Q'. By what was shown in the first part m:\.Q' is a completely isolated subsemigroup minimal over X. Since m:\.Q' c: I!!\.Q = ~ it follows by the definition of ~ that m:\.Q' = ~,i.e., .Q' =.Q. This completes the proof since .0' possesses the property required of .Q.

6.10. THEOREM. Let I!! be a semigroup satisfying the commutator condition and let X be an ideal ofl!!. Then the isolated subsemigroup minimal over X is itself an idealofm:, consists of all the elements some power of which is included in X and is equal to the intersection of all the completely isolated subsemigroups of the semigroup m: minimal over X. PROOF. (1) Let X' be the isolated subsemigroup minimal over X and let 5B be the set of all elements some power of which belongs to X. If X'" E X (X E I!t), then X E ~'. Consequently, 5B c: ~'. 5B is an ideal of m:. In fact, from X'" E X we obtain, for any I!! E I!!,

eXA)'"

= XAXAXA ... XA = XXR,A,xAXA ... XA = XXXR(R..4.,z,,Ax)A ... XA

i.e., it follows from X E 5B that XA E 5B.

= ...

=

X'" Y EX,

Sec. 6]

ISOLATED IDEALS

177

The isolation of ~ is obvious. It follows that ~ :=l X'. In view ofthe converse proved above, we obtain ~ = X'. (2) We denote by (£ the intersection of all the completely isolated subsemigroups of 21 minimal over X. We have (£ :=l X' since X' is included in each isolated su bsemigroup containing X (6.2 (~); 6.5). Let X E 21\X' . Since X' is an isolated subsemigroup we have [Xl n X' = 0, i.e., [Xl c 21\X' c 21\X. Let in be a subsemigroup containing [X] that belongs to 21\X but is not contained in any subsemigroup belonging to 21\;:£: other than in (III. 4.7). By 6.9, 9J1 = 21\in is an ideal that is a completely isolated subsemigroup of21 minimal over X. Since X E in it follows that XE 9J1. But (£ c 9J1 and hence XE (£. We have shown that 21\;:£:' and B can have no common elements. Consequently, (£ c X'. In view of the above converse we have X' = (£. 6.11. There are semigroups that have no proper isolated ideals. The condition for this for commutative semigroups was found by Thierrin [16]. We introduce the relation n into a commutative semigroup 21, where X "-' Yen) (X, Y E 21), if X21n[Y]=fi0.

If X,....., Yen) in 21 for each pair of elements X and Y, then 21 has no proper isolated ideals. To see this, let us assume that ~ is a proper isolated ideal of 21. Let B E ~ and A E 21\~. Since B "'-' A(n) for some A and some natural number n it must be true that BZ = An. But BZ E~, while An E ~ in view of the isolation of~. If there exist in 21 elements X and Y such that X ri-' Yen), then 21 possesses proper completely isolated ideals. We denote by ~ the collection of all elements B such that B ri-' Yen), and by (£ the collection of all elements C such that C......, Yen). ~ is nonempty since X E ~, while (£ is nonempty since Y E (£. ~ is an ideal of 21, since for any B E ~ and A E 21 it follows from B21 n [Y] = o that BA21 n [Yl = 0. We show that ~ is completely isolated. Let us assume that C1 C2 E 58, where C1, C2 E (£ = 21\58. The latter means that, for some Zl' 22 E 21, But then (C1 C2)(Zl Z2) = yn+m, which contradicts the fact that C1 C2 E ~. 6.12. In particular, it follows from 6.11 that if a commutative semigroup has no proper completely isolated ideals, then it has no proper isolated ideals. 6.13. It is sometimes appropriate to consider related conditions along with the conditions of isolation and complete isolation. For example, for a subsemigroup ~ of a semigroup 21 it is sometimes required that XY E 58 be possible

178

IDEALS

[CHAP.

IV

only when X, Y E 5B (MacKenzie [ID. Sometimes the requirement is strengthened so that from XY E 5B it follows that either xn E 5B or yn E 5B for some natural number n (Aubert [1]). Just as for the conditions of isolation and complete isolation, these conditions also may be given starting with the complement m:\5B of the semigroup m: to the semigroup 5B. 5B will be isolated if m:\5B, along with each element X, also contains X2. 5B will be completely isolated if m:\5B is a subsemigroup of m: or is empty. 5B satisfies the condition that XY E 5B implies that X, Y E 5B if m:\5B is a two-sided ideal of m:. 5B satisfies the condition that XY E 5B implies xn E 5B or yn E 5B for some n, if m:\5B is such that [X], [Y] c: m:\5B always implies XY E m:\5B.

CHAPTER

V

SEMIGROUPS WITH MINIMAL IDEALS 1. Two-Sided Ideals that are Groups 1.1. In the present chapter we will be concerned with the investigation of minimal left, minimal right, and minimal two-sided ideals of a semigroup (IV, 3.3). Of course, not every semi group possesses such ideals. The fact of the existence of various minimal ideals in a semigroup is not only interesting in itself but turns out to have an important influence on certain other properties of semigroups. Since many properties of the semi groups possessing minimal ideals lend themselves to deeper investigation, while the structure of the minimal ideals themselves is for the most part relatively easy to clarify, mathematicians studying semigroups have been very much interested in minimal ideals. The first results in this direction were obtained by A. K. Suskevic [3], who was mostly concerned with finite semigroups in which, of course, minimal left, right, and two-sided ideals always exist. Later the investigations were continued for infinite semigroups in the work of Rees [1; 2], Clifford [6; 7], Rich [1], Schwarz [1; 3; 4], Hashimoto [2; 3], L. M. Gluskin [3; 6], Teissier [2; 4] and Tamura [9]. 1.2. We first occupy ourselves with the question of the existence of a universally minimal ideal in the collection of all the ideals (left, right, and twosided) of a semigroup. We will show later that if such an ideal exists it is two-sided. Thus, in the case of the existence of such an ideal, it is the unique minimal left, the unique minimal right and the unique minimal two-sided ideal. As we will show, the question of the existence of a universally minimal ideal in the collection of all the ideals is equivalent to the question of the existence of two-sided ideals that are groups. We have already seen examples of two-sided ideals that are groups (II, 1.8). Of course, not every semigroup possesses a two-sided ideal that is a group. For example, the infinite monogenic semigroup does not in general have subsemigroups that are groups. THEOREM. Ifa semigroup '11 possesses a two-sidedideal:r. that is a group, then :r. is contained in every idealof'11 and is thus the universally minimal ideal in the collection of all the ideals of'11.

179

SEMI GROUPS WITH MINIMAL IDEALS

180

[CHAP. V

PROOF. Let.2 be an arbitrary left ideal (the argument for a right ideal is completely analogous). Since X is a two-sided ideal of Ill, while .2 is a left ideal of Ill, we have Here X.2 is obviously a left ideal of::t. But::t, being a group, has no ideals other than itself (IV, 1.8). Consequently, ::t =::t.2 c .2. Being contained in all the ideals of the semigroup III and being itself an ideal, by the same token ::t is the intersection of the collection of all the ideals of Ill, i.e., is the universally minimal set in this collection (IV, 2.3). 1.3. COROLLARY. is a group.

A semigroup cannot have more than one two-sided ideal that

1.4. The question of the presence in a semigroup of a two-sided ideal that is a group turns,out to be identical with the question of the existence in the semigroup of the zeroid elements (II, 1.6) of Clifford and Miller [1]. THEOREM. A semigroup III has a two-sided ideal that is a group if and only if among its elements there are elements that are divisible on both the right and left by every element of Ill. In this case, the two-sided ideal that is a group consists of all the elements that are divisible on both the right and the left by every element of Ill. PROOF. (1) In a semigroup III let X be a two-sided ideal that is a group. Let A be an arbitrary element of III and let T be an arbitrary element of ::t. Since ::t is a two-sided ideal it follows that

TA = T'E::t. Since ::t is a group, for some X E ::t we have

XT'=T, from which it follows that

(XT)A = XT' = T. It may be shown similarly that T is divisible by A on the left. (2) Let the set (£ consisting of all those elements of the semigroup III that are divisible on the left and right by every element of III be nonempty. We have already shown in II, 1.6 that (£ is a group. Let A be an arbitrary element of Ill. The identity EC& of the group (£ is divisible on the left by the element AEC&:

Ers.=AErs. Y• Multiplying this equality on the right by Err, we obtain

Err

= AEC& YErs. =

ACo'

Sec. 1]

TWO-SID1m IDEALS THAT ARE GROUPS

181

It follows from the fact that Err is divisible by every element of III on both the right and the left that the element Co = Err YErr also possesses this property (II, 1.4 (~», i.e., Co E (£:. Thus A(£: = AErr(£: = A CoCo-I(£: = Err Co -I(£: c (£:

(here Co-l is the inverse of Co with respect to Err). Consequently, (£: is a left ideal of Ill. It may be shown similarly that (£: is a right ideal. Hence we have seen that if (£: is nonempty, then it is a two-sided ideal of III and is a group. 1.5. COROLLARY. If a semigroup III possesses a two-sided ideal that is a group, then every element of9.1 that is divisible on the left by all the elements of'll will also be divisible on the right by all the elements of Ill; each element that is divisible on the right by all the elements oflll will also be divisible on the left by all the elements of Ill. To see this, let T be some element of a two-sided ideal :t that is a group, and let A be an arbitrary element of III that is divisible on the left by all the elements of 9.L For some Y E Ill, A=TY. Since :t is a two-sided ideal of 9.t, it follows from this that A E:t. Thus, by 1.4, A must be divisible on the right by every element of 9.t. 1.6. We note that in a semigroup which does not possess a two-sided ideal that is a group the property of 1.5 may not be fulfilled. For example, in a semigroup with no less than two elements in which XY = Y for each pair of elements X and Y, each element is divisible by all the elements on the left. At the same time, this semigroup contains no element which is divisible by all the elements on the right. 1.7. Using 1.4, it is not difficult to show that the sufficient condition obtained in 1.2 for the existence of a universally minimal ideal in the collection of all the left and right ideals is also a necessary condition. THEOREM. If the collection of all the ideals of a semigroup ~I possesses a universally minimal ideal, i.e., the intersection of all the ideals oflll is nonempty (IV, 2.3), then this intersection is a two-sided ideal that is a group. PROOF. Let the intersection :t of all the ideals of III be nonempty. We choose arbitrary elements A from III and Tfrom:t. Since IllA and AIll are ideals of 91 it follows that ~rA ~ :t and A~I ~ :t. This means that for some X, Y E 91,

XA= T,

AY= T.

Consequently, T is divisible on the right and left by every element of the semigroup 91.

[CHAP.

SEMIGROUPS WITH MINIMAL IDEALS

182

V

On the other hand, if Z E mis divisible on the left and right by all the elements of 'Jr, then Z is included in each ideal of the semigroup m. In fact, if Q is a left ideal of'll (and analogously for a right ideal), while L E Q, then for some X E

m

Z=XL, and since Q is a left ideal, Z E Q. It follows that Z E ::r. We have shown that ::r is the set of all the elements of that are divisible on the right and the left by every element of m. By 1.4 the desired property of::r follows from this.

m

m

1.8. If a semigroup possesses a two-sided ideal::r that is a group, then the identity Ex of the group is an element with the following properties: (oc) E:r is an idempotent; «(1) Ex is divisible on the left and right by every element of m(1.4); (y) E:r commutes with every element of m. In fact, for any A E m, and thus

AEx

= E;rJAEx) = (E;rA)E;r, =

(0) For every idempotent I of the semigroup

E;r,A.

m, we have

IE;r, = E;r/ = Ex· In fact, by (y), lEx = Ex/ and

(ExI)2 = ExI . E:r/ = E§P = E:r/ E:rIbelongs to::r. Since::r, being a group, has only one idempotent, Er/ = Ex. By 1.4 the existence in of an element with the property (13) is also sufficient in order for to possess a two-sided ideal that is a group. A semigroup possessing an element satisfying properties (oc), «(1) and (y) was called a homo group by Thienin [6]. Thus a homogroup is nothing other than a semigroup possessing a two-sided ideal that is a group. Along with the properties listed above, a whole series of other properties have been obtained for homogroups (see, for example, Thienin [19]). We note that by 1.4 every finite commutative semigroup is a homogroup since the product of all of its elements is clearly divisible on the left and the right by each of its elements. Groups are related to homogroups, a homogroup being a group if and only if it has no proper two-sided ideals.

m

m

2. Semigroups with Minimal Left Ideals 2.1. As we have already remarked, not every semigroup possesses minimal left, right, or two-sided ideals. In the present section we will consider semigroups possessing minimal left ideals. We will explain the connection between

Sec. 2]

SEMIGROUPS WITH MINIMAL LEFT IDEALS

183

different minimal left ideals and will give their relationship to a minimal twosided ideal. It is understood that analogous results are valid for semigroups possessing minimal right ideals. The results mentioned were obtained in part by Clifford [6] and in part by Schwarz [3]. 2.2. It turns out that all the minimal left ideals may be easily obtained from one of them. THEOREM. Let Q be a minimal left ideal of a semigroup~. Then for any S E ~ the product QS is also a minimal left ideal of~, while every minimal left ideal of ~r may be represented in the form QS for some S E ~L Distinct minimal left ideals ofm have no common elements. PROOF. (1) QS is clearly a left ideal ofm. Let some ideal Q' of the semigroup m be contained in 52S. We denote by 91 the collection of all the elements X of Q for which XS E Q'. Since each element ofQ' is included in QS, i.e., is representable in the form XS, where X E Q, the set 91 is nonempty. From the fact that Q' is a left ideal it follows that, for any X E 91 and Z E m, ZXS c Z52' c 52',

i.e., ZX Em. Consequently, 91 is a left ideal of m. But 91 c Q and since 52 is a minimal left ideal of m, 91 = Q. Hence Q' is contained in 91s. But by the definition of 91, 91s is included in Q'. Consequently, Q' = 91s= QS.

We conclude from this that 52S is a minimal left ideal of~. (2) Let 9J1 be a minimal left ideal of m. We choose some element Sin 9J1. Since 9J1 is a left ideal, QS c ~9J1 c 9J1. 52S is a left ideal of m contained in the minimal left ideal WC. By definition, this is possible only when 9J1 = 52S. (3) Let the minimal left ideals WC1 and 9J12 of a semigroup ~ have a common element X. The intersection 9J1 0 of the left ideals 9)11 and WC2 , being nonempty, will itself be a left ideal of~. It will be included in both WC1 and WC 2, and since they are minimal left ideals,9J1 o = 9)11 and 9J1 0 = WC 2 • 2.3. COROLLARY. If a semigroup ~ possesses minimal left ideals, then each left ideal of~ contains some minimal left idealofm. PROOF. Let 52 be a minimal left ideal of m and let 91 be an arbitrary left ideal of m. Then for any N E 91, QN belongs to 91. By 2.2, QN is a minimal left idealofm. 2.4. As we have already noted, the relation of being a left or right ideal is not transitive. Thus it does not follow directly from the definition that minimal left and right ideals themselves cannot have their own corresponding left and

SEMIGROUPS

184

WITH

MINIMAL IDEALS

[CHAP. V

right ideals. We will show that this cannot occur. It will follow in particular that a left ideal of a semigroup will be a minimal left ideal if and only if it is a semigroup without proper left ideals. THEOREM. If.2 is a minimal left ideal of a semigroup Ill, then .2 is a semigroup without proper left (and thus also two-sided) ideals. PROOF.

Let 91 be an arbitrary left ideal of the semigroup.2. Then .291 c 91 c .2.

But since .2 is a left ideal of a semigroup Ill, 'll(.291)

= ('ll.2)91 c

.291.

Consequently, .291 is a left ideal of'll included in the minimal left ideal .2 of the semigroup Ill. This is possible only when .291 =.2. But, as we showed earlier, .291 c 91. Consequently, 91 =.2, i.e., .2 has no proper left ideals. 2.5. Among the different ideals of a semigroup which we have already discussed and will discuss below, of special interest are the minimal two-sided ideals. As we have already remarked, if such an ideal exists, it is the universally minimal two-sided ideal (IV, 4.4). In § 1 we considered a special case of such ideals, namely, two-sided ideals that are groups. However, minimal two-sided ideals may also exist in other cases. One sufficient condition for the existence of a minimal two-sided ideal in a semigroup is the presence in the semigroup of minimal left (or right) ideals. In this case the minimal two-sided ideal is their union. THEOREM. If a semigroup 'll possesses a minimal left ideal, then'll possesses a minimal two-sided ideal which is equal to the union of all the minimal left ideals. PROOF. Let .2 be one of the minimal left ideals of a semigroup Ill. It follows from 2.2 that

is the union of all the minimal left ideals of'll. Obviously.2'll is a two-sided ideal of~r. Let.2 be an arbitrary two-sided ideal of'll. The product::t· (.2S) is a left ideal of'll included in the minimal left ideal .2S. Consequently, ::t.2S = .2S. On the other hand, it follows from the fact that ::t is a two-sided ideal of'll that ::r.2S c ::t. Consequently, .2S c ::t.

Sec. 2]

SEMIGROUPS

WITH

MINIMAL LEFT IDEALS

185

Thus,

£'21 c :t, i.e., £'21 is the universally minimal two-sided ideal of m:. 2.6. For the case considered in the theorem of 2.5, it is easy to explain which are the left ideals of the minimal two-sided ideal of a semigroup. THEOREM. Let a semigroup '21 possess minimal left ideals and let .Sl be the minimal two-sided ideal of'21· Then the left ideals oflff. included in.R are the only left ideals of the semigroup .R. PROOF. It is obvious that every left ideal of the semigroup '21 contained in.R is a left ideal of .R. Let :t be an arbitrary left ideal of the semigroup .R and let T be an arbitrary element of:t. Since.R by 2.5 is the union of the minimal left ideals of ~{, T is included in some minimal left ideal £ of the semigroup m:. Clearly,.RT is a left ideal of the semigroup '21 contained in £. Consequently,

£

= .RT.

In particular, T c .RT. Thus each element of:t is included in .R:t, i.e., :t c .Sl:r. On the other hand, it follows from the fact that X is a left ideal of.R that X

~

.R:r.

Consequently, But .RX is obviously a left ideal of the semigroup '21. 2.7. The question of the existence in a semigroup of minimalleft, right, and two-sided ideals, and also of the way in which they are interrelated, has great importance in the study of many of the properties of semigroups. It was shown in 2.5 that if a semigroup '21 possesses minimal left or minimal right ideals, it will also possess a minimal two-sided ideal R Here.R by 2.5 will contain all the minimal left and minimal right ideals of the semigroup m:, which by 2.6 are respectively minimal left and right ideals of the semigroup R But.R, as was shown in IV, 4.8, is a semigroup without proper two-sided ideals. Thus the question of the relationship between the minimal left and minimal right ideals of an arbitrary semigroup '2{ turns out to be equivalent to the question of the relationship between minimal left and minimal right ideals of the semigroup .Sl, which is a semigroup without proper two-sided ideals. This situation presents another reason for the interest in semigroups without proper two-sided ideals (in passing we mention that we have already indicated the fundamental importance of such semigroups in IV, 4.10).

[CHAP. V

SEMIGROUPS WITH MINIMAL IDEALS

186

2.8. The condition of the absence in a semigroup of proper two-sided ideals is clearly equivalent to that of the semi group being itself its own minimal two-sided ideal. In the presence of minimal left ideals in the semigroup, this will occur by 2.5, if and only if the semigroup is the union of all of its minimal left ideals. 2.9. THEOREM. Let a semigroup II! possess minimal left ideals. In order for there to be no proper two-sided ideals in ~r it is necessary and sufficient for each principal left ideal (IV, 3.6) to be a minimal left ideal. PROOF. (1) Each element is included in its own principal left ideal. It follows that if all the principal left ideals are minimal left ideals, then ~( is the union of minimal left ideals and thus has no proper two-sided ideals (2.8). (2) If II! has no proper two-sided ideals, then by 2.8 each element A is included in some minimal left ideal n. The principal left ideal of the element A is included in the left ideal n containing A. It follows from the minimality of n that the principal left ideal of A coincides with n. 2.10. THEOREM. Let a semigroup 'll possess minimal left ideals. In order for there to be no proper two-sided ideals in II! it is necessary and suffiCient for each right divisor of an arbitrary element of the semigroup to be itself always divisible on the right by this element. (This condition means the symmetry of the relation of divisibility on the right.) PROOF. (1) Let 'll have no proper two-sided ideals. It is necessary for us to show that from the fact that some element A is divisible on the right by B, namely, (A,B, XE~,

A=XB

it follows that B is also divisible on the right by A. If A = B, this is clear. Let A :;of B. Since 'llA U A = II!XB U XB c II!B U B,

where ~(A uA and ~lB UB are principal left ideals of II!, it follows from 2.9 that 'llA U A

= II!B

Since B :;of A, it follows from this that B B

=

u B. E

II!A, i.e., for some Y,

Y'll.

Thus

n= and, consequently,

A u 'llA = YB u II! YB c 'llB c

n'

n' = n.

2.11. We dwell in particular on the case studied by Clifford and Miller [1], where the semigroup has a unique minimal left ideal. Such an ideal will by 2.3

Sec. 2]

SEMI GROUPS WITH MINIMAL LEFT IDEALS

187

be a universally minimal left ideal. There is a necessary and sufficient condition for the existence of such an ideal that may be obtained directly from the next theorem. THEOREM. If there are among the elements of a semigroup 'n elements that are divisible on the right by all the elements of 'n, then the collection of all such elements is a minimal two-sided ideal of'n that is a universally minimal left ideal of'n· If there are among the elements of'n no elements that are divisible on the right by all the elements of 'n, then ~{ either has no minimal left ideals or it has more than one. PROOF. (1) Let the collection U of all the elements of ~{ that are divisible on the right by all the elements of'n be nonempty. For any AI' A2 E 'n and U E U in \II there exists an element X such that Thus (A 2 U)

= (A2X) . AI'

Consequently, AzU is divisible on the right by any element Al from 'n, i.e., A2 U E U. This means that U is a left ideal. If 58 is an arbitrary left ideal of 'n, then for any U E U and V E 58 there exists a Z E 'n such that U=ZV. Since 58 is a left ideal of 'n, we have ZV E 58. It follows that U is a universally minimal left ideal. Of course, there can be no other minimal left ideals of 'n; for, by 2.5, U is the minimal two-sided ideal of 'n. (2) Let £ be a unique minimal left ideal of 'n. For any A E'n the product £A is by 2.2 also necessarily a minimal left ideal of 'n. Consequently,

£A=£. But this means that for any L E £ there exists in £ an X such that XA =L.

Consequently, the elements of £ are such that each of them is divisible on the right by each element of'n. It is only in the presence of such elements that 'n can possess a unique minimal left ideal. 2.12. In 1.2 we saw that when there exists in the semigroup a two-sided ideal that is a group, the semigroup has a unique minimal left and a unique minimal right ideal. By the theorem of 2.11 it is now possible to state the converse. COROLLARY. If a semigroup 'n has a unique minimal left ideal U and a unique minimal right ideal 58, then U is equal to 58, is a two-sided ideal of'n and is a group.

188

SEMI GROUPS WITH MINIMAL IDEALS

[CHAP.

V

PROOF. By 2.11, U is a minimal two-sided ideal of ~l. Analogously, 5D is a minimal two-sided ideal of Ill. Consequently, U = 5D. Here U, equal to 5D, is the collection of all the elements divisible on the left by all the elements of Ill. By 1.4 this collection is a two-sided ideal that is a group.

2.13. Using the above corollary we can easily discuss the question of minimal left ideals of inverse semigroups (II, 7.2) which were studied in detail by Preston [2]. It turns out that an inverse semigroup can have no more than one minimal left ideal, which in this case turns out to be simultaneously a minimal right ideal. This follows from the next theorem if we keep in mind the theorem of 1.2. THEOREM.

idempotent

If in

10 is

the set of the idempotents of an inverse semigroup III the a two-sided zero of this set, then ~Uolll

= Illlo = l olll = lo~Uo

is a two-sided ideal that is a group. If none of the idempotents is a two-sided zero of the set, then there are neither minimal left ideals nor minimal right ideals in Ill. PROOF. (1) Let the idempotent 10 be a zero in the set of all the idempotents of ~(. For an arbitrary idempotent I it follows from 101 = 10 that the left ideal Illlo is included in the principal left ideal ~U. Since every principal left ideal has the form Illl (IV, 3.17), therefore Illlo is contained in every left ideal, i.e., is a universally minimal left ideal. Similarly, l o'll is a universally minimal right ideal. By 2.12, 'lllo = 1091 is a two-sided ideal which is a group. In particular, it follows that 9110~I =

9110 = 1091

= 109110,

(2) Let B be a minimal left ideal of 91. It clearly is the left ideal envelope of every element of B. By IV, 3.17, B contains an idempotent I, where

B

= IllI.

Let l' be an arbitrary idempotent of 91. The intersection of the left ideals B = 9J.I and 9J.I' is nonempty since it contains II'. But this intersection is a left ideal included in B. Consequently it coincides with B. Thus

911'

:::>

911.

By IV, 3.17 it follows that I'I = II' = I. Consequently, the idempotent I is a two-sided zero in the set of all the idempotents. 2.14. In connection with the theorem of 2.13 one should note that an inverse semigroup that has only a finite number of idempotents (in particular, any finite inverse semigroup) always possesses an idempotent that is a zero in

Sec. 3]

SEMIGROUPS WITH MINIMAL LEFT AND RIGHT IDEALS

189

the set of all the idempotents. As easily follows from the commutativity of the idempotents in an inverse semi group, such an idempotent is in this case the product of all the idempotents of the inverse semigroup.

3. Semigroups with both Minimal Left Ideals and Minimal Right Ideals 3.1. In the preceding section we saw the important role played by the very fact of the existence of minimal left ideals in a semigroup. Of course, analogous results hold for the existence in a semigroup of minimal right ideals. Even more important is the case when a semi group possesses simultaneously both minimal left and minimal right ideals. Such semigroups were studied by Rees [1], Schwarz [3], Clifford [6], and Hashimoto [2; 3]. In the present section we will consider some of these properties. 3.2. It is easy to see that the intersection of any left ideal with any right ideal is always nonempty, for it contains their product (in which the right ideal is taken as the left factor). When these ideals are minimal we obtain an important property for this intersection. THEOREM. Let.52 be some minimal left ideal and 9\ some minimal right ideal of a semigroup Ill. Then
9\ PROOF.

=

E(!j'l!,

.52

= 'llE(!j'


=

9\.52

= 9\ II .52 = E(!j'llErs,.

Since.52 is a left ideal,
=

9\.52 . 9\.52

=

9\ . .52 9\.52 c 9W

Consequently,
E

=


9\, G9\ is a right ideal of the

G9\ = 9\, from which we obtain, by multiplication on the right by .52, G
It is shown analogously that
=


from which it follows that
'llErs, = .52, which means that .52 is the set of all the elements of III for which E(!j is a right identity. We show analogously that

E(fj'll = 9\,

190

SEMI GROUPS WITH MINIMAL IDEALS

[CHAP.

V

i.e., 9t is the set of all the elements of III for which Ef!5 is a left identity. It follows that 9t n B is the set of all the elements of III for which Ef!5 is a two-sided identity. Consequently, 9t n B = Ef!5IllEf!5, from which it follows in particular that

9t n .2 But, on the other hand, if X

~ Ef!51ll . IllEf!5

E

=

9tB

= GJ.

9t n B, then

X= XEf!5

9W

C

= GJ.

Consequently, 91 n B = GJ. 3.3. Using the theorem of 3.2 it is possible to clear up completely the character of the collection of those idempotents of the semigroups we have been considering that are contained in a minimal two-sided ideal. The very fact that every minimal left ideal possesses idempotents is interesting in itself. Let r< l) be the set of all the minimal left ideals of a semi group III and let r
is always nonempty and contains a unique idempotent, namely, the identity of the group GJil,!)t, which will be denoted by Eil,'J(' By 2.5 the minimal two-sided ideal.R of the semigroup III may be represented in the form of the unions .R = U B = U 91, ilEr(t)

!)tE['(T)

where the components of each union are pairwise nonintersecting. It follows from this that (B E r
(91 E r
9t

=ilErtl) U GJ il9t ,

.R

=ilEr(lJ U GJil,!)t, 9lertr)

where the components of each union are nonempty and pairwise nonintersecting. It follows from what has been said that the idempotents of .R are ES2 .9t • Thus, we obtain a one-to-one correspondence between those idempotents of the semigroup contained in its minimal two-sided ideal .R and the pairs of ideals

ES2 ,9t ~ (B, 9t)

(B

E r
91

E

r
3.4. We note especially the following property of idempotents which we will need later.

Sec. 3]

SEMIGROUPS WITH MINIMAL LEFT AND RIGHT IDEALS

191

COROLLARY. Let a semigroup 'll have minimal left and minimal right ideals. Then no idempotent of the minimal two-sided ideal ~ can be a two-sided identity of any idempotent other than itself. PROOF. where II

Let it be true for the idempotents II and 12 of the semigroup '1t that

E~.

111211 = 12, By 3.3,

Thus

12

= 111211 C

!lUlI2 9\£ = 9\. £129\. £ c 9\£

But there is only one idempotent,

E'\),9\

= II' in

(i),\),9\'

=

(5'\),9\'

Hence 12 = II'

3.5. As we already know (2.5), the presence in a semigroup of a minimal left or a minimal right ideal implies the existence of a minimal two-sided ideal. We consider the connection between these minimal ideals. THEOREM. Let £ be a minimal left ideal, 9\ a minimal right ideal and ~ the minimal two-sided ideal of a semigroup Ill. Then ~

=

£9\ = £'ll9\ = £'ll = 'll9\ = £~ = ~9\.

PROOF. By 2.5, £ c~. Analogously, 9\ c~. It follows from this that each of the sets £9\, £'ll9\, £'ll, '1t9\, £~ and ~9\ is included in R But they are all obviously two-sided ideals. Inasmuch as ~ is a minimal two-sided ideal, each of these six ideals must coincide with ~. 3.6. COROLLARY. If £ and £' are minimal left ideals of'll while 9\ and 9\' are minimal right ideals of'll, then £9\ = £'9\'. By 3.5 both £9\ and £'9\ are equal to R 3.7. We note also that for a minimal left ideal £ and a minimal right ideal 9\ of a semigroup 'll the group 9\£ (3.2) is included in the minimal two-sided ideal ~ of the semigroup Ill. This is true since by 2.5 we have £ c R 3.8. In the case under consideration we can explain without difficulty how all the minimal left and right ideals of a semigroup may be obtained. THEOREM. If a semigroup 'll possesses minimal left and minimal right ideals, then the left ideal envelopes of the idempotents that belong to the minimal two-sided ideal, and only they, are the minimal left ideals of the semigroup. The analogous statement holds for right ideals. PROOF. (1) Let Ibe an idempotent included in the minimal two-sided ideal R Since 1 is an idempotent its left ideal envelope is 'llJ. By 2.5, I is included in some minimal left ideal £ of the semigroup Ill.

192

SEMIGROUPS WITH MINIMAL IDEALS

[CHAP. V

Similarly, I is included in some minimal right ideal 9\ of the semigroup Ill. By 3.2, 9\£ is a group. Since I = I· IE 9\£, I is the identity of this group and, by 3.2, W= £. (2) Let £ be an arbitrary minimal left ideal of Ill. We choose some minimal right ideal 9\ of the semigroup Ill. By 3.2, (£i = 9\£ is a group and £ = IllE(!j. Here 'llE(!j is obviously the left ideal envelope of the idempotent E(!j which belongs to .R by 3.5. 3.9. In IV, 4.10 we were concerned with the important role of semigroups having no proper two-sided ideals in the analysis of the structure of an arbitrary semigroup. In the general case such semi groups may be quite complicated and we do not yet have a sufficiently clear conception of their structure. However, with one additional quite natural assumption, it is possible to obtain a rather clear description of the structure of such semigroups. DEFINITION. If a semigroup with no proper two-sided ideals possesses minimal left and minimal right ideals, it is said to be a COMPLETELY SIMPLE SEMIGROUP WITHOUT ZERO.

The words "without zero" in this definition are justified by the fact that, with the exception of the trivial case when the semigroup consists of one element, the semigroup so defined cannot have a zero, since it would be a proper two-sided ideal. The necessity of inserting these words arises from the necessity of considering a related class of semigroups that have a zero. The next two sections will be devoted to that class. It should be kept in mind that for the definition of a completely simple semigroup many authors make use of a set of properties which is equivalent to the above definition. The approach to completely simple semigroups which we use in the present book is due to Clifford [6; 7]. This approach is more natural than the original approach of Rees [1], who was the first to give in the general case the description of the structure of completely simple semigroups which will be presented below. Subsequently many mathematicians (see 1.1) studied the various properties of completely simple semigroups. The starting point for all of them was the work of A. K. Suskevic [3], who obtained most of these properties for finite semigroups. 3.10. The first definition of a completely simple semigroup, given by Rees [1; 2], is based on the concept of a so-called primitive idempotent. A nonzero idempotent I is said to be primitive if it is not a two-sided identity for any other nonzero idempotent. It follows from 3.2 that a completely simple semigroup

without zero that is not the identity semigroup always possesses nonzero idempotents, and by 3.4 all of them are primitive. As we will show in 6.15, the existence of at least one primitive idempotent in a semi group that has no proper two-sided ideals is sufficient for it to be a completely simple semigroup without zero.

Sec. 3]

SEMIGROUPS WITH MINIMAL LEFT AND RIGHT IDEALS

193

3.11. We note that the various properties of minimal ideals obtained already provide us with a whole series of properties of completely simple semigroups with zero (for example, the existence for each element of a two-sided identity, which follows directly from 2.8 and 3.2, and so on). Using these properties we could already give a description of the structure of an arbitrary completely simple semigroup without zero. However, for economy of space we will leave this until § 6, where this description will be obtained from simple corollaries of theorems concerning semigroups of a related class. 3.12. If a semigroup I[( possesses a zero, then the properties of the minimal ideals of I[( considered in the present and preceding sections will become completely trivial. In fact, a subset of the semigroup Il( consisting of one element all[ will be in I[( simultaneously a minimal left ideal, a minimal right ideal and a minimal two-sided ideal. Of course, there are no other minimal ideals of 1[(. All the relationships found above of minimal ideals of different kinds will coincide in a trivial manner. A semigroup with zero that has no proper twosided ideals turns out to be merely a semigroup consisting of one element. Since the approach to the study of the properties of semigroups without zero from the point of view of properties of their minimal ideals gives valuable results, the desire naturally arises to try to transform this approach so that it will also be suitable for the study of semigroups with zero. A number of successful attempts have been made in this direction. Instead of the concept of an ideal, the concept of a nonzero ideal serves as a basis. 3.13. DEFINITION.

An ideal (left, right or two-sided), is said to be

NONZERO

if it includes elements that are not zeros of the semigroup. A minimal ideal in the set of all the left nonzero ideals is said to be a MINIMAL Minimal right nonzero ideals and minimal two-sided nonzero ideals are defined analogously. LEFT NONZERO IDEAL.

3.14. The theory of minimal nonzero ideals arising in connection with these concepts is to a great degree parallel with the theory of minimal ideals of a semigroup without zero. Many results may be transferred, though usually with various weakening stipulations and complications. The reason for this clearly lies in the fact that the concept of nonzero ideals is of course a less natural one. The first significant complication arises from the fact that the intersection of two nonzero ideals can sometimes consist of a zero, i.e., not belong to the class of nonzero ideals. In this connection, a minimal two-sided nonzero ideal may not be a universally minimal two-sided nonzero ideal. Moreover, a semigroup may have several such ideals, a fact which is evident from the example of a semigroup with a zero in which the product of any two elements is equal to zero (in this case any element X together with the zero clearly forms a minimal two-sided nonzero ideal {X,O}). This example shows that the theorem of 2.5, while important in the theory considered above, can not be transferred in full into the theory of nonzero ideals.

194

SEMIGROUPS WITH MINIMAL IDEALS

[CHAP.

V

For both these reasons (the merely partial parallelism with the theory of minimal ideals and the relatively less complete state of the theory) we will not make any special study of the theory of minimal nonzero ideals (we refer the reader in this connection to the articles of Rees [1], Clifford [7], and Schwarz [4], which contain a number of properties of such ideals). We will limit ourselves to examining in the next section certain properties in this theory (essentially obtained in the articles just mentioned) which are connected with a certain important class of semigroups. 4. Completely Simple Semi groups with Zero 4.1. F or later requirements it is necessary for us to pay some attention here to one of the directions indicated at the end of the preceding section for the theory of minimal nonzero ideals. Namely, we must consider semi groups with zero having no proper two-sided ideals other than zero. We considered the very important role of these semigroups in IV, 4.10. A complete study of their properties and construction is possible at the present time under certain limitations similar to those made in the preceding sections. 4.2. DEFINITION.

If a semigroup Q[ with zero possessing the property that

has no proper two-sided nonzero ideals and has minimaUeft nonzero ideals and minimal right nonzero ideals (3.13), then it is said to be a COMPLETELY SIMPLE SEMIGROUP WITH ZERO.

4.3.

We note that the condition

is made only to exclude the following two semigroups: the semigroup consisting of one element and the semigroup consisting of two elements A and 0 with the rule of multiplication AO = OA = A2 = 0 2 = O. Both these semigroups clearly satisfy the remaining conditions of the definition. However, if these two semigroups were not excluded by the condition Q[Q[ ~ O;a from the definition of completely simple semigroups with zero (for both of which, of course, Q[Q[ = O;a), then in the discussion of the rest of the theory we would continually have to make tedious conventions. For any semigroup other than these two, the property Q[Q[ ~ O;a is a consequence of the absence of proper two-sided nonzero ideals. In fact, a semigroup with more than two elements for which Q[Q[ = O~[ has a proper two-sided ideal consisting of O;a and any nonzero element.

Sec. 4]

COMPLETELY SIMPLE SEMI GROUPS

WITH

ZERO

195

We note in passing that in a completely simple semigroup with zero the condition under consideration implies the equality

In fact, since 'll'll :;6 equal to 'll.

O~,

'll'll is a two-sided nonzero ideal of'll and thus must be

4.4. We begin the derivation of several properties of completely simple semi groups with zero. The choice of these properties is mostly explained by the requirements of the fundamental theorem of the next section which entirely clears up the structure of completely simple semigroups with zero, and which later aids in the explanation of the structure of completely simple semigroups without zero. The properties mentioned are noticeably similar to many of the properties of ideals considered in §§ 2 and 3. However, there are quite important differences which prevent us from treating the two theories simultaneously. 4.5. THEOREM. A completely simple semigroup 'll without zero is equal to the union of all of its minimal left nonzero ideals that have pairwise no common element other than O~. PROOF. Let £ be a minimal left nonzero ideal of'll. Obviously £'ll is a two-sided ideal of'll. If the ideal £U consisted only of zero, this £ would be a two-sided ideal, i.e., would coincide with the semigroup 'll itself. But then we would have o~ = £'ll = 'll'll,

which is impossible for a completely simple semigroup with zero. Consequently, £'ll = 'll. For every A E'll, the product £A is clearly a left ideal. Let £' be a left nonzero ideal of 'll contained in £A. We denote by 91 the collection of all the elements X of £ for which XA E £'. Some of these elements are nonzero inasmuch as £' c QA and £' is a nonzero ideal. Since £' is a left ideal, for any Z E 'll and X E 91, the element ZX will also belong to 91. Thus 91 is a left nonzero ideal of'll included in £. It follows from this that 91 = £, and hence £A c £'. But £' in turn is included in £A. Consequently, £' = £A, i.e., £A has no proper left nonzero ideals. Thus each product £A (A E Ilt), if different from O~, is a minimal left nonzero ideal. The equality obtained earlier implies that'll is the union of all the sets of the forin £A. We may exclude from this union components equal to O~ since O~{ is included in those £A that are distinct from 0'l1. After this we obtain a representation of 'll in the required form.

SEMIGROUPS WITH MINIMAL IDEALS

196

[CHAP. V

The intersection of two distinct minimal left nonzero ideals, being a left ideal of'2l, cannot be different from Om. In fact, it would otherwise coincide with each of these ideals, which would contradict the fact that they were assumed to be distinct. 4.6. COROLLARY. A completely simple semigroup '2l with zero is the union of all the possible products of the form

9tn, 9t

n

where is a minimal right nonzero ideal of 12! while is a minimal left nonzero ideal of 12!. The components of this union have pairwise only the element Om in common. PROOF.

By the theorem of 4.5 (applying it also for right ideals) we have

12!

= 12!12! =

(U9t)(Un) = £,91 U 9W. £R £

9t ;6

If 91', then 91 and 91' have no elements in common other than 091. (4.5). Since, for any E and E', c 91 and

mE 9t'n' em', it follows that mE and m'n' have no elements in common other than zero. We argue similarly for n ;6 n'. 4.7. In connection with this corollary the products of the form n9t playa natural role. Let us consider their structure in more detail. THEOREM. Let 12! be a completely simple semigroup with zero, let n be a minimal left nonzero ideal ofl2! and let 91 be a minimal right nonzero idealofl2!. Then

9tn;6 Om. Ifnm ;6 O~r, then nm = ~(, while ~ = 9tn is a group with an externally adjoined zero (II, 2.12), where

PROOF.

nl2! =

~r.

(1) In the process of the proof of the theorem of 4.5 we saw that Analogously I2!m = 12!. Thus

12! = 12!12! = 12! 9t . nl2!,

mn ;6

from which it follows that Om. (2) In the future we will assume that

nm;6 O~(. Since mR is obviously a two-sided ideal of U, we have

n9t =

12!.

Sec. 4]

COMPLETELY SIMPLE SEMI GROUPS WITH ZERO

197

(3) 5 = 9t2 is a subsemigroup:

=

55

9t29tB = 9t . [(B9t) . 2] c 9t2 = 5.

Let G be an arbitrary nonzero element of 5. Since G E 9tB c 2, we have mG c 2. But mG is a left ideal ofm. Thus either mG = B or mG = Om. In the second of these cases we would have that {Om' G} is a left nonzero ideal contained in 2, i.e., {am, G} = 2. But this would contradict the fact that 9W -:;i; am. Consequently, mG = 2. G9t is a right ideal ofm included in 9t. Thus either G9t = 9\ or G9\ = am. The second of these equations would imply by the equality mG = B proved above that '11 = 29t = mG9\ = amConsequently, G9t by 2 that

=

9t. We obtain from this by multiplication on the right

G5=5· For the second nonzero element G' E 5 we have analogously that G'5 = 5, from which it follows that GG'5= G5 =5· Consequently, GG' ;.6 am, i.e., the collection of all the nonzero elements of 5 forms a semigroup which we will denote by G'5. Since Gam = Om, it follows from G5 = 5 that GG'5 = G'5. We prove analogously that G'5G = G'5. It follows that G'5 is a group, while 5 is thus a group with an externally adjoined zero. Since E(£ = ES;) C 2, it follows that mE%) is a left ideal of 2! included in B. Here mES;):3 ES;)ES;) = ES;) ;.6 am. Consequently,

2!E%)

= 2.

Analogously,

E%)m = 9t.

= '11, E%)m . mE%) = ES;)mE%).

We obtain from this, keeping in mind that '11'11 5 = 9\2 = Since

=

5

9\2

C

9\ and 5

we have But, on the other hand, 9t Thus

n

2

C

2 = 2!E%),

=

9t2

C

2,

SEMIGROUPS WITH MINIMAL IDEALS

198

[CHAP. V

Consequently, 9t n.£ = f). 4.8. COROLLARY. For every minimal left nonzero ideal .£ of a completely simple semigroup I!( with zero there always exists a minimal right nonzero ideal 9t such that .£9t = I!(. PROOF.

As was shown in the proof of the theorem of 4.5, .£I!( = I!( =;6- Om:.

By 4.5, I!( is the union of its minimal right nonzero ideals, and thus it is impossible that, for each of the 9t, .£9t = Om:. If, fot some of the 9t, then by 4.7 it follows that .£9t =

~l.

4.9. It is important that all the minimal left and right nonzero ideals of a completely simple semigroup with zero can be given with the help of the idempotents of the semigroup. THEOREM. If I is a nonzero idempotent of a completely simple semigroup I!( with zero, then I!(I is a minimal left nonzero ideal of the semigroup, while II!( is a minimal right nonzero ideal. Here each minimal left or right nonzero ideal can be given with the aid of some nonzero idempotent of the semigroup. PROOF. By 4.5, I is included in some minimal left nonzero ideal.£. Clearly, I!(I is a left ideal of the semi group contained in.£. Since I!(I:3 II = I it follows that I!(I =;6- O~[. Thus I!(I = .£. Let .£ be some minimal left nonzero ideal of I!(. By 4.8 there exists for .£ a minimal right nonzero ideal 9t such that .£9t =;6- O~[. By 4.7 it follows that 9t.£ possesses an identity E, where

.£

= I!(E,

E2

=

E.

The argument for right ideals is analogous. 4.10. COROLLARY. Each element X of a completely simple semigroup I!( with zero has a left identity and a right identity that are idempotents. PROOF. By the theorem of 4.5, X must be included in some minimal left nonzero ideal .£ and in some minimal right nonzero ideal 9t. It follows from 4.9 that there exist in U idempotents I and I' such that 9t = I'I!(.

Sec. 4]

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

199

Obviously,

XI=X, 4.11. COROLLARY. group mwith zero, then

If X

I'X=x.

is a nonzero element of a completely simple semimxm=m.

PROOF. mxm is a two-sided ideal of m. Let I be a left identity of X and I' a right identity of X e4.10). Then

mxm 3 IXI' = X ~ Om:. Consequently, mxm is a nonzero two-sided ideal ofm and thus may only be m itself. 4.12. Let m be a completely simple semigroup with zero. For any nonzero idempotent I the set of all the elements having I as their two-sided identity is 1m!. Such sets are very important in the study of~. This is connected with the fact that each subgroup (fj of the semigroup ~ is contained in a set of the form EmmEm. Here we limit ourselves to the proof of only two properties of such sets. THEOREM. If I is a nonzero idempotent of a completely simple semigroup m with zero, then ImI is a group with an externally adjoined zero. Here, for every nonzero idempotent I' and for X E 1m!' it is always true that

X· I'mI = ImI, PROOF.

(1) Since mm

I'mI' X

= I'mI'.

= ~, we have I~I = (1m) . emI).

By 4.9, mI is a minimal left nonzero ideal of ~ and JIlt is a minimal right nonzero ideal of~. Here (m!) . (Im) :3 I, and consequently (m!) . (1m) ~ Om:. By the theorem of 4.7 it follows that

(1m) . em!)

= ImI

is a group with an externally adjoined zero. (2) Since X E l'!II', XI' lX= X,

=

X.

Thus xm is a right nonzero ideal included in the minimal right nonzero ideal 1m (4.9). Consequently

xm = 1m.

[CHAP.

SEMIGROUPS WITH MINIMAL IDEALS

200

From this we obtain I9.II = X9.II

V

= X1'9.II.

The second required equality is proven analogously. 4.13. COROLLARY. Let I be a nonzero idempotent of a completely simple semigroup 9.I with zero. Then no idempotent of 9.I different from I can be a two· sided identity of 1. In fact, if the idempotent l' is a two-sided identity of I, then l' ¥: Ow. and, by 4.12, I'9.II' is a group with an externally adjoined Zero. The nonzero idem· potents 1 and l' both belong to it. This is possible only when l' = I.

4.14. The property of 4.13 implies that each nonzero idempotent of a completely simple semigroup with zero is primitive (3.10). It turns out that even in the weakened form this property, along with the requirement ofthe absence of proper two·sided nonzero ideals, gives a sufficient condition for a semigroup to be a completely simple semigroup with zero. THEOREM. If a semigroup 9.I with zero has no proper two-sided nonzero ideals and possesses a primitive idempotent, then 9.I is a completely simple semigroup with zero. PROOF.

(1) We denote by mthe collection of all the elements B such that

B9.I = am. For any X

E

9.I, (XB)9.I

= am,

(BX)9.I c: B9.I = 01){'

m

m

m

Consequently, is a two-sided ideal of 9.I, and thus = am or = 9.I. The latter would mean that 9.I9.I = a'll' which is impossible, for 9.I9.I:3 II = 1 ¥: am. Consequently, Q3 = a'll' (2) For every A E 9.I the product 9.IA9.I is a two·sided ideal. Thus one of the following is true: 9.IA 9.I = 9.I. 9.IA9.I = O~(, By the first part of the proof, for a nonzero A we have A9.I ¥: 0lJ.(' We choose in A9.I some nonzero element C. The fact that 9.IA9.I = 0lJ.( implies that 9.IC = a'll' However, it is proven in a manner similar to that of the first part that the existence of nonzero elements satisfying such a condition is impossible. Consequently, for A ¥: 0lJ.( it must necessarily be true that 9.IA 9.I = 9.I. (3) The set 9.Il containing II = 1 is a left nonZero ideal of 9.I. Let.2 be some left nonzero ideal of 9.I included in 9.I1. We choose in .2 some nonzero element L. As was shown in the second part, 9.IL2t = 9.I.

Sec. 5]

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

Consequently, there exist in

~{

201

elements U and V such that

ULV= I. We consider the element

l' = IVIUL. Since L

E

mI, LI = L. Thus

1'1' = IVIUL . IVIUL

= IVI·

= IVI· I· IUL = IVIUL = 1',

ULV· IUL

i.e., l' is an idempotent. Since furthermore,

IULI'V= IUL . IVIUL . V= I· ULV· I· ULV= I, while I is a nonzero element, it follows that l' is not a zero of the semigroup either. Since LI = Land 12 = I it follows from the expression for I' that

1'1 = I',

II'

= I'.

But by the assumption of the theorem this is possible only when l' But then

= 1.

'1(1 = '1(IVIUL c £l. We have shown that '1(1 contains no proper left nonzero ideal. Consequently, '1(1 is a minimal left nonzero ideal. We argue similarly for right ideals. Noting finally that '1('1( is different from zero since it contains the nonzero element II = I, we come to the conclusion that '1( is a completely simple semigroup with zero. 5. The Structure of Completely Simple Semigroups with Zero 5.1. For the study of the structure of completely simple semigroups Rees [1] used a certain general construction recalling the construction of Wedderburn in the theory of simple algebras. Let -S be a semigroup with zero and let rand r' be two arbitrary nonempty sets of indices. Let each pair (IX, (3), where IX E r, (3 E r', be associated with some element of the semigroup P,,-,(3 E -S. The rule giving this correspondence can be represented in the form of a matrix P (finite or infinite), where r is the set of the rows, r' is the set of the columns and all the elements belong to -S. For finite (or even denumerable) rand r', which in this case it is natural to consider as sets of natural numbers, such a representation is completely intuitive:

P=

r P21

P12 P22

u

P m1 Pm2

p,. ]

: .. :~:

(P",(3

E

-S).

202

SEMI GROUPS WITH MINIMAL IDEALS

[CHAP. V

We denote by 6(P, f) the set consisting of all possible three-tuples (H,~,

'1')

and of an element 0 6 , For uniformity in subsequent calculations we will also write the latter element in the form of three-tuples (05), ~, '1') for any; E r', 'I')

Er.

In the set 6(P, f) we define the multiplication (HI' ~1' '1')1) . (H2, ~2' '1')2) = (H1Pn,<2H2' ~I' '1')2)'

We prove the associativity of this operation:

Thus 6(P, f)) is a semigroup. The element 0 6 is clearly its zero. We will call the semigroup 6(P, f) the matrix semigroup over f) with defining matrix P. If f) is a group with an externally adjoined zero (II, 2.12) and if there are in each row and column of P elements distinct from 05), then 6(P, f) will be called a completely simple matrix semigroup with zero over f) with defining matrix P. The justification of the last definition will be found in 5.3 and 5.4.

5.2. The significance of the law of multiplication introduced into 6(P, f) and the name given above for such a semigroup become completely clear and intuitive if we turn to the matrix interpretation. We limit ourselves for this to finite rand r /, which will be considered to be sets of natural numbers:

r=

{l, 2, ... , m},

r' = {I, 2, ... , n}.

The three-tuple (H, ~,'I') (H;;.!: 05) may be represented in the form of a matrix with n rows and m columns (in contrast to P, in this matrix r' is the set of rows, while r is the set of the columns), in which the element in the ~th row and the 'l')th column is equal to H while all the other elements are equal to 05)' Here 0 6 is the matrix all of whose elements are equal to 05)' If we define addition between the element 05) and all the elements of f) by the rule 05)

+ H = H + 05) = H

(HE~),

then multiplication of three-tuples of 6(P, f) may be represented in the form of

Sec. 5]

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

203

matrix multiplication with the insertion of the matrix P:

(HI> ';1' 1]1) . (H2 , ';2, 1]2)

~ x

r~

l:

0

0

Pn

P12

PIn

(Hl)~,n,

0

P21 P22

P2n

0

0

P n1 Pn2

P~n

0

0

0

0

(H2)/;2n2

0

0

0

0

=

0

0

0

0

(HtPn,!;2H2)I;,n2

0

0

...

0 0

= (H1Pn,1j2H2' ';1' 1]2). 5.3. THEOREM. A completely simple matrix semigroup S(P, N with zero (5.1) is always a completely simple semigroup with zero (4.2). PROOF. We denote by ffi the group of all the nonzero elements of ,D. As follows directly from the rule for multiplication in S(P, ,D), for each 'V E the set £v consisting of all the elements of the form

r

(H, .;, 'V) is a left ideaL We show that this ideal is a minimal left nonzero ideaL We choose two arbitrary nonzero elements of this ideal:

For X

= (Ga, a, T), we have XLI = (GaPT/;,G1 , a, 'V).

If in X we set a = ';2' choose T so that PTIj , E ffi (such an index definition of the matrix P) and take as Ga an element of ffi

T

exists by

G3 = G2GIIP~:, then for such an X we find that

XLt

= L 2•

It follows from this that every left ideal of S(P, ,D) containing one of the nonzero elements of £. must necessarily contain every other element of Qv' This means that £v is a minimal left nonzero ideal of the semigroup. We convince ourselves similarly of the existence of minimal right nonzero ideals.

204

SEMIGROUPS WITH MINIMAL IDEALS

[CHAP.

V

The semigroup S(P, .5) has no proper two-sided nonzero ideals. For let us choose two arbitrary nonzero elements:

As before we may choose an element X such that

XS1

= (G2 , ~2' rll)·

A similar argument will yield a Y such that

(XS1 )· Y

= (G2 , ~2' 111)·

Y

= (G 2, ~2' 'YJ2) =

S2·

It follows from this that a two-sided ideal S(P,.5) containing some nonzero element of a semigroup will also contain every other element of the semigroup. It follows that in the semigroup S(P,.5) the semigroup itself is a unique twosided nonzero ideal. It is easy to see that S(P, .5)S(P, .5) =F Os. 5.4. As was shown by Rees [1] in the general case, the completely simple matrix semigroups S(P,.5) with zero (5.1) exhaust up to isomorphism the class of completely simple semigroups with zero (4.2). THEOREM. Every completely simple semigroup with zero is isomorphic with some completely simple matrix semigroup S(P,.5) with zero (5.1). PROOF. (1) Let III be a completely simple semigroup with zero. We denote by rCl) the collection of all of its minimal left nonzero ideals and by rCf') the collection of all of its minimal right nonzero ideals. We fix some pair of ideals .20 E rCI) and 9\0 E rCf') such that .209\0 =0= m(4.8). By 4.7, .50 = 9\0.20 is a group with an externally adjoined zero. We denote the group of all of its nonzero elements by (50 and its identity by Eo. By 4.7 we have .5 0 = EomEo. Since .5 0 3 Om:, we have 04'10 = Om:. By 4.9 there exist in each ideal £ E r cI) and in each ideal 9\ E rCf') idempotents II!, and IJR such that

£ = mII!"

9\ = IJRm.

We fix them for each £ E rCl) and each 9\ E rCf'). Clearly II!, is a right identity of £ while IJR is a left identity of 9\. By 4.12, II!,"llEo • EomII!, = II!,mII!, 3 II!,. Thus it is possible to choose and fix elements UI!" VI!, such that UI!,VI!, = II!" Uj3 E II!,mEO' Vj3 E EomIj3. By 4.14 it follows from UI!, E Ij3mEo that

EomIj3 . Uj3 = Eo"llEo3 Eo. Thus it is possible to choose and fix an element Wj3 such that

WI!,UI!, = Eo,

Wj3 E EomII!,.

Sec. 5]

205

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

Similarly, for 91

EO r(r)

For an arbitrary pair 2

we fix elements U'J{' V'J{' W9\ such that

EO r(!l,

P£,'J{

=

9t

EO r(r)

VSJUg;

E

we use the notation

Eo'11Eo =

~o.

The matrix over the group flo with an externally adjoined zero, the set of whose rows is r( Z), the set of whose columns is r(r), and whose elements are P£,'J{, will be denoted by P. We note that in every row (and similarly in every column) there are elements distinct from 0f>o = O~. In fact, for 2 EO r(!) we take 9t = lEU. By 4.9, 9t EO r(r). Since

= ISJI£ = USJVSJI£ EO U£V£9t = UEVEU'J{Vm'11, it follows that V£Um = PE'J{, :;6 O~, for otherwise it would IE = O~l and 2 = '111£ = O~. 1£

turn out that

(2) Let '11' be a completely simple matrix semigroup with zero over a group ~o with an externally adjoined zero, with defining matrix given by the matrix P constructed above (5.1). By 4.5, for each nonzero element A E'11 there are determined in a unique fashion ideals 2(A) EO r(l), 9t(A) EO r(r) such that

A

EO

9t(A)2(A).

Since it follows that

V'J{(A)AUE(A)

E

Eo'11Eo

= ~o.

Thus the following mapping X of the semigroup '11 is a mapping of it into '11': x(A) = (Vm(A)AU£(A), 9t(A), 2(A»), X(O~ = O~,.

We show that the mapping X is one-to-one. Let X(A)

This means that

91(A)

=

9t(B)

= x(B), (A, B EO '11). = 9t, 2(..1) = 2(B) = 2 and V9\AUE = V9\BUE·

Multiplying the last equation on the left by U9\ and on the right by V£, we obtain I9\AIE = ImBISJ· But 1m is a left identity of the ideal 9t that contains both A and B, while h is a right identity of the ideal 2 that also contains both A and B. Thus the last equation implies that A = B.

206

[CHAP.

SEMI GROUPS WITH MINIMAL IDEALS

V

The image of III under the mapping X is the whole semigroup Ill'. To see this, let A'

= (G,

9\, £)

E

(G

Ill'

E (\50'

£

E

r(l), 9\

E

r(r)).

We consider the element Since it follows that Thus

= (EoGEo, 9\, £) X(O,!!) = O'!!"

X(A) = (VlRWlRGW£U£, 9\, £)

= (G, 9\, £)

=

A',

(3) We show that the one-to-one mapping X of the semigroup III into the completely simple matrix semi group Ill' = 6(.$)0' P) with zero is an isomorphism. Keeping 4.6 in mind, let

By the rule for multiplication in Ill', we obtain X(A 1) • X(A 2) = (VlRIA, U£I' 9\1' £1) . (VlR2 A 2 U£2' 9\2' £2)

= (VlR A1 U" PE '" V"l A2U£ , 9\1' £2) = (VlR A1 UE VE UlR Vm A 2U 9\1' £2) 1

..... 1

1

1

1"\2

1

.;1\2

2

2

2

O ,

-2

= (VlRIAlE/lR2A2U£2' 9\1' £2)'

Since Al E £1 while lEI is a right identity of £1' and A2 E 9\2 while IlR2 is a left identity of 9\2' we have X(A 1) • X(A 2) = (VIR I A 1A 2 U£ 2, 9\1' £2)

=

X(A 1 A 2)·

5.5. The representation of completely simple semigroups with zero in the form of matrix semigroups 6(P,.$5) (5.1) allows us to easily obtain some of their various properties. We will discuss, for example, how to construct the systems of left and right ideals of a completely simple matrix semigroup with zero. We denote by r the set of the rows of the matrix P and by r' the set of its columns. Let ~ be a nonempty subset of the set r. As follows directly from the rule for multiplication in completely simple matrix semigroups, the collection of all the elements of the form (G, ~, 'Y)), where 'Y) E~, is a left ideal of 6(P, .$5). The semigroup 6(P, .$5) has no left nonzero ideals other than those obtained by means of subsets~. This follows from the fact that if some left ideal £ of the semigroup 6(P,.$5) contains an element (G, ~,'Y)) (G =F O~), then, as is easily seen

Sec. 5]

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

207

(in essence we have already proved this in the preceding arguments), B will also contain every other element of the form (G' , ~', 'rj). Thus, in a completely simple semigroup with zero the left ideals are in one-to-one correspondence with the nonempty subsets of the set r. The minimal left nonzero ideals are those which correspond to the subsets consisting of one element of the set r. If to the zero ideal of the semigroup 6(P,.5) we make correspond the empty subset of the set r, then we will obtain a one-to-one correspondence between all the left ideals of the semigroup 6(P, .5) and all the subsets (including the empty one) of the set r. The situation is analogous for right ideals, which turn out to be in one-to-one correspondence with the subsets of the set r' 5.6. In a completely simple matrix semigroup 6(P, .5) with zero a nonzero idempotent will be an element of the form (P;i, ~, 'Y)), where P'rJ; ¥- O$). The maximal subgroup corresponding to this idempotent (III, 1.16) is clearly the set consisting of all the elements of the form (G, ~,'Y)). The mapping rp of the group (!) = .5\0;)( into this group, rp(G)

= (GP;;il, ~, 'rj),

is one-to-one. It is an isomorphism, for rp(G1) • rp(G2 )

= (G1P;;il ~, 'Y)

. (G2P;;il ~, 'Y))

= (G1P;;il . P'rJ; . G2P~\ ~, 'Y) = rp(G1G2)· Thus all the nonzero maximal subgroups of the semigroup 6(P,.5) are isomorphic with each other, since each of them is isomorphic with the group (!) = .5\0$). 5.7. By what has been said above, it is clear how important and convenient is the representation of completely simple semigroups with zero in the form of matrix semigroups. To complete the question of their representation in such a form it is still necessary to say whether such a representation may be realized in a unique fashion, and if not, we must give the relationship between the completely simple matrix semigroups with zero that are isomorphic with a given completely simple semigroup with zero. Clearly, this question is equivalent to the corresponding question arising when two given completely simple matrix semigroups with zero are isomorphic to each other.

m r

THEOREM. Let m: = 6(P,.5) and = 6(P,:S) be two completely simple matrix semigroups with zero (5.1). Let be the set of the rows of the matrix P, let r' be the set of its columns and let rand be the corresponding sets for P; (!) = .5\0 $) and ijj = \0 $). In order for m: and iii to be isomorphiC it is necessary and sufficient that there exist: (1) a one-to-one mapping fl of r onto

r'

:s

r,

fl('Y))

=

ij ('Y) E

r, 17 E r);

208

[CHAP.

SEMI GROUPS WITH MINIMAL IDEALS

(2) a one-to-one mapping v of r' onto p(~) = ~(~ E

V

r',

r', ~' E r');

(3) an isomorphism (J of the group (£) onto (\); (4) elements Gij E (f) (ij E r); (5) elements G~ E (£) (~ E r') such that P'i~ PROOF.

by 121 and

iii.

= Gij' (J(Pn~) . G~

('f)

E

r, g E r').

(1) Let the five items in the formulation of the theorem be satisfied We define a mapping X of the semigroup minto ~:

X(G,

g, 'Y) = (Gt 1 . (J(G) . G;-l, ~, ij) E i (~ = peg), ii = p,('f)).

If G = Of" then we assume that (J(G) = Of, and thus X(Om) = 0iR' For given g and 'Y), as G runs through the group (£), by the properties of an isomorphism a(G) will run through (J«(£) = (f). Using this, we conclude that the mapping X is a one-to-one mapping of ~l onto iii. Using the connection between P and P, we show that X is an isomorphism: x(G1 , gl' 'Y)1) . X(G 2, g2, 'f)2) -, 1 1 -' 1 = (G$: . (J(G1) ' G;,., gl' iiI)' (G~-; . a( G2)

'

G~, 1 g2' ii2)

= (G~:1 . a(G1) • G;;.1 . Pijl g, • G~-;l . (J(G2) . G~l, ll' ii2) = (G:.":;1-1. a(G1)· a(P~'/1"'2 .) . (J(G2) .

G:-l, lv ii9) '/2'"

= X(GIPry,~,G2' gl' 'f)2)

= X[(G1, gl' 'Y)l) • (G2, g2' 'Y)2)]'

(2) Let there exist an isomorphism X of the semi group monto iii. For a fixed 'f) E r the set Ery consisting of all the elements of the form (G, g, 'Y) (5.5) constitutes a minimal nonzero left ideal of the semi group I2L Under the isomorphism X it must be mapped onto some minimal nonzero left ideal of the semigroup ~, i.e., onto a set Eij consisting of elements of the form (G, ~, ij) for some ij E X(Ery) = £?j' The mapping p"

r:

p,('f) = ij,

is clearly a one-to-one mapping of r onto r. We determine analogously 'a one-to-one mapping p ofthe set r' onto r' with the aid of minimal nonzero right ideals such that peg) = ~: x(9t;) = 9i~. We fix some pair (go, 'f)o) (go E r', 'Y) 0 E r) such that Pryo~o =1= 0 f,' Such a pair exists by the definition of a completely simple matrix semigroup with zero. In this case we also have P?jo~o ~ Of,' In fact, it follows from Pryo~o ~ Of, that the product of any two elements of the form (G, go, 'Y)o) (G ~ Of,) is different from

Sec. 5]

COMPLETELY SIMPLE SEMIGROUPS WITH ZERO

209

O\!!, But then by the definitions of f..l an~ '/i (f..l(~o) = ~o, '/i('Y)o) = ijo) the product of any two elements of the form (0, ~o, ijo) (0 ;r6 Os) must also be different from 0ill (since these elements are the images of the elements indicated above from m: under the isomorphism X). As directly follows from the rule for multiplication in the matrix semi groups under consideration, the latter is possible only when Pfjo~o ;r6 0ill' The one-to-one mapping cp of the group G) onto (£'10 II 91;.)\Om' cp(G) = (GP~Jo' ~o, 'Y)o)

(G

E

ffi),

as we have shown in 5.6, is an isomorphism. The analogous mapping 'IjJ of the group onto (Iiiio II 9t~o)\Oill is also an isomorphism. By our definitions of the mappings f..l and '/i the isomorphism X of the semigroup III onto brings about an isomorphism of £~o II 91$0 onto Bijo II §i~o' Thus the mapping

m

m

is an isomorphism of ffi onto We consider the elements -

-1

m.

-

N~ = (P;;;io' ~, ijo) E Ill,

-

--

Mfj = (£-(f" ~o, ij) E III

(t E r', ij E r). By the choice of ~o and 'fJo these elements are distinct from Dill' By the rule of multiplication in m: we have

Mll~

= (E"(f"

~o, ij) • (P~Jo' ~, ijo) = (E"(f,Pii~P~lo' ~o, ijo) = 'IjJ(P'74)'

By the choice ofthe mappings f..l and l' we have, for some C; E ffi and X-l(N~) = (C;, ~, 'fJo),

x-1(Mfj) = (D

Here C; and DIJ are not zeros, since Ng and For the C; and D'1 thus chosen we obtain

IJ ,

D~ E

ffi,

~o, 'Y)).

Mij are not zeros.

Pfjg = 'IjJ-IXcpcp-lX-1'IjJ(Pfj~) = 'IjJ-IXq;q;-lX-l(MfjN~) = aq;-1[x-1(M>j) . X-l(N~)]

= aq;-l[(D~, ~o, 'fJ) • (C~, ~, 'fJo)] = aq;-l(Dr;PI);C;, ~o, 'fJo) = a(D'1PIjI;C;P'1oi;o) = a(D~) . a(P'1I;)' a(C;P'1o~J Setting we obtain the desired expression for the elements of the matrix P. 5.8. The conditions for isomorphism of two completely simple matrix semigroups with zero considered in 5.7 also permit various other approaches to simple matrix semigroups with zero.

210

SEMIGROUPS WITH MINIMAL IDEALS

[CHAP.

V

For a matrix P over a semigroup i) with zero the following transformations of it, i.e., transitions from P to a new matrix P, will be called elementary transformations of the matrix P: (1) one-to-one mapping of the set of the rows r onto some set r of equal cardinality ("permutation of the rows"); (2) one-to-one mapping of the set of the columns r' onto some set f' of equal cardinality ("permutation of the columns"); (3) isomorphism of the semigroup i) onto a new semigroup 5; (4) multiplication on the left of each row of the matrix P by one of its arbitrary elements from i)\O$j; (5) multiplication on the right of each column of the matrix P by one of its arbitrary elements from i)\O$j' If for two completely simple matrix semigroups with zero, S(P, Nand S(P, 5), the matrix P can be obtained from P with the aid of one of these elementary transformations (in the case of the third transformation, of course, in the elements of the semigroup S(P, i)) itself the first component of i) is replaced by a corresponding element from 5), then by 5.7 the semigroups S(P, i)) and SCP, 5) are isomorphic. It is understood that such an isomorphism will exist only when P can be obtained from P as a result of several successive elementary transformations. In this connection, in the case of a finite number of rows or columns the last two transformations can be replaced by transformations in which only one row or one column is multiplied. As follows from 5.7, every isomorphism ofa completely simple matrix semigroup onto another completely simple matrix semigroup may be obtained as a result of successive applications of various elementary transformations (where each of the five transformations may be used only once). 6. The Structure of Completely Simple Semigroups without Zero 6.1. Our investigation ofthe structure of completely simple semigroups with zero allows us to easily obtain the corresponding properties for completely simple semigroups without zero (of course, these properties could be obtained by direct means with arguments similar to those of the preceding section). The connection (discussed below) between the semigroups of the two classes given below will serve as a basis for this purpose. THEOREM. If in a completely simple semigroup '2'( with zero the zero is externally adjoined (II, 2.12), then the set of all the nonzero elements of the semigroup '!'( is a completely simple semigroup without zero. if a zero is adjoined in an external fashion to a completely simple semigroup '!'(' without a zero, then the semigroup obtained is a completely simple semigroup with zero.

Sec. 6]

COMPLETELY SIMPLE SEMIGROUPS WITHOUT ZERO

211

PROOF. (1) Let Wbe a semigroup with an externally adjoined zero and let W' be the subsemigroup of its nonzero elements. If £ is a left nonzero ideal of W, then £\O~l' as is easily seen, is a left ideal of the semigroup W'. In this manner all the left ideals of~I' can be obtained, since it follows from the fact that £' is a left ideal of W' that £' u O~l is a left nonzero ideal of W. We argue analogously for right and two-sided ideals. (2) If Wis a completely simple semigroup with zero, £ is one of its minimal left nonzero ideals and 9{ a minimal right nonzero ideal, then by what was said above £\O~ and 9{\O~wi11 be respectively minimal left and minimal right ideals of the semigroup WI. Since there are no proper nonzero two-sided ideals in W, there are also no proper two-sided ideals in W'. (3) If W' is a completely simple semigroup without zero and £' and \"It' are minimal left and minimal right ideals of it, then £' u 02f and \"It' U O~l will be respectively a minimal left nonzero ideal and a minimal right nonzero ideal of ~r. From the fact that WI has no proper two-sided ideals it follows that Whas no proper two-sided nonzero ideals.

6.2. In connection with this theorem it is necessary, as a preliminary to our study of completely simple semigroups with zero, to clear up the question concerning when the zero in a completely simple semigroup with zero is an externally adjoined zero. By the theorem of 5.4 we may limit ourselves to completely simple matrix semigroups with zero. THEOREM. In a completely simple matrix semigroup 6(P, 4) with zero (5.1), the zero will be externally adjOined if and only if all the elements of the matrix P are different from the zero of the semigroup 4). PROOF. From the definition of multiplication in 6(P, 4) it follows that if all the elements of P are distinct from 0 fl' then the product of any nonzero elements of the semigroup 6(P, 4) is also a nonzero element, i.e., the zero 0'2; is externally adjoined to 6(P, f,). If some element of the matrix P is equal to the zero of 4),

Prye

=

0 55 ,

then for some HI' H2 E 4) \ 0 fl the following product of nonzero elements turns out to be equal to 0 6 : (HI' ~, 'f) . (H2' ~, 'f)

= (HIP~I;H2' ~, 'f) = (0 55 , ;, 'f) = 0 6 ,

6.3. Let (f) be a group and P a matrix (finite or infinite) with a set of rows r and a set of columns r', the elements Pry!; of which belong to (f). We denote by 6 (P, (f) the set of all the three-tuples of the form t

(G, ~, 'f)

(G

E (f), ; E

r', 'f) E n·

In 6'(P, G'i) we define the operation (G1 , ;1' 'f)1) . (G2 , ~2' 'f)2)

= (G1Pry,I;P2' ;1' 'f)2)'

212

SEMIGROUPS WITH MINIMAL IDEALS

{CHAP.

V

Clearly, 6'(P, (£) will be identical with the set of all the nonzero elements of the completely simple matrix semigroup 6(P, .$5) with zero, where .$5 is the semigroup obtained from (£) by adjoining a zero externally. Since all the elements of P are distinct from 0 5), by 6.2, 6(P, .$5) is a semigroup with an externally adjoined zero. By 6.1 it follows from this that 6'(P, (£)) is a completely simple semigroup without zero. We call 6'(P, .$5) a completely simple matrix semigroup without zero. 6.4. The semigroups 6'(P, (£)) described in 6.3 exhaust, up to isomorphism, all the completely simple semigroups without zero. THEOREM. Every completely simple semigroup III without zero is isomorphic to some completely simple matrix semigroup without zero (6.3). PROOF. Adjoining a zero 0 to III externally, we obtain a semigroup 58, which by 6.1 is a completely simple semigroup with zero. By the theorem of 5.4,58 is isomorphic with some completely simple matrix semigroup 6(P, .$5) with zero, where.$5 is a group (£) with an externally adjoined zero 0,:,. Since 58, and thus also 6(P, .$5), is a semigroup with an externally adjoined zero, it follows by 6.2 that all the elements of the matrix P belong to the group (£). It follows that the set of all the nonzero elements of the semigroup 6(P, ..$5) is identical with the semigroup 6'(P, (£)) (6.3). This semigroup is isomorphic with the subsemigroup of all the nonzero elements of the semigroup 58, and consequently is isomorphic with the original group \ll.

6.5. Thus, for completely simple semigroups without zero, we have obtained a complete description of their structure similar to that given earlier for completely simple semigroups with zero. The question of isomorphism, i.e., of how the various representations of the same completely simple semigroup without zero are connected, is also simply decided by means of the corresponding result for completely simple semigroups with zero. Let there be given two completely simple matrix semigroups 6'(P, (£) and 6'(P, ~) without zero (6.3). Adjoining to them externally the zeros 0 and 0, we obtain completely simple semigroups with zero 6(P,..$5) and 6(P, 5), where ..$5 is the group (£) with an externally adjoined zero 0.15' and ~ is the group ~ with an externally adjoined zero 0:5. Clearly, the semigroups 6'(P, (£)) and 6'(P, (\)) will be isomorphic to each other if and only if the semigroups 6(P,.$5) and 6(P, f)) are isomorphic to each other. A necessary and sufficient condition for the isomorphism of such semigroups was given in 5.7. Using it, we obtain directly a necessary and sufficient condition for the isomorphism of the semigroups 6'(P, (£) and 6(P, (£). It is not even necessary to reformulate it, since it coincides word for word with the condition of 5.7. It consists of the five points indicated there, which all refer to the groups (£) and (£) and the matrices P and P. 6.6. Let 6'(P, (£)) be a completely simple matrix semigroup without zero (6.3) and let 6(P,..$5) be the completely simple matrix semigroup with zero

Sec. 6]

COMPLETELY SIMPLE SEMI GROUPS WITHOUT ZERO

213

obtained from 6'(P, (l"j) by the external adjoining of a zero O. In the proof of the theorem of 6.1 we noted how the ideals of the semi group were associated. By this relationship, using the fact that in 5.5 we gave the construction of the systems of left and right ideals of a semigroup of the form 6(P, f», we directly obtain the systems of left and right ideals of the semigroup 6'(P, (l"j). Let r be the set of the rows of the matrixPand let r' be the set of its columns. For any nonempty subset 2:; c r, the set of all the elements of the form (G, ~, 1]) (G E (l"j) for which 1] E 2:; is a left ideal of the semigroup 6'(P, (l"j). This semigroup has no left ideals other than ideals of the same form for different ~ c r. The minimal left ideals will be those left ideals for which ~ consists of one element. If~' c r', then the set of all the elements of the form (G, ~, 1]) (G E (l"j) for which ~ E 2:;' is a right ideal of 6'(P, (l"j), and such ideals exhaust all the right ideals of 6'(P, (l"j). 6.7. We examine one more important property of completely simple semigroups without zero. Let r be the set of the rows and r' the set of the columns of the matrix P of a completely simple matrix semigroup 6'(P, (l"j) without zero (6.3). For ~ E r', 'Yj E r we denote by (l"j~ry the collection of all the elements ofthe form (G, ~, 1]) (G E (l"j). By 5.6, (l"j~~ is a group isomorphic with the group (l"j. Thus all the (l"j~ry turn out to be isomorphic with one another. Since 6'(P, (l"j) is their nonintersecting union it follows from 6.4 that we have the following properties for an arbitrary completely simple semigroup without zero. (Ot.:) Every completely simple semigroup without zero is the nonintersecting union of groups that are isomorphic to one another. «(3) Every completely simple semigroup without zero is completely regular. This follows from (Ot.:) and from III, 1.15. (y) Any two distinct idempotents of a completely simple semigroup without zero commute. In fact, the idempotents of 6'(P, (l"j) are identities of the groups (l"j~ry, but by the rule for multiplication in 6'(P, (l"j) elements from (l"je,ry, and (l"je.ry2 for ~1 :;t: ~2 or for 1]1 :;t: 1]2 clearly do not commute. (0) If a completely simple semigroup without zero has only one idempotent, it is a group. This follows directly from (Ot).

6.8. It follows from these properties that each element of a completely simple semigroup without zero has a two-sided identity that is an idempotent. In the semigroup 6'(P, (l"j) such an identity for the element (G, ~, 1]) will obviously be (P;;"/, ~, 1]). We note in passing that with the exception of a group, completely simple semigroups without zero never have a two-sided identity of the whole semigroup. In fact, if the set of the rows of the matrix P of a completely simple semigroup 6'(P, (l"j) contains even two elements, then (GI , ~I'

1]1) .

(G2, ~2'

1]2)

= (G1Pry ,e.G2' ~1' 1]2) :;t: (G1, ~1' 1]1)

(1]1:;t: 1)2)

214

[CHAP.

SEMIGROUPS WITH MINIMAL IDEALS

V

and (G 1 , ;1' 1]J cannot be an identity of the semigroup. The analogous statement is true when the set of the columns of the matrix P contains no less than two elements. If both the set of the rows and the set of the columns of the semigroup 6'(P, ill) consist of only one element, then S'(P, ill) is isomorphic to the group ffi (6.7). 6.9. In spite of the significant similarity between completely simple semigroups without zero and completely simple semigroups with zero, there are also differences between them. As an example, we consider the follo\Ying completely simple matrix semigroup with zero. Let ill = {E} be the identity of the group, and let the sets of the indices and r' each consist of two elements {I, 2}. As the matrix P we take the identity matrix

r

p

= [;

~]

(as a result of which the multiplication in our semigroup is obviously the usual matrix multiplication). The corresponding completely simple matrix semigroup with zero consists of the five elements

~J.

E12

= (E, 1,2) =

[~ ~J.

[~ ~],

E22

= (E, 2, 2) =

[~ ~J.

Ell

= (E, 1, 1) = [;

E21

= (E,2, 1) =

o=

[~ ~]

and, as may be directly verified, possesses the following multiplication table:

Ell

E12

E21

E22

0

Ell

Ell

E12

0

0

0

E12

0

0

Ell

E12

0

E21

E21

E22

0

0

0

E22

0

0

E21

E22

0

0

0

0

0

0

0

We see that the nonzero elements E12 and E21 not only are not included in the subgroups, but also have in general no two-sided identities.

Sec. 6]

COMPLETELY SIMPLE SEMI GROUPS WITHOUT ZERO

215

6.10. It is not difficult to explain how the property of 6.7 may be transformed to apply to a completely simple semigroup with zero. We consider a completely simple matrix semigroup 6(P,~) with zero (5.1). Its arbitrary nonzero element S = (G, ~, 'Y) is included in the minimal right ideal ~~ consisting of all the elements of the form (H, ~, J.) (H E~, J. E r) (5.5), and in the minimal left ideal £'7 consisting of all the elements of the form (H, fl, 'Y) (H E f), p, E r') (5.5). If P1)~ ;:6 0e, then

£1)9\;:3 (ESj' ~,

'Y) •

(ESj, ~,

'Y)

= (P'1<'

~, 'Y) ;:6 06'

from which it follows by 4.7 that 9t~£ry is a group with an externally adjoined zero. Our element S is included in the group 9t;B'1 \ 0e since it belongs to 9\~ n £ry = 9t~£1) (4.7). If Pry; = Sj' then

°

S2

= (G, ~,'Y)

• (G, ~,'Y)

= (GP1) gG, ~,1]) = 0 6

and S cannot be included in any group of 6(P, ~). From what has been said it follows that in a completely simple semigroup with zero all the elements whose squares are different from zero are included in subgroups and thus are completely regular (IIr, 1.18). Elements whose squares are equal to zero are, of course, not completely regular. 6.11. The consideration of completely simple semigroups without zero allows us to give the structure of completely regular semigroups in which every two elements are regularly associated with each other. We considered such semigroups in II, 6.8 and II, 6.9. Let r' and r be two arbitrary nonempty sets. We denote by)S the collection of all the possible pairs of the form (~, 1]) where ~ E r', 1] E r. By defining in)S the multiplication (~1' 'Y)1) • (~2' 'Y)2) = (~1' 1]2) we make )S into a semigroup. Obviously)S is the completely simple matrix semigroup 6'(P, G» without zero, where (l) is the identity group, P is the matrix the set of whose rows is r, the set of whose columns is r', and all of whose elements are equal to the identity E, the only element of the group (l). If either ~l ;zf g2 or 1]1 ;:6 'Y)2' then (~1' 1]1) . (g2' 'Y)2) ;:6 (~2' 1]2) . (~l> 'Y)l)'

Thus, by II, 6.9, )S is a semigroup in which every two elements are regularly associated with each other. 6.12. We will show that every semigroup III in which each two elements are regularly associated with each other is isomorphic with some semigroup )S of the type considered in 6.11. Let X and Y be two arbitrary elements of Ill. Since XYX = X, every twosided ideal of III containing Y will contain X. It follows that III has no proper two-sided ideals.

216

SEMIGROUPS WITH MINIMAL IDEALS

[CHAP.

V

For any A IS m the set of elements of the form XA(X Em) is a left ideal. Moreover, it is a minimal left ideal since every left ideal containing an element XA will also contain (by II, 6.8) the element YA = YXA for any Y IS m. Analogously, Am is a minimal right ideal. Thus mis a completely simple semigroup without zero. By 6.4, mis isomorphic with some semigroup 6'(P, (l). The group (l) must be the identity group, since on the one hand by 6.6, 6'(P, (l) is the union of groups isomorphic to (l), and on the other hand all the elements of mare idempotent (II, 6.8). The semigroup 6'(P, (l) for the identity group (l) is in an obvious way isomorphic to a semigroup of the type considered in 6.1l. 6.13. The discussion in 6.11 and 6.12 gives an exhaustive description of the structure of semigroups in which every two elements are regularly associated with each other. It also gives a classification of such semigroups, since it is easy to show directly (this also follows from the conditions for isomorphism of completely simple semigroups of matrix type) that two semigroups Q3 1 and Q3 2 of the type of 6.11 are isomorphic if and only if the corresponding sets r~ and r~ are of equal cardinality and 1 and 2 are of equal cardinality.

r

r

6.14. The semigroups of the class considered here can also be characterized as completely simple semigroups without zero, all the elements of which are idempotent. It has already been shown that they all possess these properties. Conversely, let be a completely simple semigroup without zero, all the elements of which are idempotent. It is isomorphic with some matrix semigroup 6'(P, 6) (6.4). Since all the elements of ~{are idempotent, all the subgroups of mwill be the identity group. By 6.4 it follows from this that (l) is the identity group. But we have already noted that when (l) is the identity group, the semigroup 6'(P, (l) is isomorphic with some semigroup Q3 of the type of 6.10.

m

6.15. In conclusion we will note how, with the aid of the concept of a primitive idempotent (3.10), the completely simple semigroups without zero and the completely simple semigroups with zero can be naturally combined in one common class. THEOREM. In order for a nonidentity semigroup without proper two-sided nonzero ideals to be completely simple it is necessary and sufficient that it contain a primitive idempotent. PROOF. It follows from 4.10 that every completely simple semigroup with zero possesses a nonzero idempotent. All ofthem, as was mentioned in 4.14, are primitive. By 3.10 a completely simple semigroup without zero possesses primitive idempotents. If a semigroup with zero that does not have proper two-sided nonzero ideals possesses primitive idempotents, then by 4.14 it is a completely simple semigroup with zero.

Sec. 6]

COMPLETELY SIMPLE SEMI GROUPS WITHOUT ZERO

217

If a semigroup without zero Q1 has no proper two-sided ideals and contains a primitive idempotent I, then we adjoin externally a zero. We obtain a semigroup mwhich, as is easy to see, will have no proper two-sided nonzero ideals and in which I will also be a primitive idempotent. Thus m must be a completely simple semigroup with zero. By 6.1 the set ofits nonzero elements Q1 will be a completely simple semigroup without zero. 6.16. The above description of the structure of completely simple semigroups without zero and the introduction of the above properties, as well as various other properties on which we did not dwell, may be realized without difficulty, starting from the representation of completely simple semigroups without zero with the aid of the representation of them as matrix semigroups. This approach was used by Rees [1] and by a number of other authors in later works. However, many of these properties may also be introduced directly without the use of this representation. Schwarz [3] uses such an approach.

CHAPTER

VI

INVERTIBILITY 1. Invertibility of the Product of Elements 1.1. In this chapter we return to the question which was already touched on lightly in the second chapter. DEFINITION.

An element S of a semigroup

m:

is said to be RIGHT INVERTIBLE

if it is a left divisor of every element of m:.

An element S is said to be LEFT INVERTIBLE if it is a right divisor of every element ofm:. An element which is both left and right invertible is said to be a TWO-SIDEDLY

INVERTIBLE ELEMENT.

In a commutative semigroup the properties of right invertibility, left invertibility, and two-sided invertibility coincide, and we may simply speak of invertibility. The task of distinguishing the invertible elements is a quite natural one. In particular, the abundance of elements with one or another property of invertibility characterizes the degree of closeness of the semigroup to a group. In semigroups of transformations, as we shall see, the invertibility property is essential for certain important properties of the transformations. 1.2. By definition, the element S is right invertible if for any A E m: the equation SY=A for the unknown Y is solvable in m:. It therefore follows that S will be right invertible if and only if Sm: = m:. Analogously S is left invertible if for any A Em: the equation

XS=A for the unknown X is solvable in m:, which is equivalent to the validity of the equation m:S = m:. The element S is two-sidedly invertible if and only if Sm:S

= m:.

218

Sec. 1]

INVERTIBILITY OF THE PRODUCT OF ELEMENTS

219

1.3. If the semigroup III possesses a unit element E, then for the right invertibility of S it is necessary and sufficient that S should have a right inverse relative to E. Indeed, from Sill = Ill, it follows that for some Sf we have SS' = E. In turn, from SS' = E follows III = EIll = SS'1ll C Sill, i.e., Sill = Ill. In a quite analogous way one proves that for left invertibility of S it is necessary and sufficient that S should have a left inverse relative to E. By II, 2.14, (a) itfollows from what has been said that an elementS, belonging to a semigroup III with unit element, will be two-sidedly invertible in III if and only if it has an inverse element S-l:

SS-l

= S-lS = E.

Here S-l evidently will also be two-sidedly invertible. 1.4. THEOREM. The set of all two-sidedly invertible elements of a semigroup, is not empty, forms a group whose unit is the unit of the whole semigroup.

if it

PROOF. The fact that the set (!5 of all two-sidedly invertible elements, i.e., the set of all elements which are both right and left divisors of every element of the semigroup Ill, forms a group, was proved in II, 1.5. In II, 2.15 we proved that a group always has a unit E. Suppose that A is any element of Ill. Since E E (!5 there must be in III elements A' and A" such that A = EA', A = A"E. Since E2

=

E, evidently EA

= A and AE = A.

1.5. COROLLARY. In order that the semigroup III should possess two-sidedly invertible elements it is necessary and sufficient that III should have a unit. PROOF. If III has two-sidedly invertible elements, then by 1.4 the unit of the group of all two-sidedly invertible elements will be the unit of Ill. If III has a unit E, then for any A E III

AE=A, EA=A, i.e., E is both a right and left divisor of A, i.e., a two-sidedly invertible element. 1.6. For any semigroup III we denote by (!5 the set of all of its two-sidedly invertible elements. The set of all elements which are right invertible but not left invertible will be denoted by \R; the set of all the elements which are left invertible but not right invertible will be denoted by .e. The set of all elements which are neither right nor left invertible will be denoted by 5t

220

INVERTIBILITY

[CHAP.

VI

From the definitions it follows that the sets (l), \n, n, .R are disjoint, and that

m=(l)u\nunuSt It is clear that for some semigroups some of the subsets (l), \n, n, .R will be empty. Under the notations thus adopted (l) u m is the set of all right invertible elements of mand (l) u n is the set of all left elements.

1.7. Let us see what can be said about the invertibility of the product of two elements. As E. S. Ljapin [15] proved, the invertibility of a product depends on the invertibility of the factors, although it is not determined by this alone. The results of the investigation may be put in the form of a table, to be interpreted as follows: For two given subsets:r, iB c m, their product is contained in the subset 3 indicated at the intersection of the row corresponding to :r and the column corresponding to iB (however, :riB may fail to coincide with 3). For every semigroup

THEOREM.

mwe have:

(l)

m

n

.R

(l)

(l)

m

n

.R

m

m

m

(l)umunu.R

m u.R

n

n

.R

n

.R

.R

.R

.R

nu.R

.R

PROOF. First of all we observe that if XY E (l), then X YE (l) un. Indeed, by 1.2,

E

(l) u m and

xymxy= ~r.

Hence we obtain

xm:::>

X( ym:xy) =

m:,

which means that X is right invertible and Y is left invertible. Now we shall consider successively all sixteen cells of our table. In what follows, G, R, L, K (also with primes) will mean arbitrary elements, respectively, of (l), m, n, .R. (1) That GG' E (l) follows from 1.4. (2) Since GRm: = Gm: = m:,

Sec. 1]

INVERTIBILITY OF THE PRODUCT OF ELEMENTS

221

then GR E ffi u 91. Here GR E ffi, by what was said at the beginning, is impossible, since R E ffi u 52. (3) Since '11.GL = '11.L = '11., therefore GL E ffi u 52. If we had GL E ffi, then, because G-l E ffi,

'11. = G-l'11. = G-IGL'11. = L'11., which would contradict the fact that L is not right invertible. (4) If we had GK E ffi u 91, then because G-l E ffi we would obtain

'11. = G-l'11. = G-IGK'11. = K'11., which would contradict the fact that K is not right invertible. If we had GKE (l) u 52, then we would obtain

'11.

= QWK= '11.K,

which would contradict the fact that K is not left invertible. Accordingly,

GKEst (5) Since

RG'11. = R'11. = '11., therefore RG E ffi u 91. If we had RG E ffi, then we would obtain

'11. = '11.G-l = '11.RG . G-l = '11.R, which would contradict the fact that R is not left invertible. (6) Since

RR''11.= R'11.= '11., therefore RR' E ffi u 91. But RR' E (l) would contradict the fact, which follows from what was proved at the beginning that R' E ffi u 52. (7) Since ffi U 91 u 52 u ~ = '11., therefore RL obviously belongs to that set. (8) If we had RKE (5 u 52, then we would obtain '11. = '11.RKc: '11.K, which contradicts the fact that K is not left invertible. (9) Since

QrLG

= '11.G = '11.,

222

INVERTIBIUTY

[CHAP.

VI

then LG EmU £. Here LG E m, from what was proved at the beginning, would contradict the fact that

L E (fj (10) If we had

U~.

LRE m u~,

then we would obtain

m: =

LRm: = Lm:,

which would contradict the fact that L is not right invertible. Analogously one proves the impossibility of LR E (fj U £. Accordingly, LR E~. (11) Since m:LL' = m:£' = m:, then LL' EmU £. But LL' E m would contradict the fact stated originally that L EmU ~. (12) If we had

LKEm then we would obtain

u~,

m: = LKm: eLm:.

But ~( = Lm: contradicts the fact that L is not right invertible. If we had then we would obtain

m: = m:LK = m:K, which would contradict the fact that K is not left invertible. Accordingly, LKE~.

(13) If we had

KGE

then we would obtain

mu~,

m: = KG21. = Km:,

which would contradict the fact that K is not right invertible. If we had

KG t:- (jj then, since we have

U

£,

m: = m:G-l, we would obtain m: = m:G-l = m:KGG-l = m:K,

which would contradict the fact that K is not left invertible. (14) If we had

KR E (jj U then we would obtain

~,

m: = KRm: = K21.,

which would contradict the fact that K is not right invertible.

Sec. 1]

INVERTIBILlTY OF THE PRODUCT OF ELEMENTS

If we had

KR

E

223

CfJ U B,

then we would obtain

'll = 'llKR c 'llR, which would contradict the fact that R is not left invertible. (15) If we had KL E CfJ U ~, then we would obtain 'll = KL'll c K'll, which would contradict the fact that K is not right invertible. (16) If we had KK' E CfJ u~, then we would obtain 'll = KK''ll c K'll, which would contradict the fact that K is not right invertible. Analogously, the hypothesis that KK' E CfJ u B would lead us to a contradiction with the fact that K' is not left invertible. 1.8. From 1.7 it is immediately clear which of the unions of the sets CfJ, B, St form subsemigroups. COROLLARY.

~,

The following subsets are subsemigroups of'll:

~ = CfJ u ~ u il u ft; CfJ-the set of two-sided invertible elements; ~-the set of right invertible but not left invertible elements; il-the set of left invertible but not right invertible elements; ft-the set of elements neither right nor left invertible; CfJ u ~-the set of elements which are right invertible; CfJ U il-the set of elements which are left invertible; CfJ U St-the set of two-sided invertible elements and those elements which are neither right nor left invertible; ~ U ft-the set of elements not left invertible; BUSt-the set of elements not right invertible; CfJ U ~ U ft-the set of elements which are right invertible and elements which are neither right nor left invertible; CfJ U BUSt-the set of elements which are left invertible and elements invertible neither to the right flor to the left.

1.9. Several of the cells of Table 1.7 contain not just one of the sets CfJ,

~,

il, ft but the union of several of them. We shall show by an example that it is not possible to decrease the number of sets in those cells, i.e., that there exist semigroups for which the various products of elements of the corresponding sets belong to all the sets indicated in each such cell.

224

[CHAP. VI

INVERTIBILITY

Let n be the set of all natural numbers. We consider the semigroup 60 of all transformations of n. Select the following elements of this semigroup, written in the form of substitutions:

234

n

234

n

2

(

n

1 2 3 4 1 12

=

n

2

(n - 1)

n

3 4

G 345 234 L = G 456 234 K = G 234 Ll

;

(n - 2)

C 223

=

(n

+ 1) n

2

(n

1

n n

1

;

;

;

+ 2)

n

) ... · ..) ... · ..) ... · ..) ... · ..) .

· .. ;

(n - 1)

234

Rg

...

n

3 4

123

R2 =

· ..) ;

"

· ..) ; ... · .. . · ..

)

By II, 3.3, R 1 , R 2 , Rg considered as elements of 60' are left divisors of the permutation E, the identity of 60' and are not right divisors of E. Therefore, by 1.3, R 1, R 2 , Rs E 9\. The elements L 1, L2 are right divisors of E and are not left divisors of E. Therefore, by 1.3 we have L 1 , L2 E.2. The elements Kl and K't, belong to st and E belongs to (5. We consider the seventh cell of Table 1.7. The set corresponding to this cell contains the elements:

n (n

+ 1)

: : :) = Ll E.2;

n

n

: : :) = Kl

E

st.

Sec. 1]

INVERTIBILITY OF THE PRODUCT OF ELEMENTS

225

Thus, for the semigroup 6 0 the product 91£ indeed contains elements from each of the four sets 63, 91, £, ft, whose union is written in the seventh cell of the table. Now we consider the eighth cell. In the set 91ft corresponding to this cell there are contained the elements: 2

RIKI =

G1

=

G1

2

RIK2

3 4 3

2

3 4

n (n - 1)

: : :) = Rl E 91;

n : : :) = K2 Eft.

1

Again we have discovered in 91ft elements of both the sets 91 and ft, whose union is written in the eighth cell. Finally, consider the fifteenth cell. In the set ft£ corresponding to this cell there are contained the elements:

KILl

=

G2

K2Ll

=

G

3 4

3 4

n

5

(n

2 3 4

n

+ 1) :::) = Ll E£; :::)=K2 Eft.

ft£ contains elements both of £ and of ft, whose union is written in the fifteenth cell. Finally, we observe that as the desired example we can of course take not the entire semigroup 6 0 but a countable subsemigroup of it generated by the elements E, R 1, R 2 , R s, L 1 , L 2 , K1 , K 2• It is easy to see that each of these elements will lie in the corresponding set 63, 91, £, ft of this semigroup. 1.10. We observe that the consideration of the properties ofinvertibility of elements is closely connected with the question as to whether elements lie in proper ideals of the semigroup. THEOREM. In order that the element S of the semigroup III should be right invertible it is necessary and sufficient that S should have a right identity and that it should not be contained in any proper right ideal of the semigroup Ill. PROOF.

(1) Suppose that S is right invertible: Sill = Ill.

Then Sill :3 S, i.e., for some Z ideal of Ill, containing S, then

E

III we must have SZ

III = Sill c 911ll c 91, i.e., 91= Ill.

=

S. Here, if 91 is a right

INVERTIBILITY

226

[CHAP.

VI

(2) Suppose that S has both properties indicated in the formulation of the theorem. Since S has a right identity, therefore S E Sm:. Thus the set sm which is a right ideal of mmust obviously coincide with m, which means that S is right invertible. 1.11. Taking 1.5 into account, we immediately obtain from 1.10 the corollary: COROLLARY. In order that the element S of the semigroup mshould be twosided invertible in m: it is necessary and sufficient that should have a unit and that S should not be contained in any proper left or right ideal ofm.

m

2. Invertibility of Magnifying Elements 2.1. As E. S. Ljapin [14] showed, the property of an element of being magnifying, considered in §§ 5 and 6 of the third chapter, is closely connected with the property of invertibility of that element. THEOREM. Every right magnifying element of a semigroup is left invertible but not right invertible. Every left magnifying element is right invertible but not left invertible. PROOF. If U is a right magnifying element of the semigroup some m' c: mwe have

m,

then for

m' U = m, Hence it follows that mu = m, i.e., U is left invertible.

Suppose that U is also right invertible. Then U is a two-sidedly invertible element. By 1.4, from the existence in moftwo-sidedly invertible elements it follows that mhas an identity Em, and the two-sided invertible element U has an inverse U-I. But from this follows m' = m' Em = m' UU-I = mU-I = muu-I = mEm = m, which contradicts the choice of m'. The reasoning for left magnifying elements is analogous. 2.2. The converse is not valid without some additional stipulations. Indeed, in a semigroup containing more than one element, in which for arbitrary elements X and Y we have XY = Y, it is obvious that every element is right invertible: = m.

m

xm

However, no element whatever is left magnifying, for for any X

E

mand m' c: m.

xm' = m'

Sec. 2]

INVERTIBILITY OF MAGNIFYING ELEMENTS

227

2.3. The converse to Theorem 2.1 is nevertheless valid in semigroups with unity. It turns out that (in the notation of § 1) \}\ is the set of all left magnifying elements and .2 is the set of all right magnifying elements. THEOREM. In a semigroup with unity the right magnifying elements, and only those, are left invertible and not right invertible. The left magnifying elements, and only those, are right invertible and not left invertible. PROOF. The fact that a right magnifying element is left invertible and not right invertible was proved in 2.1. Suppose that the element L of the semigroup \)! with unity is left invertible and not right invertible: IJlL = \)!, For some R E \)! we have RL = E~l' From this follows R(L\)!) = \)!, which means that R is a left magnifying element of \)!. Consider the subsemigroup .Q = [L, R]. Each element of this semigroup

if we replace RL by the identity, may evidently be brought into the form

Q = Lrt.RfJ (here and in the sequel several exponents may be equal to zero, which means

LO = RO = Ew). Let us indicate the natural method by which each element of.Q may be brought into this form. Let

If ct:l = ct: 2, then by multiplying the equations on the left by Rrt.l and recalling that RL = E'll we obtain If fJI

> fJ2'

then by multiplyIng this equation on the left by RfJ 2 we obtain

RfJ 1 -fJ 2

\)!

= E'll'

Because of III, 5.4 this is impossible, since R is a left magnifying element of and E~l is not a left magnifying element of \)!.

228

[CHAP.

INVERTIBILITY

If 1X1

VI

> 1X2' then by multiplying the equation LIXIRP1 = L"-2RP I

on the left by R"-a we obtain L IX l-"-aR!l

= RP

2•

But R is a left magnifying element. Therefore R'if! = 'if!, and from the equality so obtained there results L'if! => L"-1-"-aRP1'if! = RPa'if! = 'if!, which contradicts the fact that L'if! ~ 'if!. Thus we have shown that the subsemigroup .0 may be represented as the set of products of the form L"-RP, while two products differing in either one of the exponents are distinct. The multiplication of these products, thanks to the fact that RL = Em, is carried out according to the rule LIXI +IXa-PIRPa LIXIR!l. L"-aRP' = { L~lRPI-IX2+P2

From all ofthis it follows that the semigroup.ois isomorphic to the semigroup \p (III, 6.3). Since under the isomorphism of \p onto .0 the element L E.o corresponds to the element U E \p, it follows from III, 6.8 that L is a right .magnifying element of the semigroup 'if!. The discussion for left magnifying elements is analogous. 2.4. Because of III, 6.8 the following corollary follows from the theorem just proved. COROLLARY. Suppose that the semigroup 'if! has a unity. The element X of the semigroup 'if! is left invertible but not right invertible if and only if there exists an isomorphism q; of the semigroup ~ (III, 6.2, III, 6.3) into 'if! under which q;(E~) = Em and q;(U) = X. The element X is right invertible but not left invertible if and only if there exists an isomorphism q; of the semigroup \p into 2£ under which q;(E~) = Em and tp(.V) = X.

2.5.

COROLLARY.

If, in the semigroup 2£ with unity, for some element X E 'if!

we have X'if! = 2£,

but

n' ~ 'if!

for every 'if!' c 'if! distinct from 2£, then the element X is two-sidedly invertible. (Analogously for right multiplication.)

Indeed, X is right invertible but not left magnifying, and therefore, from 2.3 it must be left invertible.

Sec. 3]

SEMIGROUPS WITH ONE-SIDED INVERTIBILITY

229

We observe that in the formulation of Corollary 2.5 one may not drop the requirement of the existence of an identity. This is seen from the following example. Suppose that \U is a semigroup with a number of elements not less than two and such that XY = Y for every X, Y E \ll. Then in \U, for each X, we have X\U = \U, X\U' = \U'. However, it is clear that no element of the semigroup \ll is two-sidedly invertible. 3. Semigroups with One-sided Invertibility 3.1. In connection with the decomposition in § 1 above of any semigroup 'll into four subsemigroups (Yj, £', 9t, 5\, it is natural to distinguish semigroups coinciding with their subsemigroup (Yj U £, (or analogously (Yj u 9t). Such semigroups were considered in his time by A. K. Suskevic [3; 12], and therefore N. N. Vorob'ev [6] has called them Suskevic systems. They are also called semigroups with left division and left simple semigroups. DEFINITION. If all the elements of a semigroup are left invertible, then it is called a SEMIGROUP WITH LEFT INVERTIBILITY.

Analogously one defines a semigroup with right invertibility. The only semigroups with left and right invertibility at the same time are groups. 3.2. The semigroup \U is a semigroup with left invertibility if and only if it does not have any proper left ideals. Indeed, if \U is a semigroup with left invertibility, then, for any A E \U, 'llA = \U, and therefore A cannot be contained in a proper left ideal of \U. If Qt has no proper left ideals, then for any A E'll we have 'llA = Ill. We observe that, from V, 2.4 and what has just been said, minimal left ideals of any semigroup are semigroups with left invertibility.

3.3. Semigroups with left invertibility essentially divide into two classes depending on whether there are idempotents among their elements or not. THEOREM. If in a semigroup with left invertibility there are idempotents, then each of them is a right identity of the semigroup which is didsible on the right by every element of the semigroup. If in a semigroup with left invertibility there are no idempo tents, then none of its elements has a right identity. PROOF. (1) Let I be any idempotent of a semi group \U with left invertibility. Since III is a semigroup with left invertibility it follows that for any A E III there will exist U and V such that

UA = I,

VI=A.

230

[CHAP.

INVERTIBILITY

VI

Because of the second equation we obtain

AI= VII

=

VI= A.

(2) Suppose that in a semigroup with left invertibility III the element X has a right identity: (X, Y E m:). Xy=x For some Z E m: we must have ZX existence in m: of the idempotent

y2 = ZX· Y

=

= z·

Y. Hence we immediately obtain the XY = ZX = Y.

3.4. As follows from Theorem 3.3, the class of semigroups with left invertibility and having idempotents coincides with the class of semigroups having right identities which are divisible on the right by every element of the semigroup. Indeed, suppose that some semigroup m: has such a right identity I. For any A Em: there is an X E m: such that XA = I. Hence for every B E III we have

(BX)A = BI = B. Here m: :3 I and 12 = I. From what has been said it follows that when we investigate below the structure of semigroups with left invertibility and with idempotents, we at the same time obtain an answer to the question in the second section of the second chapter; namely, which semigroups have right identities which are divisible on the right by all the elements of the semigroup. The investigation of a series of properties of semigroups with left invertibility and with idempotents was carried out by A. K. Suskevic [12]. Later, developing this direction further, Munn [1] gave an exhaustive description of the construction of such semigroups and carried out their complete classification.

3.5. Let t>be a semigroup in which for any U, VEt> we have UV and let (fi be any group. Denote by ::t the set of all possible pairs (U, G)

=

U,

(U E t>, G E (fi).

Define in ::t the multiplication

(U, G) . (U', G') = (UU', GG') (U,

= (U, GG')

u' E t>; G, G' E (fi).

The associativity of the operation is evident. ::t is a semigroup with left invertibility, since for any (U, G), (U', G') E::t we have

(U', G'G-I) . (U, G) = (U', G'). Elements ofthe form (U, E(5)' as is easily seen, are idempotents, while ::t has

no other idempotents.

Sec. 3]

SEMIGROUPS WITH ONE-SIDED INVERTIBILITY

231

3.6. The structure of the semigroup just described is completely determined by the cardinality of the set 4) and the group ill. The semigroup ::r may be considered as a completely simple semigroup without zero of the matrix type 6'(P, ill) (V, 6.3), in which P consists of one column, with all its elements equal to Ery,. 3.7. The significance of semigroups of the indicated type is explained by the fact that, as we shall show below, every semigroup with left invertibility and with idempotents is isomorphic to some semigroup of type 3.5.

3.8. Suppose that m: is any semigroup with left invertibility and with idempotents, whose union we shall denote by 4). We shall deduce a series of properties of the semigroup m:. (IX) If U, V E 4), then UV = U. This follows immediately from 3.3. ((3) If U E 4), X Em: and UX = U, then X E 4). Indeed, because of 3.3, X2= XUX= XU= X. (y) If U E 4), then the set Um: is a group with U as unit.

Indeed, if X, Y E Um:, then evidently XY E Um:. The element U E Um: is evidently a left identity of the subsemigroup Um:. For each X E Um: there must exist in m: an element X' such that U = X' X. Since

u=

UU = (UX')· X,

UX' E Um:,

therefore U is divisible on the right in Um: by any element of um:. Hence, from II, 2.18 it follows that Um: is a group. (tS) For every X E m: in 4) there is a unique element I x which is a left identity for X. For any U E 4) in m: one can find an X' such that U = XX'. For Ix = XX' we obtain, using 3.4, Ix X = XX'X = XU = X, I}

=

XX' XX'

=

XUX'

=

XX'

= Ix.

If V is any left identity of X which is an idempotent, then from (IX) we obtain VXX' = XX' = Ix. The properties of semigroups m: with left invertibility and with idemV= Vlx

=

3.9. potents make it possible to prove the validity of assertion 3.7. Fix one of the idempotents U E 4). Using 3.8, (0) we associate to each element X E m: the pair

x'--' (Ix,

UX).

We shall show that to distinct elements ofthe semigroup m: there correspond distinct pairs. Suppose that, for X, Y E m:, (Ix, UX)

i.e.,

Ix

= Iy ,

= (ly,

UY),

UX= UY.

232

[CHAP.

INVERTIBILITY

VI

Then

X= IxX= IXUX= IyUY= IyY= Y. Now we shall show that for any pair (V, G), where V E.5 and G E Um:, there exists in m: an element to which this pair corresponds. As such an element we choose Z = VG. Since

VZ then I z

=

= VVG = VG = Z,

V. Since G E Um:, then UZ= UVG= UG=G.

Accordingly,

Z

t".I

(lz, UZ) = (V, G).

3.10. We denote by ::r the semigroup of type 3.5 constructed for the semigroup t) (in which, because of 3.8, (ae), the product of two arbitrary elements is equal to the left factor) and the group Um: (3.8, (y». In 3.9 there was established between the elements III and ::r a one-to-one correspondence. We shall show that this correspondence has the isomorphism property, and at the same time we shall prove the validity of assertion 3.7. Let XY = Z and

y,,-, (ly, UY), Since IXly

= Ix E.5 (3.8, (ae» (lxly)' Z

=

Z "-' (Iz, UZ).

and

(lxly)' (XY)

=

then, because of 3.8, (a), lXly = lz. Further, (U X) . (U Y) = U· XU· Y

IxXY

=

XY

= UXY =

= Z,

Uz.

Thus, because of the operation between pairs of elements of::r set up in 3.5, we find that to the element Z = XY there corresponds the following pair:

(lz, UZ) = (Ixly, (UX)· (UY» = (Ix, UX)· (ly, UY).

3.11. The isomorphic representation of the semigroups considered here in the form of semi groups of pairs not only elucidates their structure and makes it possible to deduce various properties ofthem but also amounts to a classification of semigroups with left invertibility and with idempotents. This follows from the fact that two semigroups of pairs, the semigroup ::r1 constructed from .51 and (\)1 and the semigroup ::r2 constructed from .52 and (\)2' are isomorphic if and only if the semigroups .51 and t)2 are isomorphic (for this, evidently, it suffices that .51 and .52 have the same cardinality) and the groups (\)1 and (\)2 are isomorphic. For the proof of this fact it is evidently sufficient to prove that in the semigroup ::r of pairs of type 3.5 the semigroup .5 and the group (\) are defined up to an isomorphism in a unique way. To prove this we note that.5 has the

Sec. 3]

SEMIGROUPS WITH ONE-SIDED INVERTIBILITY

233

same cardinality as the set of all idempotents :r, since only pairs of the form (H, Elf') (H E 5) are idempotents. Accordingly, 5 is defined for:r up to an isomorphism in a unique way. (!j is isomorphic to the group consisting of all the elements of:r having a given (arbitrary) idempotent (H, Err,) as its two-sided identity (such elements are, evidently, pairs of the form (H, G)). Accordingly, also (!j is defined for :r up to an isomorphism in a unique way. 3.12. It is quite evident that the consideration of properties describing the structure and classification of semigroups with right invertibility which have idempotents may be carried out in a way completely analogous to the preceding. 3.13. Now we turn to the consideration of semigroups with left invertibility which do not have idempotents. Let>n be one of these semigroups. It is necessarily infinite since every finite semigroup has idempotents. We recall further that in >n, because of 3.3, no element can have a right identity. 3.14. We consider the following important example of semigroups with the indicated properties. This example was constructed by Teissier [5]. It represents a certain development of the construction of Baer and Levi [1] and plays an important role for the whole class of semigroups under consideration. Suppose that in an infinite set Q there is given arbitrarily an equivalence n such that the number of n-classes is infinite. In the semi group 6 0 consisting of all transformations of Q we distinguish the transformations X having the following three properties; (C<) if C< ~ pen) (C<, p E Q), then XC< = Xp; (P) if C< ~ pen) (C<, p E Q), then XC< rf-; Xp(n); (y) the set of all those n-classes Q' for which XQ n Q' = 0 has the same cardinality as the set of all n-classes. It is easy to see that the collection of transformations X of 6 0 having all three properties is a subsemigroup, which we shall denote by :r~). We shall deduce a series of properties of the semigroups :r~) thus defined. (C<) :r~) has no idempotents. Indeed, suppose that for some X E :r~) we had X2 = X. By 3.14, (y) there is an n-class Q' such that Q' has no elements in common with XQ. Suppose that C< E Q'. Since X(Xex) = X 2 c< = XC< = X(c<), therefore, because of 3.14, (P), XC< ~ c«n), and consequently XC< E Q', which contradicts the fact that XQ n Q' = 0. (P) Every element of ;r~) is a right magnifying element of this semigroup. Suppose that X, Y E ;r~). We write :r~)\X = \p. The set of all n-classes Q' satisfying the condition XQ n Q' ;;f= 0 will be denoted by r I , and those satisfying the condition yQ n Q' = 0 by r 2 • We break r 2 into two disjoint subsets r 2 = r 3 U r 4' each having a cardinality equal to the cardinality of the

234

INVERTIBILITY

[CHAP.

VI

r

set of all n-classes. In each n-class Q(,;) E 1 we fix some element A,; E Qm. We choose some one-to-one mapping cp of the set r 1 onto r 3' and to each A,; E Q(o) we put into correspondence some element A~ E cp(QW) such that at least for one A~ we have A~ ;of XA,;. We introduce a new transformation S E 6 0 , putting: (1) SI:/. = fJ if there exists an element y E Q such that Xy r-..J I:/.(n) and Yy = fJ; (2) SI:/. = A~ if I:/. E Q(,;), where Q(,;) E r l . . It is easy to see that these two conditions determine S E 6 0 uniquely. By the definition of cp we have S E~. It is immediate that S satisfies condition 3.14, (IX), (fJ). Since for each Q(7) E 4 we have SQ n Q(7) = 0, therefore S also has the property 3.l4, (y). From the definition of S, for each v E Q we have SXv = Yv. Thus, SX = Y, and since Y is an arbitrary element of :rfr) and S E ~, therefore ~X = :rfr). (y) :rfr) is a semigroup with left invertibility. This follows immediately from (fJ) and 2.l. (0) In :rfr) there are no left magnifying elements. This immediately follows from (fJ) and III, 5.2. (s) No element iri :rfr) has a right identity in :rfr). This immediately follows from (1:/.), (y) and 3.3. (0 If every n-class consists of one element, then :rfr) is a semigroup with left cancellation. Indeed, suppose X;of Y (X, Y E :r\)t»). For some IX E Q we must have XI:/.;of YIX. But this means that XIXri-' YIX(n) , and therefore, by 3.l4, (fJ), SXIX ;of SYIX for any S E :r\)t). Cr;) If even one n-class contains more than one element, then :r\)t) is not a semigroup with left cancellation. Indeed, if oc r-..J fJ(n) and IX;of fJ, then we choose some transformation X E::rW such that XQ'3 IX. We construct a transformation X' E6 0 such that X'v = Xv if Xv ;of IX, and X'v = fJ if Xv = IX. It is easy to see that X' E :r\)t). Since SIX = SfJ for any S E :r\)t) (since IX r-..J fJ(n)), then SX = SX' , although X;of X'. (8) :rW is not a semigroup with right cancellation. Indeed, choose in :r\)t) two elements X;of X' for which X IX = X'IX for all (J. E Q, with the exclusion of IX, belonging to one fixed n-class Qm . We choose Z in ::r:~) such that ZQ n QW = 0. For such elements, evidently, we obtain XZ=X'Z.

r

3.16. The properties derived for :rfr) characterize it as a semigroup which is very interesting from several points of view. In particular, we have for the first time obtained an example of a semigroup, all the elements of which are right magnifying elements for it. Moreover, as Teissier proved [5], the significance of the semigroup ::r:\)t) goes far beyond the bounds of an interesting example. This follows from the considerations presented below.

Sec. 3]

SEMI GROUPS WITH ONE-SIDED INVERTIBILlTY

235

Suppose that III is a semigroup with left invertibility which does not have idempotents. Denote by 0 the set consisting of all elements of III and one further "separating" element I (I, 3.9). In the set Q we define an equivalence relation n, putting X r-J Yen) jf X = Y = lor if X, Y E III and AX = AYfor every A E Ill. Symmetry, reflexivity and transitivity ofn are evident. We note further that if X and Y of III are such that for some Z E III we have ZX = ZY, then X"'-' Yen). Indeed, let A be an arbitrary element of Ill. Since ~r is a semi group with left invertibility, then for some S E Ill, SZ = A. Therefore AX= SZX= SZY= AY.

For the equivalence n in 0 we construct the semigroup ::r~). It is a subsemigroup of the semigroup 6 0 , We consider the isomorphism rp (I, 3.9) of the semigroup III into 6 0 which is a representation of it by left translations (I, 3.10). We shall prove that
= AX= AY= A(Y).

Thus Ahas the property 3.14, (0:). If X and Y belong to different n-classes while both are elements of Ill, then, according to the remarks above, the element A(X) = AX and the element A( Y) = A Y cannot stand in the relation n to one another. If X E III and Y = I, then A(I)

A(X) = AX,

=

A

and AX A(n) is impossible, since otherwise, by the definition ofn, we would obtain AAX = AA, which contradicts the lack of a right identity for A2 (3.3). Thus A has property 3.14, «(3). . For each S E III there is in III an element R(8) such that A = R(8)S. We denote by 0(8) the set of all elements of Ill, equivalent mod n to the element R(S). The sets AO and O(S) have no common elements. Indeed, in the contrary case, for some X E Q we would have AX R(8)(n). Hence it would follow that (SR(S»SX = SR(s>, which would contradict the lack of a right identity for the element SR(S) (3.3). We have associated to each S E III an n-class 0(8) satisfying the condition AQ(S) n O(S) = 0. Now we shall show that if S1 l' S2(n), i.e., if Sl and S2 are chosen from different n-classes, then 0(8, ) r' Q(8 2). Hence it will result immediately that property 3.14, (y) is satisfied for the transformation I"-.J

I"-.J

A. Since

236

INVERTIBILITY

then for each B

E

21 we have BR(S')Sl

[CHAP.

VI

= BR(S2)S2'

+

But S1 S2(n); therefore BR(S,) ~ BR(S2). This means that R(S,) rf- R(S2)(n) and therefore the n-classes n(S,) and n(s.) are distinct. 3.17. The considerations presented show that the subgroups of type ~~) form a universal class (III, 1.9) for the class of all subgroups with left invertibility and without idempotents. Here they themselves belong to that class (3.15, (IX), (y)). If one restricts oneself to subgroups, the cardinalities of the sets of whose elements do not exceed some cardinality m, then it is not difficult to construct a semigroup ~~) which is a universal semigroup for all of that class (true, it will not belong itself to that class, as it has a higher cardinality). The obtaining of the indicated universal class for the class of semigroups with left invertibility without idempotents in a certain sense clears up the character of semigroups of that class. In studying one of the semigroups of that class we may, without leaving the limits of the class, imbed in the corresponding semigroup ~:w, the study now being contin~ed for the latter. On the other hand, of course, our present result is far from being a complete description of the construction of semigroups with left invertibility and with idempotents, a circumstance which is due to the greater complexity, in principle, of the structure of our semigroups in the case of absence of idempotents.

3.18. In conclusion we shall consider the semigroups :r~) in which the equivalence n is that of identity, i.e., each n-class consists of one element only. We obtain in this case the construction of Baer and Levi [1], which served as the initial point of the considerations presented above. In this case the semigroup ~~) will be a semigroup with left invertibility and left cancellation and without idempotents (3.15, (m. As is immediately evident from the considerations of 3.16, any semigroup having the indicated three properties maps isomorphically into some such semigroup :rW. Thus, the semigroups ~~) with the identity equivalence n not only belong to the class of semigroups with the indicated three properties but also form the universal class of that class. 3.19. In the general case the properties of cancellation and invertibility are independent. An infinite monogenic semigroup has the property of two-sided cancellation. However, it is evidently neither a semigroup with left invertibility nor a semigroup with right invertibility. On the other hand, the semigroup ~W is a semigroup with left invertibility (3.15, (y)) but is never a semigroup with right cancellation (3.15, (0)), and, with the exclusion of the case mentioned under 3.15, ('If), is never a semigroup with left cancellation. The semigroup antiisomorphic to the semigroup :r~) is a semigroup with right invertibility but is neither a semigroup with left cancellation nor a semigroup with right cancellation (with the exclusion of the case corresponding to 3.15, ('If)). There is an interesting connection, indicated in the work of Hewitt and

Sec. 4]

SUBSEMIGROUPS REGULAR WITH RESPECT TO INVERTIBILITY

237

Zuckerman [1], between these important properties in the class of periodic semigroups (III, 4.1). THEOREM.

and only

A periodic semigroup ~ has the property of right cancellation with left invertibility.

if it is a semigroup

if

PROOF. (1) Let mbe a semigroup with right cancellation. For any idempotents U and V of the semigroup ~ we have

(UV) V = UV2 = UV,

from which it follows that UV = U. Any element A of the semigroup ~ generates a finite monogenic semigroup [A], whose type (III, 3.7) will be denoted by (h, d). If now h > 1, then we would obtain Ah+d = A", Ah+d-l . A = Ah-l . A, AHd-l # An-I, which contradicts the condition of right cancellation. From this it necessarily follows that h = 1, which means that [A] is a group (III, 3.11). From what has been proved it follows that for any two elements A and B of the semigroup mthere exist A', B', U', V' such that AU' = A,

A'A

=

U',

B'B = V',

where U' and V' are idempotents. Because of this we obtain A = AU'

= AU'V~= AU'B'B = (AU'B')' B.

Hence it follows that m: is a semigroup with left invertibility. (2) Suppose that ~ is a semigroup with left invertibility. Since m: is a periodic semigroup it has idempotents. Hence, from 3.7 and 3.9, it follows that ~ is isomorphic to the semigroup described in 3.5. But the semigroup X is a semigroup with right cancellation since, for any Ti = (Ui , Gi) EX (i = 1,2,_ 3), from TlT3 = T2 T a

it follows that Tl = T 2. Indeed,

= (U1, G1) • (U3 , G3) = (Ul , GIGS), T2 T 3 = (U2, G2) . (U3 , G3) = (U2, G2GJ and TlT3 = T2Ta implies 0 1 = U2 and G1 GS = G2G3, i.e., Gl TITs

= Ga (since Gl , G2 ,

. Gs are elements of a group).

4. Subsemigroups Regular with Respect to Invertibility 4.1. Suppose that ~ is a subsemigroup of the semigroup m: and B is some element of~. The invertibility of B in ~ may differ from the invertibility of B in m:. The case is possible that B is a right invertible element of ~ but is not right invertible in m. On the other hand, the case is possible that B is not a right invertible element of 58 but is right invertible in m. Analogous cases may arise in the relations of left and two-sided invertibility.

238

INVERTIBILITY

(CHAP.

VI

4.2. We present an example of the possibilities mentioned in 4.l. We choose a commutative semigroup 'll

= {... , A-I, AO, A, A2, ... , I},

in which An and Am multiply by the rule of adding exponents, IAn = AnI = An (n = ... , -1,0, 1,2, ... ), and 12 = I. One may say that'll is obtained from an infinite cyclic group by external adjunction of the unit I (II, 2.12). We consider the semigroups Since A'lll3 A, then A is not an invertible element of'll!. At the same time A is evidently an invertible element for 'll2' which is a supersemigroup of the semigroup'lll· At the same time we observe that A is not an invertible element of'll, since evidently A'll§ I. Thus A is a supersemigroup of the semigroup 'll2 in which A is not invertible though it is an invertible element of the semigroup 'll2. 4.3. In connection with the phenomenon mentioned above there naturally arises the concept following below, introduced and discussed by E. S. Ljapin [14]. The subsemigroup Q3 of the semigroup 'll is said to be A REGULAR if every element B of Q3 is right invertible in 'll if and only if it is right invertible in Q3. DEFINITION.

SUBSEMIGROUP WITH RESPECT TO RIGHT INVERTIBILITY

Analogously one defines subsemigroups regular with respect to left invertibility and regular with respect to two-sided invertibility. 4.4. It follows from the definition that the semi group Q3 will be regular with respect to right invertibility if for every B E Q3 it follows from BQ3 = Q3 that B'll = 'll and, conversely, from B'll = 'll that BQ3 = Q3. Analogously, for left invertibility, one of the equalities Q3B

= Q3,

'llB

= 'll

must follow ftom the other. 4.5. We note several of the simplest properties of subsemigroups regular with respect to right invertibility (in an obvious way the corresponding properties hold for semigroups regular with respect to left invertibility and regular with respect to two-sided invertibility). (~) The semigroup itself, regarded as a subsemigroup of itself, is regular with respect to right invertibility. (fJ) If \Ul is a subsemigroup regular with respect to right invertibility of \U2' and \U2 is a subsemigroup regular with respect to right invertibility of \U3' then \Ul is a regular, with respect to right invertibility, subsemigroup of \U3.

Sec. 4]

239

SUBSEMIGROUPS REGULAR WITH RESPECT TO INVERTIBILITY

(y) Suppose that the semigroup III has a unit E and that the subsemigroup

5.8 of the semigroup 21 contains E. In order that lB be a regular, with respect to right invertibility, subsemigroup of Ill, it is necessary and sufficient that, for every B E lB for which the equation BY = E is solvable in I}.{, that equation should be solvable in lB. In order that lB should be a regular, with respect to left invertibility, subsemigroup of I}.{, it is necessary and sufficient that, for every B E lB for which the equation XB = E is solvable in I}.{, that equation be solvable in 5.8. (b) Suppose that the semigroup III has an identity E and that lBl and lB2 are subsemigroups of Ill, while E E lB1 C 5.8 2 C I}.{. If 5.8 1 is a subsemigroup of I}.{, regular with respect to right invertibility, then 5.8 1 is a regular subsemigroup of 5.8 2 with respect to right invertibility. (8) Suppose that the semi group I}.{ has an identity E and that the subsemigroup 5.8 of the semigroup I}.{ contains E. In order that lB should be a subsemigroup ofl}.{, regular with respect to right invertibility, it is necessary and sufficient that every element B E 5.8 which is right invertible in I}.{ should be right invertible also in 5.8.

4.6. If the subsemigroup 5.8 of interest to us in the semigroup I}.{ is not regular with respect to right invertibility, and the presence of this property for the semigroup in question is essential to us (soon we shall encounter a question in which that property becomes essential), then one may imbed lB in a supersemigroup 5.8', which as a subsemigroup of the semigroup III is regular with respect to right invertibility. That this is always possible follows already from the fact that one may take I}.{ itself to be 5.8'. But usually it turns out to be convenient to pass from 5.8 to lB' by the addition of fewer new elements not belonging to 5.8. In certain cases one succeeds in constructing a minimal supersemigroup for 5.8 which is a subsemigroup of I}.{ that is regular with respect to right invertibility; however this is not always possible. 4.7. We shall present an example for the case indicated above. Consider a commutative semigroup m: consisting ofthe elements II' 12, with the operation between them defined by the rule

Ilk = Ikli

= Ii

••• ,

In>

(i, k = 1,2, ... ; i <; k).

We shall show which of the subsemigroups of I}.{ are regular with respect to invertibility. If the subsemigroup lB = {I"" I"" ... , I"J (oc1 < OC2 < ... < ocm) is finite, then from 4.4 it is not regular, since

5.8I"m = lB,

I}.{I"m :;C

I}.{.

The last inequality follows from the fact that for no I{3 is it possible that the product I{3I"m equal J"mH'

240

[CHAP.

INVERTIBILITY

VI

If iB = {I", , 1", ' ... } (al < a2 < ...) is infinite, then it is regular with respect to invertibilitY. This follows from the fact that for no l"'k does one have either iBI(J.le = IB or 'H.ICl.k = Ill. For any finite subsemigroup IB = {I", 1 , I", , ... ,Irt.m } 2 (al < 1X2 < ... < am) of our semigroup 'H., regular with respect to invertibility, its supersemigroup can be any infinite subsemigroup of'H. containing it: IB'

= {iB, 11'1' I fJ2 , •• • , I fJ;, • •• }

(pj ¥' ai ; i

= 1,2, ... , m;j =

1,2, ... ).

Among such subsemigroups IB' there is none which is minimal. Indeed, for any IB' one may always choose a subsemigroup )E,t = {IB,If3 ., I fJ ., ••• } which is regular with respect to invertibility, contains iB and at the same time is a subsemigroup of IB' distinct from iB'. 5. Seroigroups of Transformations Regular with Respect to Invertibility 5.1. As we have already mentioned, the necessity for considering the various properties of this or that transformation arises in quite diverse branches of mathematics. Depending on the subject matter and on the purpose of the investigation, interest may lie in different properties of the transformations. Very frequently it is necessary to investigate the existence and uniqueness of a solution of the following equation. Suppose that S is some transformation of the set n, a some given element of n and ~ an unknown desired element of n. The equation in question has the form S~

=

IX.

In view of the arbitrariness of the set n and the fact that S is any of its transformations (operators), many forms of concrete equations are special cases of this very general equation. Therefore E. S. Ljapin [14], in studying certain properties of this equation, discussed below, called it an equation of general form. In any discussion of this equation there usually arises at the outset the necessity of considering the two following principal questions. First, is this equation solvable with respect to the unknown $ (problem of existence)? And secondly, if it is solvable, then is its solution unique or not (problem of uniqueness)? E. S. Ljapin [14] considered a series of properties of semigroups which turned out to be connected with the properties of the indicated equation. The following section of the present chapter will be devoted to the presentation of the corresponding results. 5.2. Most frequently in practice one needs to consider not just one or another separate equation of the type indicated in 5.1, but rather a collection of equations with the same transformation, i.e., the same operator S, but with distinct right sides IX. In this connection the problem may be formulated as follows. For a given transformation-operator S we need to clear up, on the one hand, whether for all

Sec. 5]

TRANSFORMATIONS REGULAR TO INVERTIBILITY

241

IX of Q the equation S~ = IX has a solution in Q and, on the other hand, whether for all those IX for which this equation is solvable it has a unique solution. The questions thus formulated may be put differently. We need to clear up whether the equation SQ = Q holds (which is equivalent to S~ = IX being solvable for all IX E Q) and whether SA and Sp are different for different but arbitrary A and p (which is equivalent to the uniqueness of the solution of the equation S~ = IX for all those IX for which the equation is solvable).

5.3. An affirmative or negative answer to the questions put in 5.2 is an important property of the operator-transformation S. If one considers S as an element of the semigroup 6 0 of all mappings of the set Q, then the answers to the indicated questions are obtained immediately. From II, 3.2 it follows that in order to have SQ = Q it is necessary and sufficient that S should be a left divisor of all elements of 6 0 , i.e., from 1.1, that it should be right invertible in 6 0 , . Inasmuch as 6 0 possesses a unit E, it follows from 1.3 that it suffices for this that S should have a right inverse with respect to E in 6 0 , It follows from II, 3.1 that in order that SA ".,. Sp for any A ".,. p it is necessary and sufficient that S should be a right divisor of all elements of 6 0 , i.e., by 1.1, that it should be left invertible in 6 0 , It follows from 1.3 that it is sufficient for this that S should have a left inverse in 6 0 relative to E. 5.4. The approach indicated in 5.3 to the solution of the two fundamental questions for the equations 5.1, as originally put, is too superficial and is therefore insufficiently fruitful. In the investigation of a given collection of operatortransformations it is usually natural and fundamentally important to consider them as elements of a semigroup of transformations with certain properties which the given operator-transformation also possesses. However, if one passes from the semigroup 6 0 to one of its subsemigroups III c 6 0 , the equivalence of the properties of the transformation S which are of interest to us with properties of invertibility of S may be disturbed. This follows from the fact that S might be right invertible in 6 0 but not right invertible in Ill, and conversely. We have spoken in the preceding section of the possibility of such phenomena. The considerations of this section will suggest a way of escape from the indicated difficulty. 5.5. DEFINITION. A semigroup consisting of some of the transformations of the set Q is said to be REGULAR WITH RESPECT TO RIGHT INVERTIBILITY if it is regular with respect to right invertibility as a subsemigroup of the semigroup of all transformations 6 0 (4.3).

Analogously, one may define regularity with respect to left invertibility and regularity with respect to two-sided invertibility. 5.6. From this definition and from what was said in 5.3 we have the following conclusion.

242

[CHAP.

INVERTIBILlTY

VI

Suppose that m: is a semigroup of transformations n, regular with respect to right invertibility, and that the transformation S belongs to m:. In order that the equation S~= (:/. in the unknown ~ E n should be solvable for all values (:/. E n it is necessary and sufficient that S should be a right invertible element of the semigroup m:. Suppose that m: is a semigroup of transformations of the set n, regular with respect to left invertibility, and that the transformation S belongs to m:. In order that the equation S~

n

=

IX

n

with respect to the unknown ~ E should have for each (:/. E no more than one solution it is necessary and sufficient that S should be a left invertible element of the semigroup m:.

5.7. It follows from 5.6 that the study of these important properties of operator-transformations belonging to some semigroup of transformations m:, comes down to the study of a purely algebraic question on the divisibility of elements in the semigroup m:. It is important to observe that such an approach reduces the investigation of a question related to a concrete semigroup to the study of an abstract property (in the sense of I, 1.8 and I, 1.14). Indeed, the properties of right and left invertibility are evidently preserved for isomorphic semigroups. The indicated reduction may be of fundamental interest. However, it holds only in the case when m: is a regular semi group of transformations with respect to invertibility. The question as to which semigroups of transformations are regular with respect to the invertibility properties may be resolved, of course, only on the basis of a study of the concrete properties of the given set n and of the properties of those transformations which make up the given semigroup of transformations. The nature of such a study will depend on the specific branch of mathematics which is under investigation. By way of illustration we present such investigations for several important cases. 5.8. Let n be a linear space over the field r.l In what follows, in order to avoid obvious stipulations, we shall suppose that n consists of more than just the null element e. The transformation S of the set (2 is said to be linear if for any (:/.1' (:/.2' ••• , IXn E (2 and AI, .:1.2, ..• , An E r we have

1 Definitions of basis, linear space and linear dependence, and their elementary properties are taken in the same sense as in the book by A. I. Mal'cev, Foundations of linear algebra, OGIZ, Moscow, 1948 (see Chapter II, §§ 1,2). (Russian)

Sec. 5]

TRANSFORMATIONS REGULAR TO INVERTIBILITY

243

We shall denote the set of all linear transformations by we. It is not hard to see that the product of two linear transformations is again a linear transformation, so that we is a semigroup. 5.9. LEMMA. Suppose that 0' is a linearly independent subset of the linear space 0 and that cp is any mapping of 0' into O. Then there exists a linear transformation S such that for all ~ on 0' we have S~

=

cp(~).

PROOF. The usual proof of the existence of a basis in a linear space 2 may without difficulty be sharpened in the sense that any linearly independent subset may be included in some basis. Suppose that 0" is a basis of 0, containing 0'. The mapping cp of 0' into 0" may be extended to a mapping 1p of the basis 0" by putting 1p(~) = cp(~) for ~ EO' and 1fJ(;) = for ~ E 0"\0'. For every ('J. E 0 there exists a unique linear expression in terms of the basis elements

e

('J.

=

Al~1

+ A2~2 + ... + An~n

(~1' ~2' ... , ~n EO"; ,11' ,12' ••• , An E r).

We define the following transformation S of the space 0:

It is easy to see that S

E

m.

Also, for any

Sm =

1fJ(~)

~

of 0',

= cp(~).

5.10. THEOREM. The semigroup we of all linear transformations of the linear space 0 is regular with respect to both right invertibility and left invertibility. PROOF. Since the identity transformation E, which is the unit of the semigroup of all transformations 6 0 , is linear, the proof of our theorem may be carried out with the use of property 4.5, (y). Suppose that A Y = E, where A E we and Y E 6 0 , We choose in 0 some basis 0'. Using Lemma 5.9, we construct a linear transformation BE we such that for every ~ EO' we have

B~

For any element ('J. the basis 0':

E

=

Y~.

0 one finds its linear expansion in terms ofthe elements of ('J.

=

Al~l

+ A2~2 + ... + An~n

(~1' ~2' ... , ~n EO'; ,11' ,12"'" An E r). 2 See, for example, A. I. Mal'cev, Foundations of linear algebra, OGIZ, Moscow, 1948 (see Chapter II, § 2, Theorem 2). (Russian)

244

[CHAP.

INVERTIBILITY

Since the transformations A and B are linear and A Y AB~

VI

= E, then

+ A2(B~2) + ... + An
= A[J.l(B~J

i.e., AB = E. Thus, we have proved that if the equation A Y = E is solvable in 6 0 it is solvable in 9R. From 4.5, (y) it therefore follows that 9R is a subsemigroup of 6 0 that is regular with respect to right invertibility, i.e., is a regular, with respect to right invertibility, semigroup of transformations. Now suppose that XA = E, where A E 9R and X E 6 0 , Choose some basis 0.' of the linear space Q. We shall prove the linear independence of the set AQ'. Let where Al~l

~l> ~2' ••• , ~n

are mutually distinct elements of 0.'. Then we have

+ A2~2 + ... + An~n = E(Ar~l + A2~2 + ... + An~n) = XA(Ar~1 + ~~2 + ... + An~n) = X[Al(A~I) + ~(A~J + ... + AnCA~n)] = xe = e.

For the elements of the basis ~1' ~2'

Al = 0,

~

••• ,

~n this is possible only if

= 0, ... , An = O.

For elements", of AQ' we define a mapping of q; into 0.:

C!" E AQ'). Since the set An' is linearly independent, it follows from Lemma 5.9 that there exists a transformation C E !In such that for all", E A,Q'. Suppose that IX is any element of Q. It is linearly expressible in terms of the elements of the basis ,Q' : IX

= Al~1 + A2~2 + ... + An~n

(~1' ~2' .•. , ~n EQ'; AI, ,12' ••. , An E

Since CA

CAlX

=

E

9R and XA

n.

= E, therefore

+ A2~2 + ... + An~n] = AIC(A~J + A2C(A~2) + ... + AnC(A~n) = AIX(A~I) + ~X(A~2) + ... + AnX(A~J = Ar~1 + ~~2 + ... + An~n = IX,

CA[Al~l

i.e., CA = E.

Sec. 5]

TRANSFORMATIONS REGULAR TO INVERTIBILITY

245

By 4.5, (y) it follows from what has been proved that 9J1 is a regular, with respect to left invertibility, semigroup of transformations. 5.11.

From 5.6 and Theorem 5.10 we have the following corollary.

Suppose that S is a linear transformation of the linear space the equation S~

=

n.

In order that

rJ.

in the unknown ~ E n should have a solution for all rJ. E n it is necessary and sufficient that S should be a right invertible element of the semigroup 9JC of all linear transformations n. In order that the equation S~ = rJ.

in the unknown ~ E n should have no more than one solution for every rJ. E n it is necessary and suffiCient that S should be a left invertible element of the semigroup 9J1 of all linear transformations n.

5.12. Now suppose that £ is any partially ordered set (1, 5.9). Denote by set of all those transformations B of that set which do not violate the partial order. Moreover, B belongs to mif and only if from ex: .,;;; {3 (ex:, fJ E B) and Bex: ~ BfJ it always follows that Bex: = B{3. Thus, in mthere fail to enter only those transformations S of the set £ for which in B there are ex: < fJ such that Sex: > SfJ.

m the

5.13. THEOREM. The semigroup m of all transformations of the partially ordered set £ which do not violate the partial order (5.12) is regular with respect to right invertibility. PROOF. Since the identity transformation E, which is the unit of the semigroup 6£ of all transformations of £, is evidently in m, we may make use of property 4.5, (y). Let BY = E, where B E m, Y E 6£. If we had Y E m, then for some IX, fJ E £ we would have Yex: > YfJ. ex: < fJ, However the relations Yex: > YfJ,

B(Yex:)

contradict the fact that B

E

= ex: < fJ = B(YfJ)

m.

5.14. As to regularity with respect to left invertibility, in the general case the semigroup mof all transformations not violating the partial order does not possess that property. We cite the following example. Let £ be the set of all positive rational numbers less than unity. We shall say that the number rJ. precedes fJ if the value

246

[CHAP.

INVERTIBILITY

VI

of (J.. does not exceed that of {3. Consider a transformation B such that B(J.. = (1/2)(J..((J.. E B). In 6£ the equation

XB=E is solvable (for X one may take any transformation of il such that X(J.. = 2(J.. for all (J.. with 0 < (J.. < 1/2). We choose any solution X of that equation. We write X(l/2) = p. We choose a rational number pi such that p < pi < l. We have

t > i/,

Xm = p < pi = Ep' =

XBp'

=

X(tP'),

from which it follows that X does not belong to m. From this, according to 4.5, (y), it follows that Q3 is not a regular, with respect to left invertibility, subsemigroup of 6£. 5.15. Now we shall show that under some very wide restrictions one may nevertheless assert the regularity with respect to left invertibility of the semigroup of all transformations of a partially ordered set not violating the partial order. We shall say that the partially ordered set il has separating elements if B has the following property. Let 91' and 91" be any two arbitrary (in particular, possibly empty) subsets of n such that the relation (J..' ~ (J..", where (J..' E 91', (J.." Em", is possible only if (J..' = (J..". Then there exists a separating element y such that (1) (J..' ~ y for (J..' E 91' is possible only if (J..' = y and (2) (J.." :;;;; y for (J.." E 91" is possible only if rt." = y. 5.16. THEOREM. If the partially ordered set il has separating elements (5.15), then the semigroup Q3 of all of its transformations which do not violate the partial order (5.12) is regular with respect to left invertibility. PROOF.

(1) We suppose that for some B

E Q3

and C E 6£ we have

CB=E. By 4.5, (y) it suffices for us to prove that there then follows the existence in mof an element B' such that B'B = E. (2) We shall consider partial transformations of the set il (I, 4.1). In particular, we denote by 9\ the set of all those partial transformations X for which the relations (J.. < {3 and Xrt. > X{3 cannot hold simultaneously for any (J.., (3 E TIlX. We observe immediately that m = 6£ n 9\. (3) In the set of all partial transformations of the set B we define a partial ordering relation by setting X:;;;; Y if TIlX c III Y and X~ = n for every ~ E IlIX. (4) We shall denote by Po a partial transformation of B for which IlIP 0 = BB and P o~ = C ~ for every ~ EBB. We shall denote by 9J1 the set of all partial transformations of 9\ which follow Po.

Sec. 5]

TRANSFORMATIONS REGULAR TO INVERTIBILITY

We shall prove that Po Suppose that IX < (3 and POIX (3 = BfJ'· We have rt,'

247

we, for which it suffices to prove that Po E~. > P 0(3 for some IX, (3 E IIIP 0 = B52. Then IX = BIX',

E

> PofJ = PoB(3' = CBfJ' = BIX' = < fJ = B(3'.

= Ert,' = CBrJ.' = PoBrJ.' = Port,

EfJ'

= (3',

rt,

But this contradicts the fact that B E ~. (5) Let mbe a subset of we such that for any X and Y of mone always precedes the other. We denote by No a partial transformation for which IIINo is the union of all the IIIX for X E 91, and NolX = fJ if for some X E 91 we have XIX = fJ. Evidently fJ does not depend on the choice of X in 91 since the hypothesis stated relative to mis satisfied. It is immediately evident that No is an upper bound for 91. Of course, No Po. Suppose that rt, < (3 and NolX > NofJ for some rJ., (3 E S2. Then for some X, Y Em we have XIX = NolX and Y(3 = NofJ. Suppose Z is that one of X and Y which follows both X and Y. Then IX, (3 E IIIZ and ZIX = NorJ., Z(3 = N o(3. Since the relations ()( < (3 and ZIX > ZfJ contradict the fact that Z E ~ the stated hypothesis is invalid. Accordingly, No E~, and therefore No Ewe. (6) From the fact that the properties of we just derived are satisfied, one may apply Theorem II, 4.17 to we. By that theorem there is in we an element PI which is followed in we by no element distinct from Pl' Suppose that in 52 there is an element A not contained in IIIP I, We denote by 52' the set of all elements of 52 preceding A, and by 52" the set of all elements following A. Then A' < A" for every A' E 52' and A" E 52". Since PI E ffi it follows that for no fh' E PI52' and fh" E P I52" can we have fh' > fh". Because of the condition on the existence of separating elements (5.15), for PI 52' and PI 52" there exists a separating element y. We construct a new partial transformation P 2 such that P2 PI and IIIPZ = IIIP I U A, while P2A = y. We shall showthatP2 E we, for which it suffices to show thatP2 E ffi. Suppose that rt, < (3 and P 2 IX > P 2 fJ for some rt" (3 E II IP2• If rJ., fJ E IIIPI , we would have a contradiction with the fact that P2 PI and PI E ffi. If rJ. = A, we would have y > P 2 fJ, where (3 E 52" (since (3 > IX = A), which is impossible for y. If we had fJ = A, then P2 IX > y, where ()( E 52' (IX < fJ = A), which is impossible for y. Accordingly, P z E ffi. Thus the assumption A E IIIPI has led us to the existence in we of an element Pz distinct from PI and following it. The impossibility of this means that in fact IIIPI = 52, i.e., PI E 6£. Since we also have PI E ffi, therefore PI E~. (7) For any $ E 52 we have B$ E IIIP O' Since also PI Po, therefo.re

>-

>-

>-

>-

PIB$ Hence it follows that PIB

= PoB$ =

CB$

= E, while PI E~.

= E$.

INVERTIBILITY

248

[CHAP.

VI

5.17. From Theorems 5.13 and 5.16 we have the following consequence. Suppose that 52 is a partially ordered set having separating elements (5.l5), lB is the semigroup of all of the transformations of B not violating the partial ordering (5.12), and BE lB. In order that the equation

B; =

IX

in the unknown ; E B should have a solution for all IX E B it is necessary and sufficient that B should be a right invertible element of the semigroup lB. In order that the equation

B; =

IX

in the unknown; E 52 should have not more than one solution for every IX E B it is necessary and sufficient that B should be a left invertible element of the semigroup lB. 6. Groups with Separating Group Part 6.1. We have already turned our attention to the important role of twosidedly invertible elements of a semigroup, i.e., of elements which are both right and left divisors of every element of the semigroup. In the rest of the present section the set of two-sidedly invertible elements of the semigroup III will be denoted by G5(Ill). In 1.4 we have proved, and now we must always keep this in view, that G5(Ill), if not empty, is a group, while the unit of the group (£i(Ill) is a unit for the whole semigroup Ill. We shall also introduce a notation for the set of all elements of the semigroup III not lying in G5(Ill): ~(Ill) = III \ G5(Ill). Thus, in the notation used in § 1, we have

G5 (Ill) = G5, 6.2. While the set G5(Ill) is always closed relative to multiplication, the set does not have the analogous property in every semigroup. For example, for the semigroup of all transformations of a countable set, as we have seen in 1.9, the collection of elements which are not two-sidedly invertible is not a subsemigroup. At the same time, for a number of other important semigroups ~(Ill) is a subsemigroup. As an example of this consider the semigroup 6 0 of all transformations of a finite set 0.. As follows immediately from II, 3.1 and II, 3.2, (£i(6 0 ) consists of all transformations which effect a one-to-one mapping of 0. onto itself (sometimes they are called proper permutations). ~(60), consisting of all the remaining transformations (improper permutations), is evidently a subsemigroup of 6 0 , In the semigroup 9J1 n of all complex square matrices of order n, (£i(9J1 n ), as follows from II, 3.9, is the set of all nonsingular matrices, and ~(9J1n) is the set of all singular matrices, which is also a subsemigroup. ~(Ill)

Sec. 6]

GROUPS WITH SEPARATING GROUP PART

249

Aside from the fact that many important semi groups have the indicated property, the class of semigroups for which NI[) is a subsemigroup is characterized by certain other important properties. Among these properties are certain important ones relating to the abstract theory of transformations. 6.3. That the collection ~(I[) is closed under multiplication means, in other words, that no two-sidedly invertible element can be represented in the form of the product of elements of ~(I[). Moreover, as we shall soon show, in this case the product of elements of III belongs to m(I[) only if all the multipliers belong to m(~l). Thus the case in question is characterized by the setting apart, the isolation, of the group of two-sidedly invertible elements m(I[). DEFINITION. The semigroup III is said to be A SEMIGROUP WITH SEPARATING GROUP PART if it has an identity (i.e., from 1.5, if m(I[) is nonempty) and if the product of any two of its elements not two-sidedly invertible is itself not a twosidedly invertible element (i.e., if.5(I[) is a subsemigroup of III or an empty set). We observe that the concept of semigroup with separating group part is closely connected with the concept of hypergroups introduced by Rauter [1] (see also § 51 of the book of A. K. Suskevic [12]). 6.4. THEOREM. A semigroup III with unit which is not a group is a semigroup with separating group part if and only if it can be represented in the form of two nonintersecting subsemigroups

such that IllI is a subgroup and 1112 is an ideal. In this case

while the ideal1ll2 is two-sided. PROOF. (1) Suppose that III is a semigroup with separating group part. ~{ is the union of m(~l) and .f>(I[), while m(I[), from 1.4, is a group. Suppose that HE .5(1[), A E Ill. If A E ~(I[), then by the definition of a semigroup with separating group part, HA E .5(1[) and AH E .5(1[). If A E m(I[), then neither AH nor HA can belong to m(I[). Indeed, in the case HA = G E m(I[) we would obtain H = GA-I E m(I[). We reason analogously for AH. Thus for any A E III we have HA E ~(I[) and AH E ~(I[), i.e., ~(I[) is a two-sided ideal. (2) Suppose that III decomposes in the way indicated in the theorem: III = IllI ,u 1ll2'

IllI n 1112 = 0.

Since 1112 is an ideal of Ill, the unit E of the semigroup III must lie in IllI' Since IllI is a group, for any Al E IllI there exist X, Y E IllI such that

XA I

=

E,

A1Y= E.

250

[CHAP.

INVERTIBILITY

VI

Hence from 1.3 it follows that Al is a two·sidedly invertible element of the semigroup'l(. On the other hand, elements of'l(2 cannot be two·sidedly invertible elements of'l( since 'l(2 is an ideal. Therefore (fi('!() = 'l(1' from which it follows that N,!() = 'l(2' i.e., m: turns out to be a semigroup with separating group part. From what was proved in the first part, the ideal m 2 is two-sided.

6.5. COROLLARY. In a semigroup 'l( with separating group part, ~('!() is a two-sided ideal of'l( or else an empty set. 6.6. COROLLARY. In a semigroup 'l( with separating group part the subsemigroups (fi('ll) and ~('ll) (if the latter is nonempty) are completely isolated (IV, 6.1). 6.7. In the class of semigroups with unit the semigroups with separating group part may be distinguished by the requirement that the subsets mand .£ considered by us in § 1 should be empty. THEOREM. Suppose that 'l( is a semigroup with unit. Ij'l( is a semigroup with separating group part, then in 'l( every right invertible element is left invertible, and conversely. Ij'l( is not a semigroup with separating group part, then in 'l( there are elements right but not left invertible and also elements left but not right invertible. PROOF. (1) If'l( is a semigroup with separating group part, then no element H of ~('!() can be either right or left invertible. This follows from the fact that N'll) is a two-sided ideal of 'l( (6.5). So for any A E 'l( the elements AH and HA lie in .5(,!() and are therefore necessarily distinct from Em. (2) If'l( is not a semigroup with separating group part, then in m: there are elements X, YEN'll) such that XY = G E (fi('!(). Because of 1.2, X'l(::J XY'l(

= Gm =

'l(,

i.e., X is right invertible. At the same time X is not left invertible, since otherwise X would belong to (fi('!(). Analogously one proves that Y is left invertible but not right invertible. 6.8. Theorem 6.7 explains the role of semigroups oftransformations which are semigroups with separating group part. For transformations from such a semigroup, ifthey are regular with respect to invertibility, the condition that the equation S$ = oc should be solvable, which we discussed in § 5, turns out to be equivalent to the condition of uniqueness of the solution. THEOREM. Let 'l( be a semigroup of transformations of the set n (5.5) which is regular with respect to both right invertibility and left invertibility and which contains the identity transformation E.

Sec. 6]

251

GROUPS WITH SEPARATING GROUP PART

If'11 is a semigroup with separating group part, then for each S from the solvability of the equation in the unknown ;, S$

=

E

'11 it follows

IX,

for any IX E .0, that the solution is always unique. In turn, in the case that the indicated equation is not solvable for some ()( E .0 we have the result that for some ()( E .0 it has more than one solution. If'11 is not a semigroup with separating group part, then there are Sl' S2 in '11 such that the equation Sl$= IX is solvable for all ()(

E

.0, but some

0(.

has more than one solution. The equation

S2$ = for some ()(

E

0(.

.0 is not solvable, but for every ()(

E

.0 has not more than one solution.

PROOF. (1) Suppose that '11 is a semigroup with separating group part. From 5.6, from the solvability of our equation it follows that S is right invertible in '11. Hence it follows from 6.6 that S is also left invertible in '11. Therefore, from 5.6, the equation for each 0(. E.o has no more than one solution. (2) Suppose that '11 is not a semigroup with separating group part. From 6.7 there is an element Sl in '11 which is right but not left invertible, and an element S2' left but not right invertible. From 5.6 the equation

Sl; = ()( is solvable for all ()( E .0, but for some of these has more than one solution. The equation S2$ = 0(. has at most one solution for any ()( E.o. For certain 0(. it is not solvable. 6.9. Suppose that '11 is a semigroup with separating group part, while '11 is not a group. Since two-sidedly invertible elements cannot be contained in any proper ideal of the semigroup '11 it follows that ~('11), from 6.5, is a universal maximal proper ideal of \le. 6.10. It ought to be observed that the presence in a semigroup of a universal maximal proper ideal in the general case is not sufficient for it to be a semigroupwith separating group part. We present an appropriate example. Let $ be a group and Go a fixed element of it. We consider a set '11 obtained by adjoining to 03 a new element X. Elements of 03 multiply in '11 according to the multiplication law of the group $. For X we put

= GoG,

GX = GGo, X2 = G~ (G E (3). The associativity of the operation is evident. 03 is a two-sided ideal of '11. If ::r is a left ideal of \le containing X, then for some G E $ we have (GG0 1) . X = GG01GO = G. XG

252

[CHAP.

INVERTIBILITY

VI

Accordingly X, containing X, contains also any element of $, and therefore coincides with m:. In the same way we can show the absence of proper right ideals containing X. Thus $ turns out to be the unique proper ideal in m: and thus a universal maximal proper ideal. Nevertheless, m: is not a semigroup with separating group part, since m: does not have a unit. 6.11. Questions on the existence and properties of maximal proper left ideals, and analogously of right and two-sided ideals, were considered by 8t. Schwarz [6; 7]. He gave particular consideration to the question of the structure of the set of elements not contained in one or another maximal ideal. From the results of Schwarz follow in particular certain sufficiency tests for the semigroup to be a semigroup with separating group part. 7. Subsemigroups of a Semigroup with Separating Group Part 7.1. Now we shall take up subsemigroups of semigroups with separating group part. First of all we observe that every semigroup may be imbedded as a subsemigroup in some semigroup with separating group part. Moreover, it can even be done in such a way that the imbedded semigroup coincides with the subsemigroup of all elements which are not two-sidedly invertible elements. For let $ be any group and m: any semigroup not having common elements with $. Consider the set m:' = ij) u m:. We shall define an operation in it. If both elements of Ill' are contained simultaneously in the same semigroup $ or m:, then their product is defined as the product, respectively, in ij) or Ill. For G E ij) and A E I}l we put

GA =AG =A. It follows from 6.4 that

m:' is a semigroup with separating group part, while m: = .t)Cm:').

7.2. Semigroups with separating group parts do not have subsemigroups of right and left magnifying elements (III, 5.4). THEOREM. Suppose that m: is a semigroup with unit. Ifm: is a semigroup with separating group part, then I}l has neither right nor left

magnifying elements (III, 5.1). If I}l is not a semigroup with separating group part, then magnifying elements. PROOF.

I}l

has right and left

By 2.1 each magnifying element is invertible on one side but not

inverti~le on the other. From 6.7 such elements cannot lie in a semigroup with

separatmg group part.

Sec. 7]

SUBSEMIGROUPS WITH SEPARATING GROUP PART

253

If mis not a semigroup with separating group part, then, from '6.7, mmust contain elements right invertible but not left invertible, and vice-versa. By 2.3 these elements are left and right magnifying elements of the semigroup. 7.3. Because of III, 6.8 one may immediately draw a consequence of Theorem 7.2. COROLLARY. In order that a semigroup mwith unit should be a semigroup with separating group part it is necessary and sufficient that among its subsemigroups containing E& there are none isomorphic to the semigroup \:j3 (III, 6.2; III, 6.3). 7.4. From Theorem 7.2 and III, 5.3 it follows that the semigroups belonging to various classes of semigroups are semi groups with separating group part. (IX) Every finite semigroup with identity is a semigroup with separating group part. ((3) Every commutative semigroup with identity is a semigroup with separating group part. (y) Every semigroup with two-sided cancellation with an identity is a semigroup with separating group part. 7.5. Using 7.3, it is easy to indicate still another class of semigroups with separating group part. Suppose we are given a sequence of semigroups with separating group parts in which each term is a subsemigroup of the next one:

m1 c: 'll2 c:

. . . c:

mn

c:

mn+l

c: . . . .

Their union is n=l

As we already have observed in III, 1.2, ~ is a semi group. If m has an identity, then mis a semigroup with separating group part. Indeed, if mwere not a semigroup with separating group part, then from 7.3 m would have a subsemigroup isomorphic to \:j3 and containing E'l). Both generators of this semigroup would be contained in some in which they would generate a subsemigroup isomorphic to '-l3 and containing E'l) = EM-n' But this contradicts the fact, from 7.3, that mn is a semigroup with separating group part.

mm

m

7.6. THEOREM. Let be a semigroup with separating group part. If the subsemigroup m' of the semigroup mcontains its unit E&, then it is itself a semigroup with separating group part. PROOF. E~r is evidently the unit of m'. Since in '1t, from 7.3, there are no subsemigroups isomorphic to '-l3 and containing E&, there can be no such subsemigroups in the subsemigroup m' either. But then, from 7.3, m' is a semigroup with separating group part.

254

INVERTIBILITY

[CHAP.

VI

7.7. Making use of 7.4, we may indicate a still further extensive class of semigroups with separating group parts. Recall that the semigroup 'l( is said to be a semigroup representable by matrices if for some n there is an isomorphism of m: into the semigroup ill1n of all complex square matrices of order n. THEOREM. Every semigroup with identity, representable by matrices, is a semigroup with separating group part. PROOF. Suppose that n is the smallest of the integers for which there exists an isomorphism q; of the given semigroup m: into ill1 n • We reduce the matrix q;(Ew) to the normal Jordan form. 3 This means that for some nonsingular matrix P E 9J1n the matrix has the normal Jordan form. Consider a new mapping into 9J1 n :

!p

of the semigroup

m:

!peA) = P-l[q;(A)]P.

Evidently !p is a representation of m: by matrices. Since the element E21 of the semigroup m: is idempotent, the matrix !p(E'!J.) , being in normal Jordan form, satisfies the condition As follows immediately from the definition of the normal Jordan form, this condition may be satisfied only in the case when the matrix !P(E~I) is diagonal and all its diagonal elements are equal to zero or unity. Suppose that the ith diagonal element of this matrix is equal to zero. Since for any element A E m: we have AE'!J.=A, E2lA = A, it follows immediately that

Taking into account that the matrix !p(E'!J.) is diagonal and that its ith diagonal element is equal to zero, we conclude that all the elements of the ith row and the ith column of the matrix !peA) are equal to zero. It is easy to verify that the matrix X(A), obtained from the matrix 1p(A) by striking out the ith row and the ith column, will again give a matrix representation of m: (the case when n = 1 and !p(E~ is the null matrix does not have to be considered because of its triviality). But the matrix X(A) is contained in 9J1n- 1, which contradicts the initial hypothesis concerning n. Hence it follows that q;(Em) is the identity matrix. 3 See, for example, A. 1. Mal'cev, Foundations of linear algebra, OGIZ, Moscow, 1948 (see Chapter IV, § 3). (Russian)

Sec. 7]

255

SUBSEMIGROUPS WITH SEPARATING GROUP PART

Thus we have come to the conclusion that 'IjJ is an isomorphism of ~ onto some semigroup 'IjJ(1ll) c 9J?n containing the unit of the matrix 'IjJ(Ew.) = E. In 6.2, we indicated that 9J?n is a semigroup with separating group part. From 7.6 it therefore follows that 'IjJ(Ill), and therefore ~ itself, are semigroups with separating group parts. 7.8. As to semigroups of infinite matrices, the considerations presented above do not extend to them. For example, the set of all countable matrices, in each row and column of which there is only a finite number of elements distinct from zero, evidently forms a semigroup whose identity is the countable identity matrix E. This semigroup is not a semigroup with separating group part. Indeed, 0 0 0

0 0 1 0

0 0 0 0 0 0 0

0 0 0 0

0

1 0 0

0

0 0

0 0

1 0

0 0

0 0

0

o

1 0 0 0 1 0 0

0

0 0

1 0

0 0 0 0 1 0 0

=

0

0 0 1 0 0 0 0 1 0 0 0 0

=

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

=E;

=;6 E,

0 0 0

from which, taking account of 1.3 and II, 2.14, ({J), it follows that the matrices multiplied above are elements of our semigroup invertible on one side but not invertible on the other. In semigroups with separating group part such elements, from 6.7, cannot exist.

CHAPTER

VII

HOMOMORPHISMS 1. Homomorphisms and Their Divisibility 1.1. In the first chapter we spoke of mappings preserving this or that relation between elements of the set being mapped. In the construction of the theory of semigroups it is to a high degree natural to distinguish mappings of one semigroup into another under which the relations of the operation are "preserved." In other words, one is concerned with mappings such that if in the first semigroup one has AB = C for certain elements A, B, C, then in the second semigroup the relation A'B' = C' will hold for the elements A', B', C' onto which the respective elements A, B, C are mapped. Isomorphisms are an example of such mappings. DEFINITION. The mapping rp of the semigroup U into the semigroup lB is said to be a HOMOMORPHISM, iffor any elements X and Y ofU and lB one always has

rp(X)· rp(Y) = rp(XY). If rp is a mapping of U onto lB, then one speaks of a homomorphism of U onto lB (sometimes in this case one uses a special term, epimorphism). If'2! and lB coincide, then the homomorphism is called an endomorphism of the semigroup U. This use of the term does not contradict the wider meaning of it indicated in I, 3.18, since a homomorphism of a semigroup into itself is a transformation of it which preserves relations between its elements which have the form AB= C. 1.2. It is easy to see that if the semigroup '2! is a group and if rp is a homomorphism of '2! onto the semigroup lB = rp(lll), then also lB will be a group. Indeed, for arbitrary elements of the semigroup lB,

the elements rp(X) and rp(Y) (X and Ybeing elements ofU such that XA I and Al Y = A 2 ) satisfy the conditions

rp(X) . rp(AI) = rp(XA I) = rp(A2)' rp(AI) . rp( Y) = rp(AI Y) 256

= rp(A2)·

= A2

Sec. 1]

HOMOMORPHISMS AND THEIR DIVISIBILITY

257

The concept of homomorphism for groups is one of the most important in the theory of groups. In most of the sections of this well-developed theory one uses directly or indirectly (through normal divisors) the properties of homomorphisms of groups. This use is greatly facilitated by the relative simplicity of defining homomorphisms of groups by means of the so-called normal divisors. In the theory of semigroups the structure of homomorphisms is incomparably more complicated. A complete study of homomorphisms has been carried out only for various special classes of semigroups. Of the numerous different general properties only a few have been considered. No doubt further study of homomorphisms will have an essential influence on the general theory. 1.3. The concept of homomorphism may be considered as a natural generalization of the concept of isomorphism. Indeed, it is evident that an isomorphism is simply a one-fo-one homomorphism.

However in its intrinsic character this generalization goes beyond the scope of the idea of isomorphism. As distinct from an isomorphism of one semigroup onto another, with homomorphisms there is no way of thinking of both semigroups as identical in some sense. Evidently the semigroups may be essentially different with respect to very diverse properties. 1.4. There are many reasons calling for the consideration of homomorphisms, some of which, connected with the theory of transformations, we shall now mention. In regard to various questions in mathematics and physics, particularly in the theory of differential equations, it frequently becomes necessary to consider the so-called one-parameter semigroups of transformations. Suppose that ~ is some additive semigroup of numbers and that to each number t of ~ we associate some transformation At of a given set.o. If, in addition, for any t l , t2 E ~ we have A t1 ·A t2 -A 11+/ 2 ' then the collection ~ of all transformations At (t E :2:) evidently is a semigroup, a one-parameter semigroup of transformations. The association to each number t of :2: ofthe transformation A t represents a homomorphism of the semigroup of numbers :2: into the semigroup of all transformations of the set D. The number of different important one-parameter semigroups oftransformations is exceedingly large. Their study frequently turns out to be very useful in various mathematical theories. The extensive book of HiIle [2] (see also Hille and Phillips [1]) is basically devoted to the study of extremely varied classes of one-parameter semigroups of transformations. Of course, along with one-parameter transformations one sometimes has to consider transformations given by systems of parameters. 1.5. In regard to certain questions involving the transformations of sets it is convenient to take a point of view rather different from the one to which we basically adhere in the present book.

258

HOMOMORPHISMS

[CHAP.

VII

Suppose that we are given two sets m: and Q such that for each pair of elements (A, oc) chosen from these sets (A Em: and oc EQ) their product is defined to be an element of Q, that is, Aoc = (3 E Q. In this case elements of the set m: are called operators on the set Q. The operator A Em: effects in Q a certain transformation. In its turn, each transformation may be considered as an operator on the set Q. Nevertheless, the consideration of operators is not fully identical with the consideration of transformations. The point is that two operators given as different elements of the set of operators m: may realize the same transformation in O. That in certain cases this is convenient may be seen, for example, from the following. Suppose we are given a certain family of transformations 58 of the set O. Suppose that r is a subset of Q such that xr c r for every transformation X E 58. Then transformations of 58 realize certain transformations of the set r. However, it can well happen that two different transformations of 58 realize in r one and the same transformation. Thus, considering the action of the elements of 58 on r, we have to take into account the possibility that certain of them which we cannot regard as identical (since they carry out different operations in Q) effect in r one and the same transformation. Suppose that in a set of operators m: on the set Q one is given in some way a multiplication operation such that m: is a semigroup with respect to it. One says that the multiplication in the semigroup m: is consistent with the multiplication of operators of m: on elements of Q if for any X, Y Em: and oc E0 the following condition, having an associative character, is satisfied: (XY)oc

=

X(Yoc).

We shall denote by A that transformation in the set Q which is effected in Q as a result of multiplication by the operator A Em:. Thus, we obtain a mapping of the semigroup of operators m: into the semigroup 6 0 of all transformations of the set Q. Since Aoc = AIX, it follows that for any X, Y Em: and IX EQ we have (.KY)IX

= .K(Yoc) =

=

X(Yoc)

X· Y=

(XY).

X(Yoc)

= (XY)oc = (XY)oc.

Since oc is arbitrary, this means Thus, the association to each operator A Em: of the transformation yields a homomorphism of the semigroup m: into the semigroup 6 0 ,

..IE 6 0

1.6. We have already repeatedly spoken of isomorphic representations of semigroups. In connection with the concept of homomorphism one may speak of homomorphic representations. Suppose that m: is a semigroup and 2: is a certain class of semigroups. A homomorphism of m: into some semigroup of 2: is said to be a homomorphic representation ofm: by semigroups of E.

Sec. 1]

HOMOMORPHISMS AND THEIR DIVISIBILITY

259

In connection with what was said in 1.2 on the fundamental difference in approach to the concepts of isomorphism and homomorphism, it is natural to outline the difference in the meaning of isomorphic representations and homomorphic representations. The use of the homomorphic image for the study of the properties of the original semigroup is made difficult by the fact that different elements of the original semigroup may map into one and the same element under the homomorphism. This circumstance must always be borne in mind in the study of homomorphic representations. In this connection, the following concept arises. A collection of homomorphisms
260

[CHAP.

HOMOMORPHISMS

VII

The mapping X is called the product of the mappings cP and '1jJ and is written X = cP • 1p or X = CP'1jJ. It should be noted that in considering such a multiplication of mappings we in fact pass beyond the limits indicated in § 1 of the first chapter for the concept of an algebraic operation. It is immediately clear that if for the mappings CP1' CPz, CPa the products CP1CP2 and CP2CPS are defined, then also (CP1CPJCP3 and CPl(CP2CPa) are defined, while (CPl CPJfP3 = CP1 (CP2CPg)· 1.10. Suppose that 01 and O 2 are any two nonempty sets. Denote by 9\ the set of all mappings of 01 into 02' The operation of multiplication of mappings considered in 1.9 is as a general rule simply inapplicable. Possible natural immediate generalizations of the operation of 1.9 lead to trivial results. However, in 9t one may introduce a nontrivial operation in the following way. We fix some mapping 1T' of the set 02 into 01' For X, Y E 9t we define the product X 0 Y by means of the multiplication of mappings of 1.9: (X 0 Y)et: = (X, 1T" Y)et:

=

X{1T'(Yet:)}

(et:

E

01)'

(The product X' 1T" Yis evidently always defined in the sense of 1.9, and belongs to 9\.) The set 9t, considered relative to this operation, will be denoted by 9\... The operation in 9t". is associative, since evidently (X 0 Y) 0 Z = X1T' Y1T'Z,

X 0 (YO Z)

=

X1T'Y1T'Z.

Thus 9\1/" is a semigroup. It sometimes is as useful for the study of mappings of 9\ as the consideration of the semigroup Sn is in the study of transformations of the set 0. Apropos, Sn is a special case of the semigroup 9\1/"' In fact, Sn is 9t1/" for 01 = O 2 = in the case when one takes for 1T' the identity transformation. We assign to each XE 9\" a transformation of the set 01:

°

cP".(X) = 1T' • X • .It is clear that cP" is a mapping of 9\.. into Sn. This mapping is a homomorphism: cp",(X)· CP1/"(Y) = (1T" X), (1T" y) = 1T'(X1T' y) = CP1/"(X 0 Y).

If the mapping 1T' is not one-to-one, then it is easily seen that the homomorphism cP" is not an isomorphism. If 1T' is one-to-one, then cP" is an isomorphism. If in addition 1T' is a one-to-one mapping of 02 onto 1, then cP'" is an isomorphism of 9i,.. onto 6 n , so that the semigroups 9\1/" and 6 n turn out in this case to be isomorphic. In considering transformations of some set n, interest usually centers not only in the semigroup Sn of all its transformations, but in subsemigroups distinct from it, consisting of transformations having certain given properties.

°

Sec. 1]

HOMOMORPHISMS AND THEIR DIVISIBILITY

261

Just the same, in considering the multiplication of mappings of m" introduced above, the interest lies not only in the semigroup m" itself but also in its various subsemigroups. Here, moreover, there arises further the question of clarifying the mutual relations of the semigroups m" for various Tr. An analogous construction may of course be effected for partial mappings of one set into another. 1.11. Let IDlm,n be the set of all complex matrices having m rows and n columns, and P some fixed matrix having n rows and m columns. In IDlmnone may consider an associative operation ' Ml 0 M2 = Ml . P . M2

(Ml' M2 E IDlm,n)

(where the point denotes the usual multiplication of rectangular matrices). This is, of course, just the operation considered in 1.10, since matrices of~,n may be regarded as linear mappings of vectors of n-dimensional complex space into m-dimensional complex linear space. From the point of view of such an operation over matrices it becomes natural to consider the operation in the semigroup of matrices of type S(P, ~), which played such an important role in §§ 5 and 6 of the fifth chapter (with the difference that the matrices there could be infinite as well, and their elements were not complex numbers). 1.12. Suppose that "P is a homomorphism of the semigroup III onto the semigroup ~ and that 9' is a homomorphism of ~ onto the semigroup (t. The product of transformations X = 9'''P, in the sense of the product of transformations described in 1.9, is defined and as one easily sees is a homomorphism of III onto (t. The homomorphism X is called the product of the homomorphisms 9' and "P. Only in cases of the type described will we say that the multiplication of the homomorphisms 9' and "P is possible (or defined). The homomorphism "p in this case is called a right divisor of the homomorphism x. Concerning X one says that it is divided on the right by "P. We shall in this case write "P '" X(p). If for the homomorphisms "Pl and "P2 we have simultaneously "Pl '""" "P2(:P) and "P2'""" 'ljJl(P), then we will write "PI '" 'ljJ2(Q). We note that only homomorphisms of one and the same semigroup can lie in the relation of right divisibility p. 1.13. We shall consider several properties of the multiplication of homomorphisms. (IX) If for the homomorphisms. 9'1' 9'2' 9's the products 9'19'2 and 9'29'3 are defined, then also the products (9'19'2)9'3 and 9'1(9'29'a> are defined and equal to each other.

(fJ) Iffor the homomorphisms !Pl' 9'2' 9'3 one has 9'1'" !P2(p)'-and 9'2'""" 9's(p), then one has also 9'1'" 9's(p) (transitivity ofp)·

Indeed, suppose Then, evidently,

262

HOMOMORPHISMS

[CHAP.

VII

(y) Iffor the homomorphisms qJ1' qJ2' qJ3 one has qJ1 ,....., qJ2(q) and qJ2 "-' !P3(q), then there holds also !P1"-' qJiq) (transitivity of q). (0) Iffor the homomorphisms !P1' !P2 and the isomorphism 8 one has qJ1 = 8qJ2' then there exists an isomorphism 8' such that qJz = e' qJ1' Indeed, if !P2 is a homomorphism of the semigroup \U, then 8 is an isomorphism of the semigroup qJ2(\U) onto the semigroup !P1(\U)' Denote bye' the isomorphism inverse to this of the semigroup !P1(\U) onto qJz(\U). Since 8'!P1 = 8' eqJ2

and since e'8 is the identity isomorphism it follows that qJ2 = 8' !Pl' (8) For the homomorphisms qJ1 and !P2 of the same semigroup, the relation !P1 ,......, qJ2( q) holds if and only if there exists an isomorphism 8for which !P1 = 8!P2' Indeed, if!p1 "" !P2(q), then, for certain homomorphisms 'IjJ and 'IjJ', i.e., !P1 = ('IjJ'IjJ')qJ1' The product 'ljJ1p' is the identity mapping. This is possible only in the case when the homomorphisms 1p and 1p' are one-to-one mappings, i.e., isomorphisms. Now suppose that for some isomorphism 8 we have qJ1 = 8qJ2' This means that qJ2"""" qJ1(P). But then, from (0), there must exist an isomorphism e' such that qJ2 = e' !P1' and therefore qJ1 "" qJz(p)· (0 Iffor the homomorphisms qJ1' !p~, qJ2' qJ; of the semigroup mwe have then qJ~ "" qJ~(p).

Indeed, for certain isomorphisms 81 and e2 and a homomorphism have

'IjJ

we must

Denoting bye; the inverse isomorphism, relative to 82, of the semigroup

qJ~(\U) onto qJ2(\U)' we have

, , (') , . , '() qJ1 = e1qJ1 = 81'IjJ!P2 = 81'IjJ8 282qJ2 = 811p82 qJ2' l.e., qJ2"""" qJ1 P . (n) If qJ is some homomorphism of the semigroup \U and 8 is some isomorphism of it, then e"-' qJ(p). . Indeed, denoting bye' the isomorphism of e(\U) onto \U inverse to e, we obtain qJ = !P . 8' e = (qJ8') • 8. (0) If 8 is an isomorphism of the semigroup \U and qJ a homomorphism of \U, with qJ"""" e(p), then qJ is also an isomorphism. Indeed, for some homomorphism 'IjJ we have e

=

'ljJqJ.

Sec. 1]

HOMOMORPHISMS AND THEIR DIVISIBILITY

263

Since B is a one-to-one mapping, the mapping rp must also be one-to-one, i.e., an isomorphism. (£) Let rp and'1j! be two homomorphisms of the semigroup Ill. For rp ""' 1p(:p) it is necessary and sufficient that if rp(A) = rp(B) for any A, B E Ill, then

1p(A)

= 1p(B).

Indeed, if for some homomorphism X' follows that

1p(A) = (X' rp)(A) = X[rp(A)]

1p

= xrp,

= X[rp(B)]

then from rp(A) = rp(B) it

= 1p(B). rp(A) = rp(B) it always

= (X' rp)(B)

On the other hand, if rp and 1p are such that from follows that 1p(A) = '1j!(B), then for the semigroup rp(m) one may define the following mapping X of it onto the semigroup 1p(m). Put

X[rp(A)]

= 1p(A).

This mapping is uniquely defined, independently of the choice of the representative A in the class of those elements X E III for which rp(X) = rp(A), since, if rp(A) = rp(A'), we have '1j!(A) = 1p(A') in 1p(m). From the fact that 1p and rp are homomorphisms, it follows immediately that the mapping X is also a homomorphism. Thus xrp = 1p, i.e., rp ""' 1p(:p). 1.14. From 1.13, (y), (£), it follows that in the class of all homomorphisms of a semigroup the relation q is an equivalence, and thus that this class decomposes into disjoint classes of homomorphisms which are equivalent to each other relative to the relation q. One of these classes is formed by all the isomorphisms of the semigroup. If for the homomorphisms rpl and rp2 we have rp11"-' rp2(:P), then also for any homomorphisms rp~ and rp;, taken from the corresponding classes, the relation rp~ cp~(p) is satisfied (1.13, (m. Therefore, for those same classes r 1 and r 1\ one may introduce the relation p, holding if for rpl E r 1 and rpl\ E r 1\ one has rpl ""' rp2(::P)· Because of 1.13, «(3) this relation is a partial ordering relation, and we may write r 1 <; r 1\ in place of r 1 r 2(:P). The class of all isomorphisms, because of 1.13, (n) and 1.13, (6),' turns out to be a predecessor of all the other classes. It is easy to see that the class consisting of homomorphisms mapping the semigroup onto the identity group (consisting of the unit element alone) will follow all the remaining classes. Since homomorphisms of one and the same class differ, by 1.13, (8), only by a multiplier which is an isomorphism, it is usual to say that they are identical up to multiplication by an isomorphism or that they are inessentially distinct from one another. f"oo.J

f"oo.J

1.15. Suppose we are given some collection r of homomorphisms of the semigroup Ill. It is natural to call a homomorphism rp of the semigroup III the greatest common right divisor for r if rp is a right divisor of every homomorphism

264

HOMOMORPHISMS

[CHAP.

VII

r

of and if it is divided on the right by every homomorphism which is a right divisor of every homomorphism of r. The homomorphism "P is said to be the least common right multiple for if!p is divided on the right by every homomorphism of r and if it is a right divisor of every homomorphism which is divided on the right by every homomorphism of r. Evidently the class of homomorphisms which are the greatest common right divisors of the homomorphisms of r, relative to the partial ordering 1.14 of the classes, is the greatest lower bound for the classes of homomorphisms containing the homomorphisms of r. The class ofleast common right multiples is a least upper bound. From 1.13, (8) and 1.14 it follows that every two least common right divisors of the class of homomorphisms are inessentially distinct from each other. Also every two least common right multiples are inessentially distinct.

r

r

1.16. The question as to the existence of a greatest common right divisor for a collection of homomorphisms, and of a least common right multiple, will receive a positive solution in the following section (2.11). One must, however, also keep in mind that, for a given class of homomorphisms, there is usually interest not only as to whether there exist for it greatest common right divisors, but also as to whether some of them belong themselves to the class r. This is particularly important when the class r is such that if a certain homomorphism cP lies in it, then any homomorphism divided on the right by cP also lies in r. If the class r is such, and "P is the greatest common right divisor of the homomorphisms of r which belongs to r, then r may be characterized as the class of all homomorphisms divided on the right by!p. Prescribing!p in this case accurately describes the structure of the class r. As an example we mention the class of all homomorphisms of an arbitrary semigroup m: onto a commutative semigroup. Evidently, if the homomorphisms cP, !p, X are such that CP!P = X and the semigroup "P(m:) is commutative, then also X(m:) will be commutative. In 6.7 we construct the greatest common right divisor of such homomorphisms, which will in addition lie in that class. As an example of a negative solution of the question posed above we consider an infinite monogenic semigroup m: = [X] and the class of all of its homomorphisms onto groups. For any finite cyclic group (ba = [G] with a number of elements equal to d (i.e., of type (1, d)) there exists a homomorphism CPa of the semigroup m: onto (ba:

r

CPa

= (Ak) =

Gk

(k

=

1,2, 3, ... ).

If "P is some homomorphism of m: onto the group, then, for some m, "P(Am) will be the identity of that group. Hence we easily find that !p(~O =

{"P(A), "P(A2), ... , "P(Am)}.

Since the number of elements in "P(m:) does not exceed m it follows that "P cannot be a right divisor for those CPa for which d > m.

Sec. 2]

265

FACTOR-SEMIGROUPS

1.17. We have already mentioned that the connection between the properties of a semigroup and its homomorphic image in the case of an arbitrary homomorphism is far from being as elementary as with an isomorphism. Suppose that ep is a homomorphism of the semi group ~ onto the semigroup cp(1ll) = ~'. If in ~ the element Al is divided on the left by A 2• then obviously in ~' the element ep(AI) will be divided on the left by ep(A 2). The same is true for divisibility on the right. In particular, if C is an element of ~ divisible both left and right by any element of ~ (II, 1.6), then ep(C) will be divisible in ~' left and right by all elements of ~'. However, if some element C' of ~ is divided both left and right by every element of ~', then it is not necessarily an image of some element of ~ having the same property in~. Indeed, the semigroup ~ may have no such elements at all. Nevertheless, under a homomorphism of it onto the identity group the single element of the image has the property in question. In connection with what has been said, we note that for a homogroup~, i.e., in the case when ~ has a two-sided ideal:! which is a group (V, 1.8), the following property is satisfied. For any homomorphism ep of the semigroup ~, if in Ill' = ep(1ll) the element C' is divided by all the elements of~' both left and right, then there is in III an element C divided both left and right by every element of ~ and such that ep(C) = C'. Indeed, it is obvious that ep(:!) is a two-sided ideal of~', and from 1.2 it is a group. From V, 1.4, C' E ep(:!), i.e., for some C E:! we have ep(C) = C'. But from V, 1.4, :! consists of elements divided both left and right by every element of Ill. 2. Factor-semigroups

2.1. Suppose that cp is any homomorphism ofthe semigroup Ill. We define in III a relation n by putting A r - ; B(n) if ep(A) = ep(B). Reflexivity, symmetry and transitivity of this relation are evident. The equivalence n will be called the equivalence corresponding to the given homomorphism cpo Since from ep(A) = ep(B) it follows for any X E III that

ep(XA)

=

ep(X) • ep(A)

= ep(X) • ep(B) =

cp(XB),

ep(AX) = ep(A) • ep(X) = ep(B) • ep(X) = ep(BX), the equivalence n, corresponding to the homomorphism ep, is two-sidedly stable.

2.2. Now suppose that n is any two-sidedly stable equivalence in the semigroup Ill. Denote by Ill/n the set of all n-classes (I, 5.8). Observe that for all n-classes St 1, St 2 , fts it always follows from that

266

[CHAP.

HOMOMORPHISMS

VII

Indeed, suppose Xl' YI E ~l; X 2, Y 2 E ~2 and YI Y 2 E ~a· In accordance with I, 5.18 it follows from Xl r--J YI(n), X 2 ,...., Y2(n) that X I X 2 r--J YI Y2(n).

Since YI Y 2 E .R a, any element X I X 2 of the set .Rl.R2 must also belong to .R a. Since the n-c1asses form a decomposition of the set of all elements of the semigroup (r, 5.8), it follows from the property just proved that for two arbitrary n-c1asses .Rl and .R2 there will always exist a unique n-class .Ra such that

(.R 1 . .R 2)

C

.Ra.

2.3. Because of 2.2 one may define in a natural wayan operation in ~/n. Since this operation does not coincide with the operation of multiplication of subsets, we shall temporarily use the sign 0 for denoting the result of this operation. Of course, one may in this case use the usual multiplication notation if one regards the n-c1asses not as subsets of ~ but simply as elements of a new multiplicative set ~/n. In the mathematical literature one usually does this. In the sequel we shall also go over to the usual multiplicative notation, dropping the usage o. Suppose that .RI, .R 2, .Rs are three n-classes such that

.RI . .R2 We put in

C

.Rs·

~/n

.RI Since, for any .RI , .R 2, .Ra

0

.R2 = .Rs·

E ~/n,

= .RI.R2.Ra, .Rs) ;:, .RI . (.R 2 • .Rs) = .RI .R2.Ra,

(.RIO .R2) 0 .Ra ;:, (.R I . .R0 . .Rs .Rl 0

(~ 0

the operation 0 is associative in mjn. 2.4. DEFINITION. For a two-sidedly stable equivalence n in the semigroup ~, the set oj all n-classes ~/n, considered relative to the operation 2.3, is a semigroup, called the FACTOR-SEMIGROUP oj the semigroup ~ modulo n. This construction is a generalization of the corresponding construction of the theory of groups. If is a normal divisor of the group and n is a relation in according to which X,...., Yen) if and only if X-I Y E 91, then, as one easily verifies, n is a two-sidedly stable equivalence and /n is exactly the factor-group of modulo 91 of the theory of groups: fIJ'c.

m

2.5. Suppose that n is a two-sidedly stable decomposition of the semigroup Assigning to each element A E ~ the component .R of this decomposition which contains it, we obtain a mapping of ~ onto the factor-semigroup ~/n. This mapping is a homomorphism, since from ~.

AIA2

= Aa

(AI, A 2 , Aa

E ~),

Sec. 2]

FACTOR-SEMIGROUPS

267

for ~i3Ai' where ~iE'l((n (i= 1,2,3), it follows that the intersection n ~3 is nonempty and that therefore

(~1~2)

~l 0 ~2

= ~3·

Such a homomorphism will be called the natural homomorphism oflJl onto IJljn. The obtaining of the factor-semi group IJljn from IJl by means of the natural homomorphism may be considered as the result of identifying to one another the elements of the semigroup IJl which enter into one and the same n-class. 2.6. In 2.1 we showed that to each homomorphism of the semigroup IJl there corresponds some two-sidedly stable equivalence. The construction of the factor-semigroup IJljn shows that for any two-sidedly stable equivalence n there exists a homomorphism (in fact the natural homomorphism of IJl onto IJljn) to which that equivalence n corresponds. 2.7. Suppose that cp is any homomorphism of the semigroup I2l and n the equivalence corresponding to it (2.1). We denote by 'lfJ the natural homomorphism of I2l onto IJljn. Assign to each n-class (i.e., to each element of the factor-semigroup I2ljn) that element of the semigroup cp(12l) into which all the elements of the given n-class map under the homomorphism cpo This one-to-one mapping is easily seen to be an isomorphism. Thus cp = 81p, i.e., cp 1'0.1 1p(q) (1.13, (8». On the other hand, if cp is any homomorphism and 'lfJ is the natural homomorphism onto some factor-semi group, and if they are connected by the relation cp 1'0.1 'lfJ(q), i.e., for some isomorphism 8,

cp

= 81p,

then evidently the decompositions corresponding to the homomorphisms cp and 'lfJ are identical. From what has been said it follows that to each homomorphism ofthe semigroup IJl there corresponds a natural homomorphism onto the factor-semigroup IJljn, where n is the equivalence corresponding to the given homomorphism. The same natural homomorphism corresponds to different homomorphisms if and only if these homomorphisms divide one another to the right (1.12), i.e., from 1.13, (6) and 1.13, (8), if they differ only by a multiplier which is an isomorphism. Thus, natural homomorphisms onto factor-semigroups may be considered as representatives, taken one from each class of homomorphisms, where we unite into one class all homomorphisms which are inessentially distinct from one another (1.14), i.e., which are equivalent to one another relative to q (1.12; 1.14), or in other words differ by isomorphism multipliers (1.13, (8)). Consequently we may regard the set of all homomorphisms of a semigroup as being exhausted, up to inessential distinctions, by the natural homomorphisms of the semigroup onto its factor-semigroups.

268

[CHAP.

HOMOMORPHISMS

VII

2.8. Suppose we are given two homomorphisms Cf!l and Cf!2 of the semigroup and that n l and n 2 are the corresponding equivalences. If Cf!2 Cf!l(P) holds for the homomorphisms, i.e., if Cf!2 is a right divisor of Cf!l (1.12), i.e., Cf!l = 1pCf!2' then, from 1.13, (l) the equation Cf!2(A) = Cf!2(B) (A, B E '!() always implies Cf!l(A) = Cf!iB), and, accordingly, for the equivalence relations we have n 2 nl (I, 1.14). Conversely, suppose that n 2 < n l is given. We assign to each n 2-class that nl-class which contains it. It is easy to see that we obtain a homomorphism 1p of the factor-semigroup I[(/n 2 onto the factor-semigroup I[(/n l . If ~l and ~2 are the natural homomorphisms of I[( onto the factor-semigroups I[(/n l and I[(/n 2 , then we evidently have gl = 1pg2· But gl and ~2 differ from Cf!l and Cf!2 by multipliers which are isomorphisms. Therefore, for the homomorphisms Cf!l and Cf!2 themselves, we find that for some homomorphism 1p' we have Cf!l = 1p' Cf!2' i.e., Cf!2 r-...J Cf!1(P)· It has turned out that the relation of right divisibility between homomorphisms, P (1.12), is induced by the partial ordering of the corresponding equivalences. 1[(,

f"-'

<

2.9. Suppose that 'Y is some nonempty set oftwo-sidedly stable equivalences of the semigroup 1[(. In the complete lattice of all equivalences in I[( (I, 5.16) we take the greatest lower bound f of the set 'Y (1,5.15) and the least upper bound I (1,5.16). If for A, BEl[( we have A,......, B(f), then for any n E'Y we have A,......, B(n). But n is two-sidedly stable and therefore, for any X E 1[(, AX BX(n) and XA,...., XB(n). Since the last relation is valid for any n E 'Y, we have AX,...., BX(f) and XA ,......, XB(f). Thus the equivalence f itself turns out to be two-sidedly stable. Since f is the greatest lower bound of 'Y in the set of all equivalences, f in the same way is a least upper bound for 'Y in the set of all two-sidedly stable equivalences. Now suppose that A""' B(l). Then there are Zl = A, Z2' ... ,Z2+1 = B in I[( such that Zi ,......, Zi+1(n i ), n i E 'Y (i = 1, 2, ... , s). Since all the n i are two-sidedly stable, we have, for any X E 1[(, f"-'

AX = ZlX, XA

=

XZ1,

XB

=

BX XZ S+1'

= Zs+1X, XZ i

""'

ZiX,,", Zi+1X(ni), XZi+l(n i )

(i

=

1,2, ... , s).

Thus AX,......, BX(I), XA ,......, XB(I) and the equivalence I turn out to be twosidedly stable. Here I is the least upper bound for 'Y in the set of all two-sidedly stable equivalences in 1[(. It follows from what has been said that relative to the partial ordering I, 5.14, the set of all two-sidedly stable equivalences is a complete lattice. 2.10. We note that sometimes the characterization of the set of all twosidedly stable decompositions of a semigroup is realized by distinguishing in it a collection of relations (the so-called basis) such that every two-sidedly stable

Sec. 2]

269

FACTOR-SEMIGROUPS

relation may be obtained as a least upper bound (sum) of certain relations belonging to that collection. In this way A. E. Liber [1] dealt with the problem of characterizing twosidedly stable decompositions of the semigroup un of all one-to-one partial transformations of the set Q (I, 4.5). Suppose that cD is some nonempty set of homomorphisms of the Denote by 'Y the set of two-sidedly stable equivalences corresponding to these homomorphisms (2.1). Let f be their greatest lower bound and I their least upper bound (2.9). From 2.8, the natural homomorphism of I[ onto I[lf is the greatest common right divisor of the homomorphisms of cD, and the natural homomorphism of I[ onto 1[/1 is the least common right multiple (1.15). 2.11.

semigroup~.

2.12. Let n l and n 2 be two-sidedly stable equivalence relations on the semigroup 1[, with n l n 2 • In the factor-semigroup I[/n l it is natural to define an equivalence relation n3 = n 2 /n 1 according to which two n3-classes .Rl and .R2 are equivalent when the elements of the semigroup 1[, Xl E.Rl and X 2 E.R, are equivalent with respect to n 2 (evidently this does not depend on the choice of the representatives Xl and X 2 in.R l and .R 2). It is easily seen that n3 is a two-sidedly stable equivalence relation in 1[/111 and that the factor-semigroups l[/n 2 and (l[/n l)/(n 2/n 1 ) are isomorphic (the isomorphism is defined by assigning to the n 2-class .R ofl[/n2 the (n 2/n 1)-class of (1[/n1)/(n2/nl) which contains the nl-classes .R' contained in .R as elements).

<

2.13.

Suppose that n is any decomposition of the semigroup 1[, I[

= U lB~, ~

having the property that for any two of its components lB" and lBl' there is always a component lB. such that lBA . lBl' C lB •. This decomposition may be considered as an equivalence in I[ (1,5.8). If A "" B(n), i.e., if A and B are both contained in the same component lBA of our decomposition, then for X E lB I' we have AX E lBAlBl'

C

lB.,

BX E lBAlBl'

C

lB.,

i.e., AX"" BX(n). Analogously one proves stability to the left. Thus, n turns out to be a two-sidedly stable equivalence, and the set of components lB ~ of the decomposition n is a set of elements of a factor-semigroup I[/n. This shows that the preceding considerations may be carried over in their entirety to the language of decompositions of semigroups having the property indicated above. 2.14. For groups, the characterization of the structure of an arbitrary twosidedly stable decomposition presents no difficulty. In any book on the theory of groups there is a full exposition on the question of the structure of arbitrary homomorphisms of groups. But for arbitrary semigroups the corresponding question presents significantly greater difficulty and is still far from a complete

[CHAP. VII

HOMOMORPHISMS

270

solution. Only for separate special classes of semigroups has the question of the complete characterization of all the two-sidedly stable decompositions been settled. In addition, the obtaining of the corresponding result presents significant difficulties. Below, in §§ 3 and 6, we present several of these results. As an example, we present a construction considered by Pierce [1]. Suppose that.R is any subset of the semigroup 'll. The elements A, BE'll are equivalent if for any X, Y Emit follows from XA Y E.R that XBY E.R. The two-sided stability of this relation is verified without difficulty. A particularly interesting case occurs when .R is an ideal. The consideration of such equivalences for a commutative semigroup of idempotents, adjoint to the distributive lattice £ (II, 4.3; II, 4.13), turns out to be useful in the study of lattice homomorphisms of the lattice £ (Pierce [1]). 3. Homomorphisms of Inverse Semigroups

3.1. In this section we consider homomorphisms of inverse semigroups, i.e., the semigroups which we defined and whose properties we considered in § 7 of the second chapter. The significance of these semigroups in connection with their role in the theory of partial transformations was considered by us in § 8 of the same chapter. We shall succeed in obtaining a number of properties of homomorphisms of inverse semigroups and in giving a complete description of the structure of arbitrary homomorphisms of these semigroups. The fundamental property of homomorphisms was obtained first by V. V. Vagner [3]. Somewhat later a detailed description of the structure of homomorphisms was published by Preston [1]. 3.2. As usual, in the consideration of inverse semigroups we shall use the bar to indicate an element regularly adjoint to a given element. LEMMA. If P is a homomorphism of an inverse semigroup m, and if peA) (A E m) is an idempotent of the semigroup p(m), then mcontains an idempotent I for which p(I) = peA). PROOF.

Since the element peA) is an idempotent, we have

peA) • p(AA)

=

peA) • peA) . peA)

=

peA) . peA) = p(AA).

Because AA and AA are idempotents of the inverse semigroup 'll (III, 7.3), they commute (II,7.4). Therefore, using the equation just obtained, we find p(AA) . p(AA)

= peA) . p(AA) • p(AA) = peA) • p(AA) . p(AA) = p(AAA) . p(AA) = peA) . p(AA) = p(AA).

On the other hand, p(AA) . p(AA)

= p(AA) . p(AA) = p(Al' peA) . peA) . peA) = peA) . peA) . peA)

=

p(AAA)

=

peA).

Sec. 3]

HOMOMORPHISMS OF INVERSE SEMIGROUPS

Thus,


Hence for the idempotent I

271

=
= AA we obtain


=
3.3. With the aid of this lemma it is easy to see that the class of inverse semigroups is closed with respect to the operation of taking homomorphisms. THEOREM. If