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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
<
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
= 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
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 ';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)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 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 corresponding to the homomorphism cpo We shall show that st is a normal complex of the semigroup m. Suppose that for some KI , Kz Est and X and Y which are elements of m or empty symbols,
XK1Y= K2 and K3 is an arbitrary element of st. Denote by K2-1 the element inverse to K2 relative to I in (£)1' We have
Xl· KI YKil
=
XKI YKil
= K2Kil = I,
Xl· I
= Xl.
Hence, from III, 4.11, XI E 65 l' Analogously we see that IY E 65 l' Therefore
XKa Y = Xl· Ka . IY E 651 and
cp(XKaT ) = cp(Xl) . cp(Kg) . cp(IY) = cp(XI) . cp(K1) . cp(IY) i.e., XKg Y E 5t.
= cp(XIK1IY) = CP(XKI Y) = CP(K2) = cp(I),
Sec. 5]
EXTENSION OF HOMOMORPHISMS
289
Denote by 'ljJ the homomorphism of the semi group m which is the greatest common right divisor of those homomorphisms for which st is the complete preimage of one of the elements. The homomorphism 1p induces in G5 [ a homomorphism'ljJ'. The homomorphisms 'ljJ and 'ljJ' of the group G5 I are such that the complete preimages of the identity under these homomorphisms are equal to st. From 4.13, it therefore follows that 4, aside from the two trivial divisors, there is only one normal divisor; for n = 4 there are two nontrivial normal divisors; and for n = 1,2 there are no nontrivial normal divisors). For 91 we define in 6 0 a relation nm by putting A'"" B(n90 (A,B E 6 0 ), if one of the three following conditions is satisfied: (a) A = B. (fJ) peA) < n, pCB) < n. (y) peA) = pCB) = n; = BD; A~ = An implies B~ = Bn (~, nED), and conversely; and if ~1' ~2" .. , ~n E D is such that A~i ¥= A~j (i,j = 1,2, ... , n; i ¥= j) and cp is some mapping AD = {A~l' A~2' ... , A~n} onto {I, 2, ... , n}, then
An
We may at once observe that in case (y) the fact that the latter permutation belongs to in does not depend, as one easily sees, either on the choice of the elements ~1' ~2' ••• , ~ n or on the choice of the mapping cp (this last follows from the fact that for any automorphism of the group (ij its normal divisor in always maps onto itself). The relation n9dust defined is obviously symmetric and reflexive. It is easy to see that it is transitive as well. Let
300
[CHAP.
HOMOMORPHISMS
VB
In cases (IX) and ((3) the situation is evident. In case (y) we have AQ = BQ = Co.,
S= (
X2= XY= y2.
But then, by hypothesis, it must follow that X
= Y.
Sec. 1]
309
BANDS OF SEMIGROUPS
1.4. We note that the relation n which we constructed in the proof of Theorem 1.3 for a commutative semigroup 12!, satisfying the condition indicated in Theorem 1.3, is the greatest lower bound for all partitions of 12! into subsemigroups. In other words, n is the finest of all possible partitions of ~r into nonintersecting subsemigroups (n is a refinement of every such partition). Indeed, if m is one of these partitions and X'""" Y(n), i.e., xn = ym, then evidently X,....., Y(m) in view of the fact that X and Y, because of X" = ym, cannot lie in different m-classes, since different m-classes cannot have common elements. Among the properties of the partition n we note that each of its components St has no more than one idempotent. Indeed, if 11' 12 ESt and If. = II' Ii = 12 , then from If = I:;: follows II = 12 , We observe further that, from IV, 6.11, St is a semigroup having no proper isolated ideals. Indeed, if X,...... Y(n), then it immediately follows that X and Y stand to each other in the relation considered in IV, 6.11. A similar decomposition was accomplished by Thierrin [16], not only for commutative semigroups but also for semigroups satisfying a certain generalized commutativity condition considered in III, 4.5, (13). 1.5. It is easy to see that the condition indicated in Theorem 1.3 is a necessary condition for a partition into subsemigroups with cancellation, not only for commutative but also for arbitrary semigroups. The question as to its sufficiency for certain classes of noncommutative semigroups was also considered by Schwarz [16]. 1.6. It is quite natural that among the various partitions of a semigroup into subsemigroups a particular interest lies in the two-sidedly stable partitions (I, 5.17). DEFINITION. A two-sidedly stable partition of a semigroup components are subsemigroups of it, is called a BAND.
m,
all of whose
From 1.2 a two-sidedly stable partition n of the semigroup mwill be a band if and only if for any X E mwe have X,......, X2(n). If the semigroup mhas a band whose components are its subsemigroups 18"" IS Fl , ••• , then one also says that mrepresents the band of its subsemigroups 18"" 5B p, " " or, more simply, that mis itself the band of these subsemigroups. Such a usage of the same term in different but evidently related senses will usually not lead to a misunderstanding. 1. 7. As follows immediately from Definition 1.2 and I, 5.18, a band of the semigroup m is defined by a collection of pairwise disjoint subsemigroups 18"" IS Fl , ••• of that semigroup, wl}ose union is equal to m, while for each pair of these subsemigroups (IS~, IS,) one may always find a subsemigroup IS" ofthis collection such that 5B~ . 1S1] cIS". Incidentally it is clear that this IS" is uniquely defined for a given pair (18 U (£)r· Here (£) r is a group with identity!. The converse is also true. Every commutative band of groups is a completely regular inverse semigroup. Indeed, its complete regularity follows from the fact that each of its elements is contained in some subgroup (III, 1.15). Suppose that II and 12 are two of its idempotents. Since the semigroup is a commutative band of groups, for some subgroup (£) of our semigroup we have
Il/z = Gl
E (£),
Hence it follows that
G1G2Gl = Il/z • 12/1 . 1112 = 11/2/1/2
= GlGl ·
Sec. 3]
COMPLETELY REGULAR INVERSE SEMIGROUPS
323
For elements of a group this is possible only in the case when G2 = EfD. Analogously one verifies that also G1 = EfD. From the commutativity of any two idempotents we conclude that our completely regular semigroup is inverse. We have shown that in a commutative band of groups the idempotents commute. It therefore follows that the set of these idempotents fl forms a commutative semigroup. As we proved in § 4 of Chapter II, the structure of fl may be given by the corresponding partial ordering in fl. We observe that the multiplication of idem po tents in the semigroup fl defines the multiplication of components in our commutative band. Indeed, if UV= W,
(U, V, WE
fl),
then and accordingly 3.4. If UV = U for U, V E fl, we consider the mapping CfJu,v of the group (fiv into the group (fiu, namely,
is any nonempty set of idempotents of W. We consider the left ideal W,f>. For some idempotent I,
W,f>
= WI.
Accordingly, for some I' E ,f>,
I c WI'. Since Q{J
I
C
W,f>
= WI,
WI c WWI' c WI',
we have 1)11' = WI and, from what has been said above, I' = I. Thus the idempotent I' = I, being the identity of the ideal of WI = W,f>, is also a two-sided identity in the set ,f> to which it belongs. (2) Select any element A of the semigroup W. A u WA is a left ideal of W, and therefore for some idempotent
WA
U
A = WI,
while I is the identity of WI. Denote by .l! the union of all those left ideals of the semigroup WI which do not contain l. Evidently.l! is a proper left ideal of the semigroup Wor .B = 0. Suppose that .B ¥- 0. Since
W.B
= wm = (WI).l!
c .l!,
it follows that .B is a left ideal of W. Consequently, for some idempotent 1', .l!
= WI' .
.l!Q{J is a left ideal of Wbelonging to WI and containing .B (since .BI = .l!). If now .l!WI contained I, then for some X E.B and A E W we would have 1= XAI
= I' XAI = l' . I = 1',
which is impossible since I' E.B and IE.l!. Accordingly,
.BWI c .l!. Thus .l! is either empty or is a two-sided ideal of the semigroup WI. Write
WI \.l!
Jf
= ~.
K E~, then WK is a left ideal of Wlying in WI and containing K. Since K E .B, then lllK must be equal to llll. Conversely, if X E lllI and lllX = lllI, then
Sec. 3]
COMPLETELY REGULAR INVERSE SEMIGROUPS
329
every left ideal of the semigroup m:I containing X will also contain m:IX = 'l(X = m:I, i.e., coincide with m:I. Therefore X cannot lie in .52, i.e., X E St If K 1 , K2 E .R, then from the given property of .R,
m:K1 K2 = m:K1 . K2 = m:I· K2 = m:K2 = m:I, Thus.R is a subsemigroup of m:. Since IE.52 it follows
i.e., KIK2 E.R. belongs to .R and, of course, is the identity in it. Suppose that K E R If.52 is a two-sided ideal of m:I, then
m:I· (.52 u .RK)
= (.52
that I
u .R) . (.52 u M) c .52 u .RM c .52 u M,
:.e., .52 u.RK turns out to be a left ideal of m:I. If.52 = 0, then .RK is also a left ideal of.R = m:I. Also it is not contained in .52 because it has an element K . K = .R not belonging to.52. By the definition of .52 this ideal must contain I. Since IE .52, for some K' E.R we must have K'K= 1.
We have shown that any element K of the semigroup .R is a right divisor of the identity I of that semigroup. From II, 2.18, 5t is a group. The original element cannot be contained in .52, since either .52 = 0 or .52 is a left ideal of m: not containing I, while A u m:A contains I. Since A E m:I, A must belong to the group.R. From III, 1.15, it therefore follows that the element A is completely regular. 3.12. Taking 3.9 into account, we have a corollary of the theorem just obtained. COROLLARY. In order that every ideal in a semigroup m: should contain its identity it is necessary and sufficient that m: should be completely regular and that every nonempty set of its idempotents should contain its identity.
The structure of such semigroups and some of their properties were described in 3.9. In particular, it is evident that the second of the conditions of the corollary may be replaced by the requirement that all the idempotents of the semigroup should form a completely ordered set in which for any
'Y)
< g < p.
3.13. Among the semigroups we are considering there are those in particular which satisfy the stronger condition of the existence in each subsemigroup of an identity. THEOREM. In order that each subsemigroup of the semigroup m: should contain its identity it is necessary and sufficient that m: should be a periodic completely
330
DECOMPOSITION OF SEMIGROUPS INTO UNIONS
[CHAP.
VIII
regular semigroup in which every nonempty set of its idempotents contains its own identity.
m
In addition, every element A of must generate a monogenic subsemigroup [A], which must contain its identity. For this it is necessary that [A] should be finite. Thus the semigroup mmust be periodic. (2) Suppose that mis a periodic completely regular semigroup in which each nonempty set of its idempotents contains its own identity. From 3.9, (fJ), mis the union of a completely ordered set of groups
m= U (S~. Ii
Any subsemigroup lB of the semigroup lB
= U (S~,
(S~
mis equal to the union
= (Sii
n lB.
!;
Some of the sets (S~ may turn out to be empty. Suppose that (S; is the first nonempty set (such a set must exist, because of the complete ordering). Let X E (S;. Since (SV is a periodic group, [X] '3 Iv, where Iv is the identity of the group (Sv' By property 3.9, (8), Iv is the identity of Xv and is therefore also an identity of lB contained in 3::v•
4. Successively Annihilating Bands 4.1. A commutative band of a semigroup determines a homomorphism onto a commutative semigroup of idempotents (1.8; 1.12). Among such semigroups one may particularly distinguish semigroups adjoint to linearly ordered sets (II, 4.11). In this connection it is natural also to distinguish the corresponding class of commutative bands. The corresponding construction was considered by A. M. Kaufman [1] under the name of successively annihilating sum. The consideration of this construction makes it possible to study the structure of the so-called holoidal semigroups. It is natural to distinguish the class of these semigroups for their interest alone. But they also playa definite role in the study of linearly ordered groups, to which we shall devote our attention in § 3 of Chapter X. 4.2. DEFINITION. A partition of the semigroup minto subsemigroups is said to be a SUCCESSIVELY ANNIHILATING BAND if for any two distinct components lB and lB' of this decomposition either X1B'
is satisfied for each X
E
lB or else
is satisfied for each Y
E
lB'.
= lB'X=
X
YlB = lBY= Y
Sec. 4]
331
SUCCESSIVELY ANNIHILATING BANDS
Evidently, a successively annihilating band is indeed a band in the sense of the definition of 1.6, and in addition is commutative (1.12). 4.3. It follows immediately from the definition that the set of all components of a successively annihilating band is a linearly ordered set relative to the partial ordering relation defined in the following way: m m' if m= m' or
x consists of all right zeros of the element X. If X E 'll", then t> x is the set of all elements U E 'll such that XU = U,
Uxo
=
U.
5.8. LEMMA. If'll E II, X E Ill, while X has finite type (h, d) (III, 3.7), then X E 'll" and O(X) is a divisor of d. PROOF.
Suppose that XU = U. From Xh+ d = Xl/, it follows that X(UXlL)
Accordingly, X
E
=
UXll,
(UXlL)Xd
=
(UXh).
'll". Let XV= V,
Vxo = V.
We carry out a division with remainder, that is, d
= qo + r,
0
< r < o.
Since X· (VXlL) = VXll, (VXh). xr = vxrXlL = vxqo. xr. Xh = VXd. XlL
=
VXMd = VXlL
it follows from the definition of 0 = o(X) that r, which is less than 0, cannot differ from zero. Accordingly, d is divided by o. 5.9. LEMMA. PROOF.
If'll
E
II and X
(1) Suppose that X
E
X(U1 U2)
E
Ill, then t>x is a semigroup lying in II.
Ill'. If U1 , U2 E t>x, then
= (XU1)U2 =
U1 U2 '
i.e., U1 U2 E t>x. For U E t>x there must be in 'll a right zero V. But then XV= XUV= UV= V,
i.e., VEt> x' and accordingly U has a right zero in t> x. (2) Suppose that X E 'll". If U1> U2 E t>x, i.e.,
340
DECOMPOSITION OF SEMIGROUPS INTO UNIONS
[CHAP.
VIII
then i.e., U1 U2 E~x. For U E fyx there must be a right zero V in
m.
Since
= VU, = X· UV· U= UV· U= 6 (VU)X = V· Ux6 = vu, U(VU) X(VU)
vu,
VU is a right zero for the element U lying in fyx.
5.10. Suppose that [X] is a monogenic semigroup and that s is some natural number which is a divisor of d if [Xl is finite and has type (h, d) (III, 3.7). Observe that from xa = xa', X b = X b' and a == b (mod s) it follows that a' == b' (mod s). For an infinite semigroup this is trivial. For a finite [Xl it follows from the definition of type that a' == a (mod d), b == b' (mod d), and therefore a' == a (mod s), b' == b (mod s). We define in [Xl a relation ns by setting xa r-..I Xb (ns) if a == b (mod s). Evidently, ns is a two-sidedly stable equivalence. If [Xl is infinite, then by no we shall understand the identity relation in [X]. We observe that if s ,t. 0 the factor-semigroup [Xl/ns (VII, 2.4) is a group.
5.11. LEMMA. Suppose HE fyx. If X E m', then the equality Hxa = HXb is possible only if xa", X b (no), i.e., when xa = Xb. If X Em", then the equality Hxa = HX b holds if and only if xa Xb (na)' r-..I
PROOF. (1) Suppose that c = a - b > 0 and Hxa = HXb. Since evidently HXb E fyx, it follows from (HXb)XC = HXa = HX"
that X Em". Divide c by 15 with a remainder, c
= q15 + r,
0 .;;;; r
< 15,
and suppose that r ¥= O. We have
(HXb)
= (HXO)X
C
= (HXO)(xa)qxr= (HX")X'.
But since r < 15 this is impossible. Accordingly, r = 0 and c is divisible by 15, i.e., a == b (mod d) and xa,...., X b (na). (2) Suppose that xa ,....., X b (na) and a > b, i.e., a = q15 + b. Since H X 6 = H it follows that
Hxa = H(X~q . X"
= HX".
5.12. Suppose that 91 and !In are two nonintersecting semigroups and that m is a two-sidedly stable equivalence in In the set of all possible pairs ~N, M) (N E 91, ME !In) we define a relation n, putting (Nl • M 1),....., (N2' MJ (n) If Nl = N2 and Ml r-..I M2 (m). Evidently n is an equivalence. The corresponding
m.
Sec. 5]
341
BASIS CLASSES
n-class, containing the pair (N, M), will be denoted by (N, M)m. We denote by (91 x 9]()m the set consisting of all the elements of 9]( and of all the n-classes. In (91 X 9]()m we define an operation, as follows. The product of two elements of 9]( will be taken to be their product obtained in 9J( by means of the operation defined in the semigroup 9](. In the three remaining cases the operation is defined in the following way: (NI' MI)m . (N2' M 2)m = (NI N 2, M 2)m, (NI , MI)m . M2 = (NI , M I M 2)m, MI . (N2, M 2)m = (N2, M 2)m. Since m is two-sidedly stable in 9](, this operation is single-valued, i.e., the result of the operation does not depend on which of the pairs (N, M) was taken as a representative of the class (N, M)m. The associativity of the operation is verified without difficulty. The semigroup (91 x 9J()m is called a right annihilating product of the semigroups 91 and 9](. If 91 belongs to the class II, then also (91 x 9J()m belongs to II. Indeed, for ME 9)1 the right zero in (91 x 9)1)m will be (N', M')m, while for any N' E 91, M' E 9]( and for (N, M)m the right zero will be (N', M')m, where N'is a right zero of the element N in 91. 5.13. Suppose that in a semigroup we are given a monogenic subsemigroup [X] which is infinite or finite nonholoidal (i.e., having type (h, d), where d> 1) and a subsemigroup 91 all the elements of which are right zeros for X. In addition we suppose that in 91 for every N I , N2 E 91, NI ¢ N2 there always exists an No E 91 such that NINo ¢ N 2 N o. Suppose that the following condition is satisfied in Ill: NMI = NM2 (N E 91; MI , M2 E9J() always implies N'M1 = N'M2 for any N' E 91. Denote by m the relation in 9]( according to which Ml t"-.J M2 (m) holds if and only if NMI = NM2. Evidently m is a two-sidedly stabJe equivalence in 9](. Consider the following mapping fP of the semigroup (91 X 9J()m into ~r: fP(X lc) = X lc, fP(N, Xlo)m = NXlc (k = 1, 2, ... ).
9](
=
This is single-valued because of the property of m. We shall show that r:p is one-to-one. Suppose that X" = NXI. Since XN = N, we therefore obtain X"+l = XNXI = NXI = X", which is not possible in a nonholoidal monogenic semigroup. Suppose that NI X"l = N 2 X"'. Multiplying on the right by an arbitrary element N' E 91, we obtain NIN' = N 2N'. But the validity ofthis equality for any N' E 91, from the hypothesis made above, means that NI = N 2 • In this case Xlcl X k • (m), but then also (NI , Xlcl)m = (N2 , Xlc·)m· ro.I
342
DECOMPOSITION OF SEMIGROUPS INTO UNIONS
[CHAP.
VIII
From the fact that the elements of 91 are right zeros for the elements of 9J1 it immediately follows that the mapping rp is a homomorphism, and therefore also an isomorphism. Thus it turns out that the set of elements of which have the forms X k and NX k (N E 91; k = 1,2, ... ) is a subsemigroup isomorphic to (91 X 9J1)m'
m
5.14. Denote by 110 the class of semigroups consisting of all finite monogenic holoidal semigroups and of the semi groups isomorphic to the semigroups (53 X 9J1)m' where (1) 9J1 = [X] is a monogenic semigroup, infinite or finite holoidal; (2) Q3 is the unit semi group or the semigroup m(II, 5.12), or the semigroup m(II, 5.13); (3) m is a relation n8(s 0) of type 5.10. Since a finite monogenic semi group has a zero, and 53 in all three possible cases is contained in TI, it follows from 5.12 that each of the semigroups of the class 110 belongs to the class TI.
>-
mo
5.15. LEMMA. If m= (Q3 x [X])n belongs to the class TIo and is a is isomorphic to subsemigroup of it containing X and belonging to TIo, then 8
mo
m.
PROOF. ~Io cannot be a holoidal monogenic semigroup, since every monogenic subsemigroup of such a semigroup is itself holoidal, while contains a nonholoidal monogenic subsemigroup [Xl. Accordingly, is isomorphic to a semigroup of type
mo
mo
From the definition of a right annihilating product it immediately results that in mthe elements of [X], and only those, are not right zeros for any element whatever. In mo, containing [X], such elements will be elements of [X]. In iii' they will be the elements of [X]. Thus [X] and [X] are isomorphic, while under the isomorphism ofm: onto 'lIo the set [X] maps onto [X]. The relation ns is completely determined by the number of different elements in the set A . [X] (A E m\[X]), while this number is the same for any choice of A. The same holds also in m: for nr' Since m: is isomorphic to the subsemigroup of the semigroup m, we therefore conclude that the relation ns in [X] is identical with the relation TIr in [X], i.e., s = r. According to the definition of the class TI 0' each of the semigroups 53 and lB is a unit semigroup either equal to U or equal to m. If Q3 is the unit semigroup, and only in that case, all the right zeros of the subsemigroup [X] are idempotents. The same holds for ill relative to lB. Inasmuch as m: is isomorphic to a subsemigroup of it therefore follows that 53 is the unit semigroup if and only if 53 is the unit semigroup. Suppose that in mthe element P is a right zero for [X] and Q is a right zero for P. If 53 = U, then P = (U"f\ xa)n8 and Q = (U~2 Xb) ns ' where Cf.1 < Vw2· 1 C(2' Therefore there exists in a pair of elements p* E p. [X] and Q* E Q' [X]
mo
m
N
m
Sec. 5] (namely, P* zero for P*.
343
BASIS CLASSES
= (ug
1,
1
XC)n 8 and Q*
= (U,,'\ 2
XC)n)s such that Q* is a two-sided
If 513 = 'D, then no element of Q . [X] can be a two-sided zero for any element of p. [X]. (This follows from the fact that in 'D no element in general has a two-sided zero (II, 5.13).) Since ill: is isomorphic to the subsemigroup ~o of the semigroup ~ containing [X], and since there is in ~o a right zero P for [X] and a right zero Q for P, it follows from what has been said that if lB = U one cannot have lB = 'D, and if 513 = 'D one cannot have lB = U. ' We have sh0.2Yn that for ~ = (lB X [X])n, and = (lB x [X])n r the semigroups [X] and [X] are isomorphic and the relations n. and nrin them are identical in 513 = lB. Accordingly 'It is isomorphic to which is isomorphic to ~o.
m
m,
5.16. THEOREM.
The class ITo (5.14) is a basis class (5.1) for the class TI.
PROOF. (1) Suppose that Xis any element of the semigroup 'It belonging to the class IT. If [X] is holoidal, then [X] is a subsemigroup of the semigroup 'It containing X and lying in ITo. Suppose that [X] is not holoidal. Since i>x (5.7) belongs to IT (5.9), there exists in i>x, from II, 5.14, a subsemigroup lB which either is the unit semigroup, or is isomorphic to U, or is isomorphic to 'D. Denote by 'lto the set of all elements which belong to [X] or have the form BXk (B E lB; k = 1,2, ...). Each element of lB will be a right zero for X. As follows from the structure of U and 'D, for any Bl , B2 E lB with Bl ~ B2 there always exists a Bo E 513 such that BlBO ¢: B2Bo. If for some B E lB,
Bxa
= BXb,
xa
¢:
Xb,
then, from 5.11, xaf"oo.l Xb(n b) (15 = b(X)). But then we would also have for any B' E 513 that B' xa = B' Xb since B' X b = B'. The converse is also evident. If X E 'It" and a == b (mod b), i.e., xa f"oo.I X b(nb), then Bxa = BXb for any B E lB. In view of these properties we may apply the considerations of 5.13 to 'lto. If X E ~', then from 5.13 it follows that m coincides with no (5.10). If X E ~", then m coincides with nb (5.10). From 5.13, 'lto is isomorphic to the semigroup (513 X [X])m which belongs to the class TIo. Thus, for TIo we have proved that the property 5.1, (ex) is satisfied. (2) Suppose that the semi group 'It is represented in the form of a union of subsemigroups belonging to the class TIo. Each element A of'lt is contained in some subsemigroup ~o of the semigroup 'ltlying in TI 0 and therefore has a right zero since 'lto belongs to ITo (5.14). Thus property 5.1, «(3) holds {or ITo. (3) Suppose that the semigroup 'lto belongs to TI o' If'lto is a finite monogenic semigroup ~ = [X], then the element X is not contained in any of its subsemigroups distinct from ~o. Suppose that 'lto is isomorphic to the semigroup §fo = (513 X [X])n.. From 5.15 the element X of
344
[CHAP.
DECOMPOSITION OF SEMI GROUPS INTO UNIONS
VIII
mo is not contained in any subsemigroup of the semi group mo belonging to IIo and not isomorphic to mo. Accordingly, in \U O itself there is an element with the analogous property. From this it follows that the class IIo has property 5.2, (y'). Taking into account what was proved in the first part of the proof, we conclude from 5.2 that IIo is a basis class for II.
5.17. It should be noted that in the class II there is not just one basis class. All the basis classes of the class II were discovered and characterized by E. S. Ljapin [12]. Among them the class IIo considered above turned out to be the simplest in construction and the most convenient for study. The simple structure of the semigroups of the class II 0 makes it possible to regard the local structure (5.4) of the semigroups belonging to the class II as having been cleared up. 5.18. Of course, it is by no means true that every class of semigroups has a basis class. Indeed, a class having a basis class must evidently have the following property. It must contain every semigroup which is a union of semigroups contained in it. But many important classes of semigroups do not satisfy this condition. For example, the class of commutative semigroups does not. As a matter of fact, every semigroup is the union of certain of its commutative subsemigroups (for example, the monogenic subsemigroups), but not all semigroups are commutative. 5.19. In this connection, the following generalization of the concept of basis class is in some cases useful. Suppose that r 0' rand 2: are three classes of semigroups, where 2: contains the class r and the class r contains r o' One may say that the class r 0 is a basis class for the class r relative to the class 2: if conditions 5.1, (IX), 5.2, (y') are satisfied in addition to the following condition (P')· (fJ') Every semigroup of 2: which can be represented in the form of the union of subsemigroups belonging to the class 0 must belong to the class In the sense of this generalization, the class of all monogenic semigroups, taken relative to the class of all commutative semigroups, is a basis class for the class of all commutative semigroups. The class of all cyclic groups (II, 3.17) relative to the class of all groups is a basis class for the class of all groups. The class of all finite cyclic groups relative to the class of all groups is a basis class for the class of all periodic groups.
r
r.
5.20. The class of all inverse semigroups has no basis class in the sense of 5.1 since it evidently does not satisfy the property mentioned in 5.18. From II, 7.4 the class of all inverse semigroups is contained in the class of semigroups having' the property of commutativity of idempotents. L. M. Gluskin [8] showed that relative to the class of semigroups with commuting idempotents the class of all inverse semigroups has a basis class.
CHAPTER
IX
RELATIONS IN SEMI GROUPS 1. Defining Systems of Relations
1.1. We have already drawn attention to the fact that the definition of elements of a semigroup in the form of a product of elements of a certain generating set of the semigroup is in general not unique. Products which are different in appearance can often be equal, i.e., represent one and the same element of the semigroup. The consideration of the corresponding relations is of evident interest. In particular, what is essential is the singling out of systems of these relations from which all other relations follow as necessary results. It is possible to define semigroups by means of systems of such relations. A special role is played by the identities, i.e., the relations which are true for all elements of the semigroup. With the notion of an identity, there is connected the notion of a free semigroup in a class. The consideration of the corresponding problems is the aim of the present chapter. 1.2. Let 1n be an arbitrary nonempty set, which, in view of our further constructions we will call an alphabet. Any finite sequence of elements of 1n written in the form of a product
we will call a word in the alphabet 1n. The number of members in the word (n) we will call the length of the word W. A word of length one X (X E 1n) we will identify with the element X itself. We will denote the set of all words in mby 21391, The use of the term alphabet is borrowed from the theory of associative calculi, although there it is considered only for finite alphabets, while here m can be infinite. 1.3. For words in an alphabet mthere is defined an operation of attachment, which we will call multiplication of words, namely,
W
=
X 1 X 2 ••• XnXn+1 ••• X m,
W=UV. 345
346
[CHAP.
RELATIONS IN SEMIGROUPS
IX
Since the operation of multiplication of words is evidently associative, the set WjR consisting of all words in 91 is a semigroup with respect to this operation, which we will call the free semigroup over 91. Free semigroups over various alphabets will also be simply called free semigroups. 1.4. Let 91 be some subset of a semigroup
X I X2 ••• Xn
(Xl' X 2 ,
m:.
•• • ,
To any word in 91,
Xn Em),
there corresponds an element S E m: which is the product of the elements Xl' X 2 , ••• , Xn defined by the multiplication operation in ~r. This element S is called the value of the word X I X 2 • •• Xn in m:. Each element S from [91] can be represented in m: in the form of a product: S = XI X2
• ••
Xn
(Xl> X2 ,
••• ,
Xn Em).
Thus S is the value of some word in 91. The element S is completely defined by this word. 1.5. If in the semigroup m: an element S belongs to [91] (91 c m:), then it is evident that'for S there can exist different words in W 91 , each of which has the value S. The fact that in the semigroup m: the values of two words
in W91 are equal means that
X I X2 ••• Xn = Y1 Y2
•••
Ym
in m:. Such an equality is called a relation in m: with respect to in. If it is clear what set 91 is being discussed, the words "with respect to 91" will usually be omitted. It is necessary to keep in mind that by a relation with respect to 91 we understand a connection between certain words of WjR' Each of the two parts of a relation is defined not by that one element S from m: which is equal in m: to the value of the corresponding word, but by the word itself in W 91 , i.e., the sequence offactors forming the given product. Thus a relation in m: with respect to in is a pair (W, V) of words in W91 whose values in m: are equal. In view of this fact it would be more precise to write the relation not with the symbol of equality (since it is not at all the case that the two words Wand V are identical) but in some other way, for example, W ~ V or the like. But usually, in view of the equality of the values of the words in m:, the equality sign is used in writing the relation, and we will conform to this notation. In this connection, for equality of words in 91 as elements of WjR we will sometimes use the identity sign =, denoting identical, or, in other words, graphical equality of words in distinction from the equality of the values of these words in .m:. It is necessary to keep in mind that in the literature such distinctions in notation are sometimes neglected, on the principle that it is clear from the context which equality is being discussed.
Sec. 1]
347
DEFINING SYSTEMS OF RELA nONS
1.6. Let (n). 3.5. Now let there be chosen in an arbitrary semigroup 2{ some generating set.ft and let there be given some defining system cI> of relations with respect to .ft. Taking.ft as the initial alphabet, we construct the free semigroup over it W.!to In 5ill.!t we define the relation n by putting W'" Yen) (W, V E W~ ifand only if in 2{ W = V is a relation belonging to cI>. By Definitions 1.8 and 1;9 the words T and U in 5ill.!t have the same value in 2{ if and only if T,,-, U(n"). But in 5ill~ the equality T = Uholds ifand onlyifT '" U(n"). Thus, between the elements of2{
356
RELATIONS IN SEMIGROUPS
[CHAP.
IX
and the elements ofWNt a one-to-one correspondence is established in an obvious way. This correspondence has the property of a homomorphism and is therefore an isomorphism. Thus it follows that any semigroup, up to within isomorphism, can be given as
a semigroup over some alphabet defined by some defining relation. 3.6. Let us return now to the question raised at the beginning of this section. The role played here by the construction of a semigroup over an alphabet defined by a defining relation consists of the following. Let there be given some set 91 and a set 'If of formal equalities with respect to 91, i.e., a pair of words in 91 connected with an equality sign. It is asked whether it is possible to construct a semigroup such that (i) each element of 91 is an element of the semigroup, (ii) the elements of 91 form a generating set in the semigroup and (iii) the relations of'If are valid in this semigroup and form a defining system of relations with respect to 91. 3.7. For the answer to this problem we consider the free semigroup Wm over the alphabet 91. We shall take in it, as a defining relation, a relation n'F such that W"" V(n'F)( W, V E Wm> holds if and only if the pair of words Wand V is connected by an equality sign contained in the given system 'If of formal equalities. We assume that for two different elements X, Y E 91 we cannot have X,...., Y(n~). Then, constructing the semigroup W~'F, we identify each of its elements of the form X (X E 91) with the element X itself. In this semigroup the set 91, coinciding with ill in the above identification, is a generating set. All the formal equalities of 'If are valid relations. In this connection they form a defining system of relations with respect to 91, since the relation W = V (W, V E WgJ in W~'F holds if and only if W = P, i.e., if W V(n;). But the latter means that W = V is a corollary of '¥. f"'o.J
3.8. Now let X,...., Y(n';') hold for some X:¢ Y (X, Y Em). If 9't is a generating set of some semigroup, in which all the formal equalities of'Y are valid relations, then from these, as a corollary, follows the relation X = Y. Thus different elements of 91 must be different elements of such a semigroup. 3.9. From the arguments of 3.7 and 3~8 follows the complete answer to the problem posed in 3.6. If the system of formal equalities '¥ is such that for the relation n'F (3.7) defined by it the inequality X:¢ Y (X, Y E 91) always implies X r-J- Y(n~), and only if this condition holds, then there exists a semigroup m: for which 91 is a generating set and '¥ is a defining system of relations with respect ~~
.
3.10. In connection with this result we naturally make the following convention, which is always made in such cases. Let 91 be a set and let 0/ be a system of formal equalities of certain words in W!jl' The semigroup W~'F, which can also be denoted by '¥m, is called the semigroup given by the defining system oj relations 0/ (or by the defining relation n'F)' Its elements are classes of words
Sec. 3]
SEMIGROUPS GIVEN BY DEFINING RELATIONS
357
in 91 which are equivalent to each other with respect to n~.. The class X, where X E 91, is identified with the element X itself. If for any two distinct elements X, Y E 91 we have X = Y (i.e., X "'" Y(n~.,)), then X and Y are regarded simply as different symbols for one and the same element of the semigroup W~':p'. As we have already shown in (3.2), the set 91, and consequently 91, if the corresponding identifications are made, is a generating set in W~':I'", and 0/, coinciding with ¢(no/) (3.4) is a defining system of relations with respect to this generating set. 3.11. From the given relation n in Wm, or, what is the same thing, from the form of the formal equalities 'Y', it is not immediately evident when two words from 91 have the same value in W~b i.e., represent one and the same element of that semigroup. The elucidation of this problem for each concrete semigroup W;i\ is not only of great practical value for the investigation of its properties, but it has great theoretical importance in view of the fact that as long as it is not settled, we have no clear idea of the nature of the given semigroup, i.e., of the set of its elements. However, the solution of this problem involves profound theoretical difficulties. The problem can be formulated more precisely for the case when the alphabet 91 is finite and the defining relation n in W 9l is given as a finite number of pairs of words in 91, lying in the relation n (or, what is the same thing, the semigroup is given by a finite number of relations). In this case we may pose the problem of finding a general algorithm by means of which for any pair of words Wand V of W~l we can decide whether or not they are in the relation nil (the second derived relation with respect to n). But this last is equivalent to a solution of the problem of whether or not the words Wand V are one and the same element of the given semigroup. The problem of the construction of such an algorithm is usually called the problem of the identity of words of the given semigroup. In some cases the solution of this problem, i.e., the finding of such an algorithm, turns out to be possible. However, as was shown for the first time by A. A. Markov [1], there exist semigroups over finite alphabets, defined by a finite number of relations, in which such a general algorithm as a rule does not exist. Such semigroups were constructed by A. A. Markov [1; 6] and later by some other mathematicians, for example Post [1] and Kolmar [1]. In some cases the external form of the corresponding semigroups can appear quite simple. An example is the semigroup found by G. S. Ceitin [1], which is constructed over the alphabet consisting of the five elements {A, B, C, D, E} and given by the system of the following seven relations:
AC= CA,
AD = DA, BC= CB, EDB=BE, ECA = AE, ABAC = ABACE.
BD
=
DB,
For this and other semigroups, the proof of the fact that the problem of identity is insolvable is very difficult and will not be taken up here. By their very
358
RELATIONS IN SEMIGROUPS
[CHAP.
IX
character, the corresponding problems belong to a self-contained branch of mathematics, namely, the theory of algorithms (cf., for example, the book by A. A. Markov [6]). 3.12. It must be noted that for a semigroup over a finite alphabet, given by a finite number of relations, other algorithmic problems naturally arise. An example is the very important algorithmic problem concerned with the question of divisibility, in which it is required to construct an algorithm by means of which for any two words Wand V over a given alphabet it is possible to decide whether there exists a third word U over that alphabet such that WU,...., VenIt).
Evidently this is the problem of constructing an algorithm by means of which it would be possible to decide when one of two elements of a semigroup defined by words in a given alphabet is a left divisor of the other (and analogously for right divisibility). As was proven by A. A. Markov [1; 6], for certain semigroups such an algorithm is not possible (cf. also the work of S. 1. Adjan [1]). 3.13. It must be emphasized that the impossibility of an algorithm for the problem of identity or the problem of divisibility in these or other semigroups shows that there does not exist any general algorithm for the solution of the problem for any pair of words in the given alphabet. But this does not at all mean that there exists at least one particular pair of words for which the problem cannot be solved in principle. From the proof of the impossibility of showing for a particular pair of words Wand V that they represent one and the same element, it would evidently follow that for these words we could not find a chain of words such that W,...., V(n"). But this would mean that such a chain does not exist, i.e., that W 1-' Ven"). Consequently this would mean the solution of our problem. The results mentioned above only show the absence in some cases of a single general algorithm, which would solve the problem for the entire infinite system of pairs of words in the given alphabet. 3.14. In conclusion it is necessary to note the close connection of the theory of semigroups defined over a given alphabet by means of a defining relation with the theory of associative computations (the latter is confined to the finite case in the sense indicated above). The theory of associative computations can essentially be considered as a constructive approach to the theory of such semigroups. Let us point out that a very precise exposition of the foundations of this theory and the results mentioned above about the impossibility of certain algorithms is given in the book of A. A. Markov [6]. In this book, as in many other works on this problem, the above results on the impossibility of certain algorithms are formulated as corresponding assertions about various associative computations.
Sec. 4]
359
IDENTITIES IN SEMIGROUPS
4. Identities in Semigroups 4.1. Let T and T' be two words in a countable alphabet :2: = {c;l' c;2' ... }. A relation connecting the words rand T' by the sign "'"' (sometimes the identity sign == is used, but we have already used it for other purposes), T"'-'T',
is called an identity in the semigroup III if, for any mapping cp of :2: into Ill, the values of the words cp(T) and cp(T') are equal. In other words, T"'-' T' is transformed, by the substitution of any of the elements of III for the elements of :2: in the words T and T', into an equality in Ill. It is evident that the choice of elements of the alphabet :2: does not affect the problem of identities in the semigroup. In particular, from the validity in III of some identity T "'-' T' follows the validity of any identity To "" T~ obtained from T ~ T' by a replacement of the elements of:2: in that identity by other elements of the same alphabet. In this case we can consider that T T' and To "'"' T~ are one and the same identity. It is possible to consider simultaneously several identities in a semigroup. We assume that we start from one and the same alphabet. Since in each of these identities there appear only a finite number of elements of the alphabet, but the number involved (for all the identities) can be unbounded, we must therefore, in order to consider all these identities, take the alphabet :2: to be countable. ."-.J
4.2.
In any commutative semigroup we have the identity c;l~2 ~ ~2~1'
The existence ofthis identity in a semigroup is the condition of its commutativity. A semigroup of idempotents is characterized by the identity ~r
"'"' ~l'
The unit semigroup, i.e., the semigroup consisting of one element, is characterized by the identity ~l ~ c;2'
We have often drawn attention to the semigroup in which the product of any two elements is equal to the left factor. Such a semigroup is defined to within isomorphism by the cardinality of its elements. The class of such semigroups is evidently characterized by the identity ~1';2 ~ ~l'
Analogously the identity ';1~2"'-'~2
characterizes the class of semigroups in which the product is always equal to the right factor.
RELATIONS IN SEMIGROUPS
360
[CHAP.
IX
4.3. It is possible to characterize each identity T.::::::: T' by the sum of the component lengths of its words T and T'. For the identities in which this number is equal to two, there will be the identity ~l ' " ~2' which we have already considered, and the identity ~l = ~l> which is trivial and true in any semigroup. In addition to the identities considered in 4.2, in which the sum of the component lengths of their words is no more than three, only two other such identi ties are possible. The identity ~1~2
'"
~3
holds only in the unit semigroup, since for any elements X and Y of the semigroup we have
xx=x,
xX= Y.
Analogously the identity is satisfied only in the unit semi group. But for an increase of the sum of the lengths there are new nontrivial identities. 4.4. A semigroup 'll, in which for some natural number n we have the two identities is a group, and the order of each of its elements is a divisor of n. In fact, for any X E 'll the element xn is a left unit of'll. Here each element A E III has the twosided inverse An-l with respect to the unit An = xn. Conversely, in any group, the elements of which have orders which divide n, both of the above identities evidently hold. An important problem in the theory of groups is the so-called Burnside problem, on whether there exist infinite groups having a finite generating set, in which the order of each of the elements is a divisor of one and the same number n. For n < 4, it is known that such infinite groups do not exist. For n ~ 72, P. S. Novikov 2 proved the existence of such infinite groups. As was shown in the work of Green and Rees [1], the Burnside problem in the theory of groups is equivalent to the following problem in the theory of semigroups: does there exist an infinite semigroup with a finite generating set in which the identity ~+1 gl holds. In one direction the connection between the two problems is evident. If for some n, any semigroup with the indicated identity and with a finite generating set is finite, then from this would follow at once the finiteness of all groups having finite generating sets, the order of whose elements is a divisor of n. This follows from the fact that the identities ~~ r v g2' ~g2 ,. . . , ~2 evidently imply the identity '"'-J
~~+1~~1' 2
On periodiC groups, Dokl. Akad. Nauk SSSR 127 (1959), 749-752. (Russian)
Sec. 4]
IDENTITIES IN SEMIGROUPS
361
The converse is far from evident. The proof of the fact that from the existence of an infinite semigroup with finite generating set, satisfying the identity ;~+1 ,. . . . ;1' follows the existence of an infinite semigroup with finite generating set, satisfying the identities ;~ :::::::;~, ;~;2::::::: ;1' is not simple, and we will not give it here. 4.5. Using the concept of an identity, A. 1. Mal' cev [6] was able to introduce into the theory of semigroups the notion of nilpotence, widely used in the theory of groups. 3 We can approach the concept of nilpotence in the theory of groups from various points of view. For example, we. can use the following inductive definition. A commutative group is said to be I-power nilpotent. A group (\) is said to be n-power nilpotent (n = 2, 3, 4, ... ) if the factor group by its center (\)/3 is (n - I)-power nilpotent. 4.6. For an alphabet 3 and Vn in 3. We put
= {;1' ;2, ...} we define by induction the words Wn
Further, we put Wn
==
(n = 1,2, 3, ... ).
W n- 1;n+2 Vn-1'
According to A. 1. Mal'cev, a semi group we have the identity
12( is
said to be n-power nilpotent ifin it
4.7. We note that evidently any n-power nilpotent semigroup will necessarily be k-power nilpotent for any k n. Commutative semigroups are all I-power nilpotent. However, in the class of I-power nilpotent semigroups there are some noncommutative semigroups. In fact, the class of I-power nilpotent semigroups evidently includes all semigroups in which the identity
>
;1;2;3::::::: ;4;5;6
holds. An example of such a semigroup is the set of elements which consist of 0 and words of length one and two in some alphabet 91. The operation of multiplication consists of attachment of words. If one of the factors is 0 or if as the result of attachment we obtain a word of length greater than two, then the product is considered to be O. In such a semigroup the indicated identity clearly holds, because the product of three factors is always O. If the alphabet 91 contains more than one element, then the semigroup is noncommutative, since for all X, Y E 91 (X ~ Y) the product of X by Y is the word XY, and the product of Yby Xis the word YX, where by definition these words are different elements of the semigroup under consideration. S Cf., for example, A. G. Kuros, The theory of groups, 2nd ed., GlTTL, Moscow, 1953; § 62 (Russian); English transl., Chelsea, New York, 1955, 1956.
362
[CHAP. IX
RELATIONS IN SEMIGROUPS
4.8. The value of the concept defined above lies in the fact that it is a direct generalization of the corresponding concept for the theory of groups. THEOREM. A group is n-power nilpotent in the sense of 4.5 n-power nilpotent semigroup (4.6).
if and only if it is an
PROOF. (1) We prove by induction on n that a group (fi which is n-power nilpotent in the sense of 4.5 is an n-power nilpotent semigroup. For n = 1, (fi is commutative and is therefore a I-power nilpotent semigroup. Suppose n > 1. We denote by cp the natural homomorphism of (fi onto the factor group of (fi by its center: (fi/3· We take arbitrary elements Xl' X 2, ... , X n+1' X n+ 2 from 05. We denote by W~ and V~ the words in 05 obtained from the words Wk and Vk in :3 (4.6) by replacing ~i with Xi' Using the inductive assumption, we can consider that in cp«(fi) = 05/3 we have the relation
CP(W~_I)
=
CP(V~_I)'
By the property of the factor group this means that in (fi we have
From this we obtain equalities in 05 for the values of the words in (fi, namely, W:
= W:_IXn+2V~_1 = V:_IZXn+2Vn_1 = V~_IXn+2V~_IZ = V:_IXn+2W~_1 =
The fulfilment of these equalities for arbitrary Xl' X 2 , that in 05 we have the identity
V:.
••• ,
X n+ 2 in (fi means
i.e., 05 is an n-power nilpotent semigroup. (2) Now we prove by induction on n that a group 05 which is an n-power nilpotent semigroup will be n-power nilpotent in the sense of 4.5. If n = 1, then the equality XIE(fiX2
=
X 2E(fiXI ,
obtained from the identity WI VI by replacing ~I with Xl' ~2 with X 2 , and ~3 with E(fi, where Xl and X 2 are arbitrary elements of 05, shows that 05 is commutative. Let n > 1. Using the notations of the first part, we obtain from the identity Wn Vn> taking as X n+2 the identity E(fi' the relation r..J
r..J
W~_l V~_l
=
V~-l W~_l
in 05. For arbitrary X n+ 2 it follows from Wn
r..J
Vn that
W~-IXn+2V~_1 = V~-lXn+2W~_I'
Sec. 4]
IDENTITIES IN SEMI GROUPS
363
Thus we obtain (V~=i W~_1)Xn+2
= Xn+2(W~_1 V~=D = Xn+2(V~,=t V~-l W~-l V~=D = Xn+2(V~=i W~_l V~_l V~=D= Xn+2(V~=i W~_l)'
But this means that (V~=i W~_l)
E
3, and therefore in (J)/3 we obtain
A
=
X I X 2 ••• X n •
Finally, we show that there do not exist nontrivial relations with respect to the generating set R Let
X I X 2 • ~. Xn
=
Y1 Y2 ··· Ym
(Xl' X 2,.··, X n, YI ,.··, Y m E.R).
If we had Xl ¢ YI , then by (y) either Xl would have to be divisible from the left by YI , or Yl would have to be divisible from the left by Xl' both of which are impossible since Xl, Yl E R But then, by (/3),
X2",Xn= Y2 ••• YmRepeating the argument, we obtain X 2 = Y2 , etc., so that finally we see that the initial relation was the identity. The fact that there do not exist nontrivial relations with respect to 5t shows that 5t is a free set with respect to the class of all semigroups (5.4). Since [5t] = ~, it follows that ~ is free in the class of all semigroups. 5.8. We will say that an identity is valid in a class of semigroups if it is valid in each of the semigroups of that class. Let r be a class of semigroups which is closed with respect to isomorphisms (i.e., r together with each of its semigroups contains every semigroup isomorphic to it) and let $ be a system of identities in the class such that all the other identities of that class are corollaries of $ (4.10). If the semigroup m3~ (4.11) over some alphabet 91, given by the system of identities $, belongs to the class r, then it is free in that class. In fact, 91 is a generating set for m3~. The relations with respect to 91 induced by the identities of c,p (4.9) form a defining system in m3~ with respect to 91 (4.9; 4.11). Since for any map of 91 into an arbitrary semigroup ~ E r these relations are mapped onto valid relations in ~, it follows from 1.11 that this map extends to a homomorphism of [91] = m3~ into ~. We remark that if 911 and 91 2 have the same cardinality, then for two sets of identities $1 .and $2 of the type in question the semigroups m3~: and m3~: are isomorphic.
370
RELATIONS IN SEMIGROUPS
[CHAP. IX
5.9. The free semigroups in the considered class r are exhausted by the semigroups W~ and semigroups isomorphic to them. In fact, let m= [St] be a free semi group in r and let St be a free set with respect to r. By 5.4 there are no relations with respect to St besides those induced by the identities which are valid in every semigroup ofr, i.e., identities which are corollaries ofC]). We consider the semigroup W~. By the definition of this semigroup, from Theorem (1.12) immediately follows the existence of an isomorphism between mand W~. 5.10. The arguments in 5.8 and 5.9 give a necessary and sufficient condition for a class of semigroups which is closed with respect to isomorphisms to have free semi groups. As we have already remarked, the role of free semigroups in a class is defined by the possibility of obtaining from them by means of homomorphisms the remaining semigroups of that class. We examine this problem in detail. Let Cj) be a system of identities in an alphabet 3. Let 9JC be a nonempty class of cardinals such that if the cardinal number n is less than the cardinal number mE ill1, then n itself must belong to ill1. We denote by r~ the class of all semigroups in which all the identities of C]) are valid and which have generating sets whose cardinal numbers belong to ill1. Of course any such class is nonempty, since it always contains the unit semigroup. r~ always has the following three properties: (0;) r~ is closed with respect to homomorphisms, i.e., a homomorphic image of any semigroup of r~ always belongs to r~. (fJ) There exist free semigroups in r:%\. (y) Each semigroup in r~ is the homomorphic image of a free semigroup in r~. The validity of (0;) follows from the fact that under a homomorphism of a semigroup onto a semigroup a generating set is carried onto a generating set and any relation is carried onto a valid relation. (fJ) follows from 5.8. Let mbe a semigroup in r~. Let St be a generating set of III whose cardinality belongs to m. The semigroup W~ belongs to the class r~ and is free in it (5.8). By 5.1 the identity mapping of St extends to a homomorphism of W~ into m. In this case the homomorphic image must coincide with all of Ill. 5.11. Properties 5.10, (oc), (fJ), (y) show that the class r~ (5.10) consists of all semi groups obtained by means of homomorphisms from the free semi groups in that class. It turns out that no other class which is closed with respect to homomorphisms has these properties. If some class of semigroups 2: has the properties (oc), (fJ), (y) formulated in 5.10 for r~, then 2: is one of the classes r~ (5.10). We denote by mthe class of cardinals m such that there exists a free semigroup in ~ having a generating set which is free with respect to 2: and whose
Sec. 5]
FREE SEMIGROUPS
371
cardinality is not less than m. By +l X 7>+2' ••
X 7J+q
== X7l+q+1 X 7J+Q+2 ••• X7J+2q
is impossible. However, it turns out that such a form of a given element is indeed not yet canonical. What is wrong is that although the semigroup [13~ itself is finite, for a number of elements greater than two the set of words in 91 satisfying the above condition turns out to be infinite. This circumstance was discovered by S. E. Arson. 4 It follows also from results in the work of Morse and Hedlund [1].
6. Determination of Free Semigroups by the Subsemigroup Characteristic 6.1. To the properties examined in the preceding section of semigroups which are free in the class of all semigroups, we add another important property. Namely, we show that these semigroups are defined by the subsemigroup characteristic (III,7.6). This concept and some of the ideas connected with it were examined in § 7 of the third chapter. • S. E. Arson, Proof of the existence of n-valued infinite asymmetric sequences, Mat. Sb. (N.S.) 2 (1937), 769-779. (Russian)
Sec. 6]
DETERMiNATION OF FREE SEMIGROUPS
375
Because of the results obtained in 5.5, in what follows we can consider the free semigroup sro 91 over some alphabet. The theorem about the definability of free semi groups by the semigroup characteristic is due to R. V. Petropavlovskaja [2]. 6.2. For the elements Xand Yofasemigroupm we call the element PE [X, Y] their special product (R. V. Petropavlovskaja called P the maximal element for X and Y), if for any Z which is an empty symbol or an element of [X, Y]\{X, Y, P}, in the semigroup ~(m) the element [P] is not a unit for [X] 0 [Z] 0 [y2] 0 [yS] or for [Y] 0 [Z] 0 [X2] 0 [XS], i.e., in other words, in m
PE [X, Z, y2, ys, ... J U [Y, Z, X2, Xs, ... ].
By the way, we mention (in what follows we will constantly make use ofthis) that for any element S it is always true that [S2] 0 [SS]
=
[S2, SS]
= {S2, Sa, S4, S5, ... }.
From the definition it immediately follows that the special product of X and Y is also the special product of Yand X. The special product P of the elements X and Y of the semigroup m will be their special product with respect to any supersemigroup of m and any of its subsemigroups containing X and Y. 6.3.
The terminology introduced above is justified by the following property.
LEMMA. P= yx.
If P
is the special product of X, Y E m, then either P
=
XY or
PROOF. (1) We suppose that XY = X. Then evidently any element of [X, YJ has the form ynxm m = 0, 1, 2, ... ; Xo,yo denote the empty symbol). Let P = P xq. If it were true that p > 1, then we would have
en,
P = yP xq E [X, y2, ys, ... ],
which would contradict the definition of special product. Analogously we see that q > 1 is impossible. Since P ¥ X and P ¥ Y, it follows from the above that in the case considered, p= yx. (2) Analogously we consider the cases XY = Y, YX = X, YX = Y. (3) Since it is evident that X" yfl E [X, XY, y2, Y3] (a:, fJ = 1,2, ... ), it follows in the case where P has the form X"l yf3 1 X"2 yfl2 ... X'" y.Bs or X"l y.B 1 X"2 yfl2 .. , X"s (a: i , fJi = 1,2, ... ), that P E [X, XY, y2, ys, ... J.
Taking into account the definition of special product, we conclude from this that XY E {X, Y, P}, i.e., either P = XY or we have one of the two cases already discussed: XY = X, Xy = Y. Analogously we consider the case where in the expression for P in the form of a product of X and Y, the first factor is equal to Y.
376
RELATIONS IN SEMIGROUPS
6.4.
COROLLARY.
[CHAP. IX
For two elements there exist not more than two special
products.
6.5. It is understood that the product XY is not always equal to the special product of X and Y. One sufficient condition for this is the requirement that XY cannot be represented in a form different from XY and YX as a product of factors equal to X or Y. In fact, for any Z which belongs to [X, Yj\{X, Y, XY} or is the empty symbol, any element of [X, Z, y2, y3, ... J U [Y, Z, X2, X3, ... J
is evidently different from XY because of the indicated condition. 6.6. For the problem of interest to us here, concerning isomorphisms of subsemigroup characteristics, the concept of special product turns out to be essential because of the following. THEOREM. Let every element Xl' Y l , PI oj the semigroup WI have infinite type, where PI is the special product oj Xl and Yv andjor some semigroup W2 suppose that there exists an isomorphism qJ OjZ:(W1) onto Z:(W 2). Then there exist unique elements X 2, Y 2, P 2 in 'R: 2 such that
where P 2 is the special product oj X 2 and Y 2' PROOF. The existence of the unique elements X 2, Y 2 , P 2 connected with Xl' YI , PI in the indicated way follows from III, 7.10, (y). We suppose that P 2 is not the special product of X 2 and Y 2 in W2 • Then for some Z2' which is the
empty symbol or an element of [X2' Y2J\{X2 , Y2 , P}, we must have [P2J c [X2 , Z2,
yg,
Yl, ... J
u [Y2, Z2'
xg, Xl, ...J.
But then, applying the isomorphism qJ-l and taking III, 7.10, (8), minto account, we obtain a relation which contradicts the fact that PI is the special product of Xl and Yl , namely, [PI]
C
[Xl' Z1' Yf, Yr, ... ] U [Yl , Zl'
Xr, Xi, ... ].
Here Zl is the empty symbol if Z2 is the empty symbol. But if Z2 belonged to [X2 , Y 2 ]\{X2 , Y 2, P 2}, then Zl is an element such that qJ[Zd = [Z2J. By III, 7.10, (8), ZI belongs to [Xl' Y1 ]· Since [Z2] =/:- [X2 ], [Z2J =/:- [Y2 ], [Z2] =/:- [P 2 ], it follows that Zl ~ Xl' Zl ~ Y1 , Zl ~ Pl' 6.7. We consider the problem of special products of elements of free semigroups. We, recall (we will repeatedly need this) that in a free semigroup two elements permute if and only if they are both contained in some singly generated semigroup, i.e., they are powers of some third element (5.6, (8».
DETERMINATION OF FREE SEMIGROUPS
Sec. 6]
377
LEMMA. If the elements V and W of the free semigroup Will do not permute, then both VWand WVare special products of V and W. PROOF. Since Will is evidently a semigroup with two-sided cancellation in which none of the elements have left or right units, from an equality of the form VW = vn we would conclude that n > 1 and W = vn-l, i.e., V and W would permute. Also VW = wn is impossible. Let VW be represented in the form of a product of factors equal to Vand W. By the above, Vand W must occur in this product. We assume that the number of factors in this product is greater than two. V and Ware words in 91. So the word VW, with length equal to the sum of the lengths of the words Vand W, turns out to be equal to a word with length greater than the sum of the lengths of Vand W. But this is impossible by the definition of a free semigroup over an alphabet 91. From what was said it follows that VW cannot be represented in the form of a product in the indicated way, except for VWor WV. But then, according to 6.5, VW is the special product of V and W. The argument for WV is similar. 6.8. We can now proceed directly to consider isomorphisms of subsemigroup characters of free semigroups. Let Wm be a free semigroup over 91 and let U be a semigroup for which there exists an isomorphism cp of 2::(Wm) onto 2::(U). For any V E Wm the singly generated semigroup [V] is infinite. From III, 7.10, (y), it follows that cp[V] must be an infinite singly generated subsemigroup of U, that is, cp[V] = [S] c U. The element S, generating the infinite singly generated semigroup [S], is uniquely defined by this semigroup. We set S = V. From the fact that all the elements of Will have infinite type it follows that in U the types of all the elements are infinite (III, 7.9). For any T E U under the isomorphism cp the subsemigroup [T] must be the image of some uniquely defined singly generated subsemigroup [Q] C Will' i.e., T = Q. Thus the isomorphism cp of 2::(W91) onto 2::(U) defines a one-one transformation 'IfJ of the set of all elements of Wm onto the set of all elements of U 'IfJ(V) = V
m,
We note that, by III, 7.10, cp[V] = [V] (VEW m) implies cp(vn) for all natural numbers n. This means that
vn 6.9.
=
Vn.
In the free semigroup Wm we define a relation n by setting
V,-..,; Wen) if in U we have
VW= Vw.
(Thus the relation n depends on U and cp.)
=
[Vn]
RELATIONS IN SEMIGROUPS
378
[CHAP. IX
Analogously we define the relation m in W9l by setting V,-...; W(m)
(V, WE
W91),
if
VW= WV. Evidently the one-one transformation 'IjJ of W91 onto U (6.8) pairing off each V E Wm with 'IjJ(V) = V E U will be an isomorphism if and only if any two elements of W91 are related to each other with respect to n. 'IjJ will be an antiisomorphism (I, 1.17) if and only if any two elements of Wm are related to each other with respect to m. In what follows we will make use, without special mention of the fact, of the notations and results introduced here and in 6.8. Also we will constantly use the uniqueness of the expressions for elements of W 9l in the form of products of elements of 91.
6.10. We note a few properties of the relations nand m. (0:) If the elements Vand W of Wm are permutable, then V I ' - ' Wen),
V I ' - ' W(m).
In fact, Vand W must both be contained in some singly generated semigroup (IX, 5.6, (8» V, WE [U], By 6.8, VW= WV= UPH = (jPH , from which follows
VW= VW= WV. (f3) nand m are reflexive relations. This immediately follows from (0:), since each of the elements permutes with itself. (y) Any two elements of W91 are related with respect to n or with respect to m. If Vand W permute, then our assertion follows from (0:). If Vand W do not permute, then VW, by 6.7, is the special product of V and W. From this, because of 6.6, it follows that VW must be the special product of P and W in U, i.e., we must have VW = VW or VW = WV. In the first case we have V Wen), in the second V,-...; W(m). (15) nand m are symmetric relations. In the case where Vand W (V, WE W m) permute, this immediately follows from (0:). Suppose Vand Wdonotpermute. By 6.7, VWisthespecialproduct of Vand W. By 6.6, WV must be the special product of Wand P, i.e., in U we must have either WV = wP or WV = VW. I'-'
Sec. 6]
DETERMINATION OF FREE SEMI GROUPS
379
If V,......, Wen), then VW = VWand therefore WV:;If VW, since VW:;If WV. Consequently, WV = WV, i.e., W,....., V(n). If V,......, W(m) , then VW = WV and therefore WV:;If WV. Therefore WV = VW, i.e., W,....., V(m). (8) If V and W do not permute, then V,....., Wen) and V,....., W(m) cannot occur simultaneously. Indeed, if V,......, Wen), then by (0), W,....., V(n). Therefore WV = WV. Since VW:;If WV, it follows that VW cannot equal WV, i.e., V'" W(m) is impossible. 6.11.
If X, Y Em (X:;If
LEMMA.
Y) and X'" Yen), then
xn,...., yen)
for all positive integers n.
If n
PROOF.
xn Y
=
> 1 and xn-1,...., XY(n), then
xn-1 . XY
=
We suppose that for some n xn
xn-1. XY
=
xn-1 . X' Y
=
xn Y
=
xn Y,
=
xn Y.
> 1 we have
+ Y(n) ,
X,,-l
1-- XY(n).
Then by 6.10, (y), xn,......, Y(m) ,
xn-1,...." XY(m),
and we obtain Xy· xn-1
=
xn-1 . XY
=
xn-1 . X· Y
y. xn = xn. Y = xn y,
which contradicts the fact that xyxn-1:;1f yxn. 6.12.
LEMMA.
If X,
Y
Em (X:;If
Y) and X,...., Yen), then
XY,...." Wen)
for all W
E
[l\ll'
If XY and W permute, then our assertion follows from 6.10, (a). Suppose XYand W do not permute. We suppose that XY rf.; Wen), i.e., by 6.10 (y), Xy,...., W(m). It will be necessary for us to consider six different cases, in each of which our assumption will reduce to a contradiction. In our arguments we will constantly use the properties of 6.10. PROOF.
(1) y,...., Wen), X,...., YW(n).
In this case we obtain W· XY= Xy· W= XYW, X· YW= Xy· W= XYW,
which contradicts the fact that WXY:;If XYW.
380
[CHAP. IX
RELATIONS IN SEMIGROUPS
(2) Y r-...; Wen), X r-.I YW(m). Then W· XY= Xy· W= X'YW,
YW' X= X'. YW= X'Yw,
and this contradicts the fact that WXY rE YWX. (3) Y""' W(m), X YW(n), YX r-.I Wen). Then YW·X= yw·X= wYX, f'J
W· YX= W· YX= wYX,
from which follows YWX = WYX, and therefore YW = WY, since W91 has two-sided cancellation. But since YW = WY, we obtain Y Wen), i.e., we arrive at one of the first two cases. (4) Y W(m), X r-.I YW(n), YX r-.I W(m). Then YX' W= WYX= wYX, f'J
f'J
YW·X= YW·X= WYX,
from which follows YXW = YWX and therefore XW = WX. Since X E 91 does not belong to any singly generated subsemigroup of Wm other than [X], it follows that WE [X], i.e., W = Xk. Using 6.11 we obtain Xk+1 Y
=
Xk+l Y= X'k.-j-l Y= X'k X'Y = WXY = XYW = XYXk,
which contradicts the fact that Xk+l Y rE XYXk. (5) Y""' W(m), X r-...; YW(m), X'-' WY(n). Then W· XY= XYW= X'YW, X· WY= XWY= X'Yw,
from which follows WXY = XWY, i.e., WX = XW. Repeating the arguments of the preceding case we arrive at a contradiction. (6) y,-, W(m), X'-' YW(ni), X'-' WY(m). Then W·XY= XYW= XYW, WY·X= X'WY= X'YW. This contradicts the fact that WXY rE WYX.
6.13. Using the properties obtained for the above relations in free semigroups, we are able, finally, to show that a free semigroup is defined by its subsemigroup characteristic.
Sec. 6]
DETERMINATION OF FREE SEMIGROUPS
381
THEOREM. Let U be a semigroup in which the subsemigroup characteristic 2:(U) is isomorphic to the subsemigroup characteristic 2:(213 91 ) of the free semigroup 213 91 , Then U and 213 91 are isomorphic. PROOF. (1) If the alphabet 9l consists of a single element, then 213 91 is an infinite singly generated semigroup and the truth of the assertion of the theorem in this case immediately follows from III, 7.9. So in what follows we can assume that 9l contains at least two distinct elements X, Y E 9l eX;;t!:: Y). Let cp be the isomorphism of 2:(213 91) onto 2:(U). As was shown in 6.8, cp induces a one-one transformation 7.p of 213 91 onto U, 7.p(W) = W
by means of which in 21391 the two relations nand m (6.9) are defined. (2) First we consider the case where X,...,., Yen). Let V and W be any two elements of 213m. We assume that V,...,., W(m). Applying 6.12 we obtain Xy· VW= Xy· VW= XYWV.
If XYW,....", Yen), then again using 6.12 we obtain XYW' V
=
XYWV
=
XYWV = xYWV.
But XYVW = XYWV implies XYVW = XYWV and since 21391 has twosided cancellation, VW = WV. From this, on the basis of 6.10 (oc), we conclude that V"-' Wen). The case where XYV,....", Wen) is completely analogous and we obtain V"-' Wen). If neither XYW N Yen) nor XYV,....", Wen) holds, then by 6.10, (y), XYW,,-, V(m) and XYV,...,., W(m). In this case, using 6.12 we obtain V· XYW
=
XYWV
=
XYWV
=
xYWV.
This would mean that VXYW = XYVW, and hence VXYW = XYVW, and since 213m has two-sided cancellation, VXY = XYV. Since V and XY turn out to permute, they must lie in some singly generated subsemigroup of 213 91 , But the only singly generated sub semi group W91 containing XY is [XY]. Therefore V = (XY)P. In a completely analogous way we see that in this case W = (XY) q. From this it follows that V and W permute and therefore by 6.10, (oc) we have V Wen). By 6.10, (y), V and W must be related by n or m. As we have shown in this Wen). case, the latter implies the former. Thus for any V, W we must have V By 6.9, 7.p is an isomorphism of Wm onto U. (3) By 6.10, (y), there remains for us to consider only the case where f',J
f',J
X
f',J
Y(m).
We consider the one-one map
~
of Wm onto itself,
382
RELATIONS IN SEMIGROUPS
[CHAP. IX
and the one-one map rJ ofl:(W91) onto itself, rJ{Ur:., Up' ... } = {~(UJ, ~(Up), . .. } ({ U", Up, ... } E l:(W91 Evidently ~ is an anti-automorphism of lID91 (1, 1.17) and r; is an automorphism of l:(lID91 ). The mapping cp* = CPrJ is an isomorphism of l:(lID91) onto l:(U). For this isomorphism we define by the method of 6.8 a mapping 1j!* of W 9l onto U and
».
relations n* and m* (6.9). Setting 1j!*(W) = W, we evidently have cP*[W] = CPrJ[W] = cp[~(W)] = [~(W)],
W=
~(W).
Since X,-...., Y(m), therefore XY = ~(XY) = YX X= ~(X) =
X,
=
= xl',
Y = ~(Y>'= Y,
Xy= XY.
Thus for cP* we have X,...", Y(n*). By the reasoning in the preceding part of the proof it follows that '1jJ* is an isomorphism of W91 onto U.
CHAPTER X
EMBEDDING OF SEMIGROUPS 1. Some Cases of Embedding 1.1. One of the methods of studying a given semigroup is to embed it in some supersemigroup which has certain additional properties that make possible a more successful study of the semigroup. Such an embedding can be carried out in many different ways. Which of them to use in each separate case, and to what extent it will be used, is determined by those properties of the initial semigroup which interest us and by those properties of the enveloping supersemigroup which contribute to the study of the given properties. The problem of the embedding of semigroups of a certain class into semigroups of another class, besides the fact that it often serves as a useful device for the study of certain properties of semigroups, is of interest in itself, since it elucidates the interrelations of semigroups of different classes. 1.2. In order to consider the problem we must bear in mind that for a proof of the possibility of embedding a semigroup ~ into a semigroup having some property, it is often sufficient to obtain an isomorphic mapping of ~ into some semigroup ~' with the given property. In fact, by the method described in III, 1.8, we can then in ~' replace by the elements of ~ those elements of~' onto which the elements of ~ are mapped under the given isomorphism. As a result of this we convert ~' into a new semigroup ~", which is a supersemigroup of~. Of course this achieves its aim only in the case where the property we are interested in has an "abstract character" and is not destroyed as a result of the indicated substitution. 1.3. We will repeatedly use the important property of regularity (II, 6.1). As we know (II, 6.1), the semigroup 6 n of all transformations of the set is regular. In I, 3.9 it was shown that any semigroup can be isomorphically mapped into a semigroup 6 n for some fl. Because of Remark 1.2 it follows that any semigroup can be embedded into a regular semigroup. We must remark that in the embedding of a semigroup it is often desirable to add to it as few new elements as possible to obtain a supersemigroup of the kind required. In this respect the embedding of ~ into the semigroup of all transformations, as in I, 3.9, can turn out to be very uneconomical. In the case of embedding into a regular supersemigroup that we are now considering, the
n
383
384
EMBEDDING OF SEMIGROUPS
[CHAP. X
situation can be somewhat improved in the following way. Having embedded III into a regular supersemigroup Ill', for each A E III we fix A E 2{' which is regularly adjoint to A (II, 6.6; II, 6.7) and then we consider the subsemigroup IllI generated by all the elements A of III and all their fixed regularly conjugate elements A. We obtain a supersemigroup IllI of III in which all the elements of III will be regular (of course IllI itself may not be regular). Then for IllI we construct in the very same way a supersemigroup 1112 in which all the elements of IllI will be regular, and so on. As was remarked in III, 1.12, the set ~
= U Illn n
is a semigroup. Each of its elements will be regular, since it belongs to some Ill n and is therefore regular in Illn+I' If III was countable, then evidently IllI will be countable; also 1112 will be countable, etc. As a result ~ will be countable. At the same time the semi group of all transformations 6 0 for countable .0 (by the device of I, 3.9 we embed III into such a semi group) has the cardinality of the continuum and therefore from the point of view of the principle formulated above it forms a less suitable supersemigroup for embedding III than the ~ just constructed. Of course, we can apply a similar method in many other embedding problems. 1.4. In some problems the interest lies in the existence in a semigroup of a generating set of the smallest possible cardinality. If the cardinality m of a semigroup III is more than countable, then any generating set for III has cardinality m and the cardinality of a generating set of any of its supersemigroups is not less than m. Thus in this case the method of embedding to obtain a reduction of the cardinality of the generating set is impossible. However if III is finite or countable, then, as was shown by Evans [2], it can always be embedded in a semigroup with two generators. We prove this using a construction somewhat different from that of Evans. THEOREM. Any finite or countable semigroup is contained in a supersemigroup with a generating set oj two elements. PROOF. Let AI' A 2 , A 3 , ••• be all the elements of the semigroup III (we consider the countable case; the finite case is analogous; further, a finite semigroup can always be embedded in a countable semi group and thereby reduced to the countable case). We define a function cp(i,j) (i,j = 1,2,3, ... ) by letting
AiAj
= A 1, let X En~H), and let Y Em:. Using the inductive assumption and the property of partial transformations proven above we obtain
whence
(H2H3' .. Hm)(XY) En~Hl),
i.e., XY E n~H). Thus niH) is either a right ideal or the empty set. In order to eliminate the second possibility, we choose elements By the hypothesis of the theorem there must exist U, V E m: such that AU= BV= W.
Since n~H2H3'"
Hm) and niH1 ) are right ideals, we have W E n~H2H3'" H >, WE niH1 ). m
The first of these inclusions means that for some C E m:. Taking into account the second inclusion, we conclude that C belongs to niH1Hs ... Hm) = niH), which shows that niH) is nonempty. Clearly n(H1Hs,,·Hm) -
2
-
ncHm".H;;H1)
1
,
and hence n~H) is also a right ideal of m:. Since the partial transformation il = ilm . .. il2il1 , which is the inverse of H = H1 H 2 . .. H m, also belongs to ,f), it follows that in ,f) by the method of 2.6 we can define the two-sidedly stable relation n. :ay 2.6 the factor semigroup ,f)/n is a group. We denote the natural homomorphism of,f) onto ,f)/n by X. We consider the map "P of m into ,f), "P(A) = S.,4
Since for any A, B, X
E
(A
E
m:).
m:
(S.,4SB)X = S.,4(SBX)
= S.,4(BX) = ABX = SaX
it follows that "P is a homomorphism of m: into,f). We consider the product of the homomorphisms = X"P. Clearly is a homomorphism ofm into the group ,f)/n. Let us suppose that for some A, BE m: we have
e
e
e(A)
= eCB),
A ¥= B.
394
EMBEDDING OF SEMI GROUPS
[CHAP. X
This means SA""" SB(n), i.e., there exists a partial transformation T E 5 such that T SA and T SB. We take some element Z E IIiT). For this element we have
Y holds in G) if and only if X is divisible by Y from the left and right in G)+. PROOF. Let X, Y E G)+ (X:;t= Y). We must have either X> Yor Y> X. Suppos.e the first holds. Then, by 3.6, XY-1 E G)+ and by 3.7, Y-1X = y-1(XY-1) Y E G)+. Therefore X is divisible by Y from the left and right in G)+ i.e., X= y. (Y-1X), x = (XY-1). Y.
Sec. 3]
LINEAR ORDERING IN GROUPS
399
As regards Y, it can not be divisible by X from either side. In fact, if Y = XS, S E G)+, it would follow that X-I Y = S E G)+, andby3.7, YX-I = X(X-Iy)X-l E G)+. But this is impossible since YX-I = (XY-I)-I and by 3.8, YX-l E G)-. Since a linear order in G) induces a linear order in G)+, the theorem follows from the proven coincidence of partial orders in G)+. 3.11. From what has been said above, we can formulate in a natural way a criterion for the possibility of introducing a two-sidedly stable relation of linear ordering in a group. THEOREM. In order that a group G) can be transformed into a linearly ordered group by means of a two-sidedly stable relation of linear order it is necessary and sufficient that it be the union of two of its subsemigroups which are holoidal semigroups with intersection equal to E CD • PROOF.
(1) If G) is a linearly ordered group, then
is a holoidal semigroup by 3.10. G)- is anti-isomorphic to G)+ by 3.8, and therefore is also a holoidal semigroup. (2) Let (D = m u m:', m: n m:' = ECD , G)+
where m and m:' are holoidal subsemigroups of (D. We note that for the two elements X and X-lone always belongs to m: and the other to m:'. In fact, let X, X-I Em: (the case when X, X-I E m:' is completely analogous). Since X = ECDXECD' ECD = XX-I = X-I X, it follows that X precedes E CD , and ECD precedes X in m:. Therefore X = E CD , and in this case, ECD Em, Em l = ECD E m:'. Let us introduce in (D a partial ordering relation by setting X ~ Y if Xy-I E m. For any X, YE (D one of the two elements Xy-I and (Xy-I)-I = YX-1 belongs to m. Therefore either X ~ Yor Y ~ X. Here the two relations hold simultaneously only when Xy-I = E CD , i.e., X = Y. Let X ~ Y, Y ~ Z, i.e., XY-l, YZ-1 Em. Then we have XZ-1 = Xy-1. YZ-1
E
m:,
i.e., X~ Z. Thus the relation we defined is a linear ordering relation. It remains to be shown that this relation is two-sidedly stable. If X ~ Y, then for any Z E (D, (XZ)( YZ)-l
i.e., XZ
~
yz.
= XZ . Z-l y-1 = Xy-1 E ~(,
400
EMBEDDING OF SEMI GROUPS
[CHAP. X
For the proof of left stability we prove as a preliminary that the condition (oc) Xy-I E '21 is equivalent to the condition «(3) y-I X E '21. In fact, if X, Y E '21 then (oc) means that X is divisible by Y from the right in '21, and «(3) means that X is divisible by Y from the left. By 3.3 one implies the other. If X, Y E &' then X-I, y-I E '21. Condition (oc) means that y-I is divisible from the left by X-I in '21, and «(3) means that y-I is divisible by X-I from the right. Again by 3.3 one implies the other. If X E '21 and Y E '21', i.e., y-I E '21, then (oc) and «(3) must be true. If X E '21' and Y E '21, then neither (oc) nor «(3) is true except in the case X= y=E(jj'
Now let X> y. By definition, Xy-I E'21 and therefore, by what has been proven, y-I X E '21. For any Z E (fi we have (ZY)-I(ZX)
=
y-IZ-IZX
=
y-IX E '21,
which in view of the proven equivalence gives i.e., ZX
> ZY.
(ZX)(ZY)-I E '21,
3.12. In the process of proving the second part of the theorem we have shown that in (fi it is possible to introduce a linear order such that '21 will be the positive part and '21' the negative part of the linearly ordered group so obtained. Thus, by 3.8 we have the following. If a group (fi is the union of two of its subsemigroups which are holoidal semigroups with intersection equal to E(jj, then these holoidal semigroups must be anti-isomorphic. The representation of (fi as such a union of two of its holoidal subsemigroups can be converted into a decomposition of (fi since the identity E(jj, by 3.3, can be picked out as a third component. 3.13. As we have seen, the positive part of a linearly ordered group is a holoidal semigroup with two-sided cancellation and a unit. It turns out that the presence of these properties in a semigroup is also sufficient for it to be the positive part of some linearly ordered group. THEOREM. Ij& is a holoidal semigroup with two-sided cancellation and a unit, then there exists a linearly ordered group (fi for which '21 is its positive part. PROOF. We take a semigroup '21', anti-isomorphic to '21 and having no elements in common with '21 with the exception of a common unit E = Em = Em.,. Clearly &' will also be a holoidal semigroup with two-sided cancellation. The element of &' corresponding to the element X E '21, under some antiisomorphism of '21 onto '21' which we are considering to be fixed, will be denoted by X'. In the set (fi = & u '21' we define a multiplication. If two elements of (fi both
Sec. 3]
LINEAR ORDERING IN GROUPS
401
belong to ~ or both belong to ~', then their product is defined according to the operation in these semigroups. Let A E~, B' E ~'. If A precedes B in the holoidal semigroup ~, i.e., for some X, YE~, then we put
AB'
=
B = AX,
B= YA,
A . ( YA)'
= A . (A' Y') = = (X'A')' A =
B'A = (AX),' A
Y', X'.
In this connection we should remark that X and Yare uniquely defined, since ~ has two-sided cancellation. If A = B, then X = Y = E. In what follows we must keep in mind Lemma 3.3, according to which for A to precede B it is sufficient that B be divisible by A from the left or right. If B precedes A, i.e., for some U, V E~,
= BU, A = VB, = (VB) . B' = V, B' . A = B' . (BU) = U. A
then we put
A . B'
We now show that this multiplication is associative. Let G1 , G2 , Ga E denote
(§.
We
If G1 , G2 , Ga all belong either to ~ or ~', then Sl = S2 by the associativity of the operations in ~ and ~'. Let G1 , G2 E~, Ga = H' E ~{', where G2 is divisible by H from the right in ~, i.e., G2 = XH (X E~. Then Sl = (G1XH)' H' = G1X, S2 = G1 [(XH) . H'] = G1x.
Let G1 E~, G2 E~, Ga = H' E ~', where H is divisible from the right by ~, i.e., H = XG 2 (X E ~). If G] is divisible by X from the right, i.e., G1 =
G2 in
UX, then
= S2 = Sl
(G1G2) • (XG 2)'
= (U'
XG 2) • (XG 2 )'
=
U,
G1 ' (G 2 • G~X') = UX' X' = U.
If X is divisible from the right by G1, i.e., X Sl = (G1G2 ) • (XG 2)' S2 = G1 [G 2 • (XG2 )']
=
VGv then
= G1G2 • (V, G1 G2)' = (G1 G2)[(G1 G2), . V'] = = G1 ' [G 2 • (G 2 X)] = G1X' = G1(G~ V') = V'.
V',
The case G1 E ~', G2 E~, Gs E ~ is dealt with analogously. Let G1 E~, G3 E~, and let G2 = H' E ~', where H is divisible by G1 from the right in ~{, i.e., G1 = XH (X E ~). Then, whether G3 is divisible from the left by H or H is divisible from the left by G3 , in both cases it is easy to see that (XH . H')Ga = XG a, S2 = (XH) . (H'G3 ) = XG 3 • S1
=
402
[CHAP. X
EMBEDDING OF SEMIGROUPS
Let G1 Em, Ga Em, G2 = H' Em', where H is divisible by G1 from the right in m, i.e., H = XG1 (X E Ill), and X = Ga Y (Y Em). Then [G1 • (XG1)']Ga = X'Ga = Y', S2 = G1 [(XG 1)' • Ga] = G1 • [(G~ X') . G3 ] = G1 . (G~ Y') Sl
=
=
Y'.
Let G1 Em, Ga Em, G2 = H' Em', where H is divisible from the right by G1 in m, i.e., H = XG1 (X E Ill), and Ga = XY (Y E Ill). Then, whether Yis divisible from the left by G1 or G1 is divisible from the left by Y, in both cases, it is easy to see that Sl = [G . (XG )'] • (XY) = X' . (XY) = Y, 1 1 S2
=
G1 ' [(XG1)' . (XY)]
=
G1[(G~X')' (XY)]
=
Y.
We need not consider the cases when two of the factors G1 , G2 , Ga belong to Ill' and the third to m, in view of the fact that the subsemigroups m and m' in the multiplicative set (D play entirely the same role. We have thus seen that (D is a semigroup. E is evidently an identity for (D. The elements X and X' (where X E Ill, X' E Ill') are inverses of each other. Therefore (D is a group. It is the union of two of its subsemigroups mand m' which are holoidal semigroups, where III n m' = E. By 3.11, (D can be converted into a linearly ordered group by introducing into (D some relation of partial ordering. As was remarked in 3.12, this can be done so that the holoidal subsemigroup mis the positive part of the linearly ordered semigroup (D. 3.14. We note that the result of Theorem 3.13 is also interesting from the point of view of the preceding section. The condition of two-sided cancellation turns out to be not only necessary but also sufficient for embedding into a group all the semigroups of a certain class, namely the holoidal semigroups with a unit. 3.15. In conclusion we remark that many of the various types of ordering are not possible in holoidal semigroups which are positive parts of linearly ordered semigroups. For example, from the theorem mentioned below it follows that the only nontrivial type of well-ordering which is possible in this respect is the type of ordering by magnitude in the set of all natural numbers. THEOREM. If all the elements of the positive part (D+ of a linearly ordered group (D are well-ordered with respect to a relation of linear ordering in a holoidal semigroup, then (D is the unit group or an infinite cyclic group.
PROOF. The first element of (D+ is evidently E(fJ" If (D ¥:- Ery" then by 3.8 ¥:- Ery,. Let X be an element of (D+ following Ery,. Put (D' = [Ery" X]. We suppose that (D' ¥:- (D+. We denote by Ythe element of (D+\(D' preceding all the elements of (D+\(D'. Since Y follows X, for some U E (D+ we have (fj+
Y=XU
since otherwise Y E
(D'.
(UE (D'),
Therefore for some V E U= VY.
(D+,
Sec. 4]
POTENTIAL INVERTIBILITY OF ELEMENTS
403
We obtain Y = XVY, i.e., XV = E(Jj. But by 3.3 this would mean that X precedes E(Jj in (fj+. But this is impossible since E(5 is the first element. This contradiction means that (fj+ = [E(5, X]
=
{E(Jj, X, X2, ... , Xn, ... }.
By 3.8, n;:\lj -
{Effi' X-I , X-2 ,
... ,
x-n , . . .}
and therefore by 3.9, (fj is an infinite cyclic group. 4. Potential Invertibility of Elements 4.1. In considering the properties of invertibility of elements it often happens that an element A of some semi group III is not invertible from the left in III (VI, 1.1) but there exists a supersemigroup Ill' of III such that A is invertible from the left by an element of Ill'. Thus embedding III into Ill' transforms A from an element which is noninvertible from the left into an element which is invertible from the left. In view of the significant importance of the property of invertibility the problem of when this possibility occurs is worthy of attention. DEFINITION. FROM THE LEFT
An element A of a semigroup III is POTENTIALLY INVERTIBLE exists a supersemigroup Ill' of III in which A is invertible
if there
from the left. The situation for invertibility from the right is analogous. 4.2. The concept and term "potential" for various properties in semigroups is due to E. S. Ljapin. We can speak of various potential properties of separate elements or of subsets of semigroups, meaning by this that the given property of an element or a subset is realized in some supersemigroup of the given semigroup. E. G. Sutov [1; 2; 3; 4] has investigated the problem of potential realization of a number of properties in semigroups. In particular, he completed and then generalized a certain problem of potential invertibility which was first taken up by E. S. Ljapin. The statement of this problem has a definite similarity to the problem of divisibility for two elements in some supersemigroup, which was considered in 1.6 and 1.7. However, the method of solving it turns out to be more complicated. 4.3. We note the following property of invertibility. Let.R be a generating set of a semigroup III and let A E Ill. If for each K E .R there exists Z:K. E III such that Z:K.A = K, then A is invertible from the left. In fact, an arbitrary element S of III can be represented in the form
S = KIK2 ... Xn
(Kl' K 2 ,
••• ,
Kn
E
.R).
Thus we have (K1K 2
• ••
K n- 1Z X 11 ) ' A
=
K 1K 2
• ••
Kn-1Kn
= S.
404
[CHAP. X
EMBEDDING OF SEMI GROUPS
4.4. Let an element A of the semigroup III be potentially invertible from the left; that is, in some supersemigroup Ill' of I2t it is invertible from the left. If for some natural number n, and for some X and Y which are elements of I2t or empty symbols, we have An X = An Y, then for any S E I2t it follows that SX = S Y. In fact, in 12t' there must be elements Zl' Z2' ... , Zn-1' Zn such that Therefore ZnAn X = Zn_1An-1X = ... ZnAny
= Z2A2 X = ZIAX =
= Zn_IAn-1Y= ... =
SX,
Z2A2Y = ZlAY = SY,
SX= ZnAnx= ZnAny= SY.
4.5. In connection with 4.4 we note that if in the semigroup 12t, for some A E 12t, A2X = A2 Y implies SX = SY for any S E 12t, then for any natural number n, An X = An Y implies SX = SY for any S E 12t. In fact, if AX = AY, then A2X = A2Yand therefore SX = SY. If n > 2, then An X = An Y implies A2(An-2 X) = A2(An-2 Y) and therefore A(A n-2 X) = A(An-2 Y), i.e., An-IX = An-l Y. Repeating the argument, we obtain A2 X = A2y, whence, by assumption, SX = SY. 4.6. In the case when an element A of a semigroup Ilt is of finite type it is at once evident that the above necessary condition for potential invertibility of A from the left is also sufficient. But more than that, this condition in the case considered simply means that A is invertible from the left in Ilt. In fact, let A satisfy condition 4.4 and Ah+d = All.
From All. Ad = All it follows that for any S E I2t we have SAd = S. Thus (SAd-I) . A = S, i.e., A is invertible from the left in 12t. We also note that in this case h = 1, i.e., [A] is a group. In fact, by VI, 1.8 for some X E 12t, XA II = A. Therefore from Ah+d = A\ multiplying by X, we obtain AHd = A, whence it follows that h = 1. 4.7. Let A be an element of a semigroup I2t such that for any natural number n and any X, Y, S E 12t, An X = An Y implies SX = SY. We consider pairs in which the first component is an arbitrary element S of Ilt and the second is an arbitrary whole non-negative number k. Such a pair we will denote by S(k). The totality of all such pairs we denote by.lt. We denote by jD all words W
=
Sik1)Sik2) ...
S~k,)
in 5t in which no two neighboring k i and ki+1 are both zero. In jD we define multiplication by setting WI' W2
=
Wa
(Wl , W 2, Wa
Wi = Sitli)S~12i) ... S;~Sii)
(i
=
E jD),
1,2,3),
Sec. 4]
POTENTIAL INVERTIBILITY OF ELEMENTS
405
°
°
where Wa s the ordinary product of• the words WI and w'2 if k 8 , 1 -.4 or k 12.,.... ...,l. • r If kSli = k12 = 0, then WS IS obtamed from such a product by replacing the neighboring terms Si~{ and sig) by the element (SS,I . SI2)(O) (where Ss 1 • S12 is the product of SSI I and S12 in 121). 1 The indicated operation in 5B is evidently associative. 4.8.
In 5B we define the following relations: (W,
w' E 5B)
if where k
>
°
and R can also be the empty symbol; W""' W'(n 2 ),
if (k?> I> 1),
where R again can be the empty symbol. If W""' W'(n 1), or W' ""' Wen l ), or W""' W'(n2)' or W' W', we write
w""'
r-.J
W(n) , or W =
W'en 3).
We note that W,......, W'(n i ) implies WS(O),......, W'S(O)(n;) (i = 1,2,3) for all SE Ql.
4.9.
For words in 5B we define an operator (J. Let W
=
Sikl)S~k2)
..• S;:,m)
E
5B
and let S?t) be the element of this word which is farthest to the right and such that t < m, kt+1 = 0, and let there exist an R which is an element of Ql or an empty symbol such that ASH1 = Akt+1. Then (JeW) is obtained from W by replacing Sikt)Si~l by (StR)(O) (if, in this connection, k t - l = 0, then St(~l is also adjoined to these elements: (St_lStR)(O). If there is no element S?,) in Wwith the property in question, then we put (J(W) = W. We note that the operator (J is single-valued. In fact, if ASt+l = Ak t +1R = A k t+1R', then by the assumption on A we have XR = XR' for all X E Ql, and in particular, StR = StR'. Evidently W '" (J(W)(n~), where n~ is the first derived relation for nl (I, 5.20). For X E Ql we always have a(X(O» = X(O). 4.10. Continuing with the notation introduced, we consider some properties of the relation n 3 • LEMMA.
If W,......,
PROOF.
Let
W'(n~), then a(W) ""' a(WI)(n~).
406
[CHAP.
EMBEDDING OF SEMI GROUPS
X
where U,-....; U'(ni) (i = 1,2) (the case W = W' is trivial, and the case U'r-..I U(n i ) is symmetric to the ones considered). By what was stated at the end of 4.8 we can consider that in the word T2 for the first factor SIP) we have p ¥- o. If a(T2) ¥- T2, then evidently a(W)
= TI . U· a(T2),
a(W')
= T I · U' . a(T2)
and therefore a(W) ,-....; a(W')(n~). In what follows we will assume that a(T2) = T2. If U,-....; U'(n1) , then, as it is easily seen, a(W) = TI · a(U) . T2, i.e., a(W) = W' (since a(U) = U'). Also, since W' '""" a(W')(n~), it follows that a(W) r-..I a(W')(n~).
Let U,-....; U'(n 2); - 1 S(k)S(O) AS2 - A~R U' U2' '
= S(k-Z+l)R(O) 1
(k
~
",
I> 1).
If for some P which is an element of mor an empty symbol, AS2 = A k+1P, then aeW) = Tl . eSIP) 1. We may assume that V2 ¥- VI. By 4.9 and 4.10, X(O)
= a( X(O» =
a( VI)'
y(O)
=
a( Y(O» = a( Vs),
a(Vi ),-....; a(Vi+I)(n~).
Since V2 '""" X(O)(n~), we evidently have V2 '""" VI(n~), where V2
=
SiO)S~k)S~O>,
= Ak+1R, SIS2R = X ASs
and therefore a(V2) = (SIS2R)(0) = X(O). By the inductive assumption for the chain X(O) a(Vs) = Y(O) we conclude that X = Y.
=
a(V2 ), a(Va), .•• , a(Vs-l),
m
4.12. THEOREM. In order that an element A of a semigroup be potentially inuertible from the left (4.1) it is necessary and sufficient that for any X and Y
Sec. 4]
POTENTIAL INVERTIBILITY OF ELEMENTS
407
which are elements of'11. or empty symbols, A2X = A2YimpUes SX SE'11..
=
SY for all
The necessity was proven in 4.4. We now show the sufficiency (taking 4.5 into account). We construct the semigroup m (4.7), considering it as a supersemigroup of '11. (for which we identify S(O) with S for each S E '11.). We consider the factor semigroup 55 = In this semigroup, by 4.3, the element A (i.e., the class of words in m which are equivalent to A mod n;) is left invertible, since for any SUe) we have PROOF.
min;.
and hence in 55
min;.
We consider the natural homomorphism rp of monto 55 = rp when restricted to '11. is an isomorphism, since by 4.11 for S ;;6 R (S, R E Ill) we have S(O) rj.; R(O)(n;), i.e., S(O) ;;6 R(O). Here rp(A) = A. Replacing in mthe elements of rp('11.) by the corresponding elements of '11., we obtain a supersemigroup of rp in which A is left invertible. 4.13. Along with the properties of potential left invertibility and potential right invertibility of elements it is natural to consider potential two-sided invertibility. Evidently an element which is potentially two-sidedly invertible is potentially invertible from the left and potentially invertible from the right. However the fulfillment of this necessary condition is, generally speaking, not sufficient for an element to be potentially two-sidedly invertible. Below we mention an example of a semigroup constructed by E. G. Sutov [1] patterned after a construction of A. I. Mal'cev [1], in which, since it is a semigroup with two-sided cancellation, each element is potentially left invertible and potentially right invertible. At the same time we shall indicate in it an element which is not potentially two-sidedly invertible. Hence, in particular, it would follow that this is a semigroup with two-sided cancellation which is not embeddable in a group, since it is clear that all the elements of any semigroup which are embeddable in a group are potentially two-sided1y invertible. 4.14. We consider the free semigroup X 3 , X 4 , X 5 , X 6 } (IX, 1.3). We denote
[)Jolt
over the alphabet .R = {Xl' X 2 ,
The set of words X 1 X 5' X1 X 6 , X 2 X 5 , X 3 X1 ' X 3 X2' X4 X 1 we denote by note that if XiXi E m, then Xi E .Rl and X; E .R2 • 4.15.
We denote by 0 the following map of.R1 onto itself:
m.
We
408
[CHAP. X
EMBEDDING OF SEMI GROUPS
and by ,. the map of St z onto itself, ,.(X1)
=
X 5,
,.(X2)
We note that (j2(Xi) 4.16.
=
=
X 6,
Xi and ,.2(Xi )
=
Xi'
We introduce in :'illft a relation n by setting
V", yen) if V = V or V = XiXi E 'n and V = (j(Xi ) . ,.(Xi ) E 'no The relation n is reflexive, and by the above properties of (j and,. it is symmetric. 4.17. We shall indicate a number of properties of the relation n. The validity of some of these follows immediately from the definition of n. (C/..) If XiXi , XiXk E 'n, then the equality Xi = (j(X,,) is impossible. (~) Let Xi 1 Xi 2 ... Xi s '" Xi 1 Xi 2 ... Xi s(nil)
and Xi, E St1 • Then Xi, = Xi,' (y) Let
Xi 1 Xi 2 ... Xi oS '" Xi 1 Xi 2 ... Xi s(nil)
and Xi, E St1 • Then either Xi, «(j) If
= Xi, or Xi,
= (j(Xi)'
then Xi 1 Xi 2 ... Xi s "'" Xi 1 Xi 2 ... Xi (n'). (8) If oS
then Xi 1 Xi 2 ... Xi 8 "'-' Xi 1 Xi 2 .,. Xi 8(n"). In fact, let VI""'" V2(n'),
V 2",-, V 3(n'), ... , V m- 1 ",-, Vm(n')
(VI' V 2 ,
••• ,
(X!T E St; I UI
=
X"Xi,Xi •. .. Xi"
Vm E [(5ft),
= 1,2, ... , m; r
= 1,2, ... , s),
Um = X"Xi,Xi • ... Xis
If XIcXi, E 'n or XkXj, E 'n, then by the definition of derived relations, the validity of the indicated relation follows immediately. Therefore, in what follows we can consider that both the indicated words belong to m. Thus XIc
E
St1.
The proof will be by induction on m. For m = 2 we have the case «(j). Let m > 2. If in some V! (l = 2,3, ... , m - 1) we have Xo = XIc' then by the inductive assumption applied to U1 , V2 , ••• , V! and U!, V!+!, ... , Um we obtain Xi1Xi •. .. Xi, "'-' X n X!2' .. X;.(n"), X n X!2' .. Xl. "'-' Xi1Xj •
whence follows the indicated relation.
•••
Xi,(n"),
Sec. 4]
POTENTIAL INVERTIBILITY OF ELEMENTS
409
Here we may assume in what follows that XIO :p X k (l = 2,3, ... , m - 1), i.e., by (y), that X zo = o(Xle)' Hence it follows in particular that X 20 :p X lO • Therefore X 20 = o(Xk ), X 21 = T(Xi ), Analogously,
From U2 ,......, Um- 1(n"), by the inductive assumption it follows that X 21 X 22 • •• X 2s Xm-1.1Xm-1.2' .. Xm-l.sCn"). By «(3) and (y) either X m- 1.1 = X 21 or X m-1.l = O(X21 )· But X 20 X 21 and Xm-l.0Xm-1.1 belong to m, and by (ex.) the second equation is impossible. Therefore X m -1.1 = X 21 . So we can apply the inductive assumption to the words X 21 X 22 · .. X 2S and Xm-l.lXm-1.2' .. X m-1.s> with the result that ("Oo..J
But as was proven, X Z1
= T(Xi ,),
Therefore, multiplying the relation so obtained on the left by Xi" which equal to X;" we obtain the required relation.
IS
WIf Xi 1 Xi 2 ... Xi s Xle
r-J
X;"1 X;"2 ... Xj s Xin"),
then X i1 X i2 ••• Xis X;lX;2 ... Xj,(n"). The proof is completely analogous to (e). (17) X 2 X 6 ~ X 4 X 2(n"). This follows from the fact that the word X 2 X 6 is congruent only to itself modn. ("Oo..J
4.18. We consider the semigroup ID3~ = W5l./n" (IX, 3.2). The elements of W~ being nil-classes, we will write them in the form W (where W is the nil-class containing W, WE ID35l.)' From 4.17 (e), W it follows that in ID3~,XleO = XTcV implies 0 = V and OXle = VXle implies 0 = V. Therefore W~ is a semigroup with two-sided cancellation. 4.19. We deduce a necessary condition for an element of a semigroup to be potentially two-sidedly invertible. LEMMA.
If in a semigroup III elements Zl,Z2' Z3' Z4' Z5' Z6 satisfy the equalities ZlZ 5 = Z3 Z 1' Z2 Z 5 = Z4Z 1, Zl Z 6 = Z3 Z 2'
410
EMBEDDING OF SEMIGROUPS
[CHAP. X
and if there exists a supersemigroup I!{' ofl!{ in which Zl is two-sidedly invertible, then Z2Z6 = Z4Z 2' PROOF.
In
I!{'
there must exist U, V such that
Using the relations between the elements Zi we obtain
Z2Z6
=
VZ~6
=
VZ3Z 2 = VZ3Z 1 U = VZ~sU = Z2 Z 5 U = Z4 Z 1 U = Z4Z 2'
4.20. In the semigroup with two-sided cancellation lillj\ the elements Xl' X2 , X3 , X4 , Xs, X6 satisfy the equalities
= lill St./n"
(4.18)
X1 X5 = X3 X1, X2 X5 = X4 X1> X1 XG = X3 X2• At the same time, by 4.17 ('YJ) we have the inequality
From this, by 4.19, it follows
X2 X6 ¥= X4 X2 • that Xl is not potentially two-sidedly invertible.
4.21. In conclusion we note that, by IX, 3.2, 3.10, the semigroup lill5\ = lillSt./n" can be defined as a semigroup with generating set {Xl' X 2, X 3' X4 , X 5 , Xs} and with the defining system of relations
X1 X s = X 3 X1 , X 2 X5
=
X4 X1,
X1 X 6 = X 3 X 2 •
5. Free and Direct Products 5.1. Along with the problem of embedding a given semigroup into a supersemigroup with given properties there naturally arises the related problem of the embedding of entire systems of semigroups. But we must at once point out that even if we do not impose any requirements at all on the desired supersemigroup, we sometimes obtain a negative answer to the question of the possibility of such an embedding. Consider the following example. We take the two sets of rational numbers, I!{
58
la, t, t, t, 1,2,4, 8, ... }, =. { ... , i"a, -t, t, -t, 1,2,4,8, ... }.
= { ... ,
Sec. 5]
411
FREE AND DIRECT PRODUCTS
We will consider mwith respect to the usual operation of multiplication of rational numbers. In Q3 we define the operation, denoted by the symbol 0, in the following way: rxfJ (if rxf3 E Q3), rxofJ= { - rxfJ (if - rxf3 E Q3). Both these operations are associative. The semigroups mand Q3 have the common part {I, 2, 4,8, ... }. The operations in ~( and Q3 are compatible, i.e., the product of two elements of the common part defined according to the multiplication in mcoincides with the product of those elements defined according to the multiplication in Q3. Nevertheless there does not exist a common supersemigroup for both these semigroups. In fact, in such a common supersemigroup, if it existed, we would obtain
t = t· 1 = t·
[2 0
(-t)] = Ct· 2)
0 (-t)
=
1 0 (-t)
= -t.
Evidently the impossibility of embedding both semigroups in a common supersemigroup in the above example depends on the presence of common elements in mand Q3. As we will show in 5.4, with the absence of common elements, embedding into a common supersemigroup is always possible.
m
5.2. DEFINITION. A semigroup is the FREE PRODUCT of its pairwise disjoint subsemigroups Q3"" Q3p, ..• ifQ3 = iB" U Q3 p U ... is a generating set ofm, where the aggregate of all the relations between the elements of Q3", all the relations between the elements of Q3p, and so forth, is a defining system of relations for m with respect to Q3. The representation of a semigroup in the form of a free product is usually called a decomposition of the semigroup into a free product. 5.3. A semigroup which is decomposed into a free product is determined up to isomorphism by giving up to isomorphism the components of that product. This immediately follows from IX, 1.12. 5.4. Let mbe a free product of subsemigroups Q3", Q3,'1, . . •• An arbitrary word W in the alphabet Q3 = Q3" U iB,'I U ... can be represented in the form W
=
UOl UI;2 ... U;k ... UI; ..'
where UI;;(i = 1,2, ... , n)is a word in Q3l;i and Q3l;i-l and Q3;i are always distinct. If we transform W by means of any of the relations in the defining system of relations 5.2, then W is transformed to a word of the form W = U. U"'-2 • .•. ~l
U~"ik
'" U."'-n ,
where the words UOk and U~k have the same value in Q3;k' As the result of a sequence of such mappings W is transformed into a word of the form W
where the words Ul;i and
U~i
U~P~2 •.. U$k ... U~", in Q3l;i have the same value in Q3l;i (i = 1,2, ... , n).
=
[CHAP. X
EMBEDDING OF SEMIGROUPS
412 5.5.
Any A
E I.!(
can be given in the form of a product A
=
B!;,B!;2 ... B~n'
where B;; (i = 1,2, ... , n) is an element of lB;i' .and t~le lB"i-l and ~;i are distinct. This form of an element of I.!( is canomcal, smce no two dIfferent products in such a form can be equal to each other. This follows immediately from 5.4. The indicated canonical representation of elements of a free product I.!( is very convenient inasmuch as elements given in this canonical form are easily multiplied. Let A'
If ~n =/=
1')1'
= B~ B~ 'II
'12
... B~ . '11n
then the canonical form of the product will evidently be AA' = B."'1 B ""2 • ... B/;on B~'/1B:'/2 .. , B~'1m .
If ~n
= 1')1 and in lB'11 we have BenB~, = B~"
then evidently
AA' = B.'il B "'2 • ... Be"'n-l B; B~ .,. B~ . '/1 '12 '1m
The description of I.!( as a set in the indicated canonical form compatible with the above rule for multiplication can evidently serve as a definition of free product. 5.6. It is not hard to solve the converse to the above problem of decomposing a given semigroup into a free product. THEOREM. For semigroups lB", lB p, ••• which do not have pairwise common elements, there exists a common supersemigroup I.!( which is decomposed into the free product lB", lB/!, .... PROOF. Let I.!( be the semi group given by the generating set 91 = lB" u lB j3 U ... and the defining system 0/ consisting of all relations of the semigroup lB", all relations of the semigroup lB/!, and so forth (IX, 3.10). The relation 110/ in the free semigroup Wm corresponding to the system of relations 0/ (IX, 3.7) is such that for X E lB" the condition X "-' Y(I1~) (Y E 91) is valid only for Y E lB" and X = Y. In fact, the transformation of X by means of any relation ofo/ gives an element which again belongs to lB" and is equal to X. The same result is obtained by successively applying several of these transformations. By IX, 3.7, in I.!( all the elements of 91 are distinct. Thus I.!( turns out to be a supersemigroup for each lB", lB/!, .•. and is evidently their free product. 5.7. The construction of a free product is not simply one of the solutions of the problem of embedding given semigroups into a common supersemigroup, but it plays a special role for this problem. Let the semigroups lB", lB/!, ... be pairwise disjoint and let I.!( be a common supersemigroup of them. We denote by I.!(' the subsemigroup of I.!( generated by lB = lB" u lBp U .. '. In m', 5B is a
Sec. 5]
FREE AND DIRECT PRODUCTS
413
generating set, and in ~' we have the validity with respect to ~ of all the relations between the elements of ~'" all the relations between the elements of ~.B' and so forth. By IX, 1.11 there exists a homomorphism of the free product IJ of the semigroups ~'" ~i3' ••. onto ~' which is an extension of the identity maps of ~'" ~.B' ... , considered as subsemigroups of IJ, onto ~'" ~.B' .•• , considered as subsemigroups of ,!!:'. Thus any supersemigroup of the semigroups ~'" ~P' ..• can be obtained from their free product IJ by means of a homomorphism of IJ, which is an extension of the identity maps of~", ~.B' ••• onto themselves, and a subsequent embedding of the semigroup so obtained into the arbitrary supersemigroup. That this argument is similar to the arguments about free semigroups (§ 5, Chapter IX) does not happen by chance. The fact is that, as is easily seen, free products of infinite singly generated semigroups are free semi groups in the class of all semigroups. 5.8. Let a semigroup ~ have subsemigroups ~'" ~.B' ••. , where each ~< is given by a generating set ~< and defining system of relations 0/ with respect to ~< (g = 0:, (J, ... ). If ~'" ~P' •.. are pairwise disjoint, and ~ = ~" U ~p U ... is a generating set for ~, and 'Y" U o/p U ... is a defining system of relations with respect to ~, then ~ is the free product of ~'" ~P' .... In fact, since ~ is a generating set for ~, the system ~" U ~p U ... will all the more generate Ill. For an arbitrary word W in ~o:, if we transform it by relations of'Y" U 'Yp U ... , we will always obtain a word in ~". From this it follows that no element of [~o:J = ~" can equal any element of [~.B] = ~p (0: :;:6 (J). Therefore ~o:' ~P' ... are pairwise disjoint. Since 0/0: U 'Yp u ... is a defining system for ~ with respect to ~ it follows from IX, 2.7 that the system of all relations between the elements of ~'" all relations between the elements of ~.B' and so forth, will be a defining system of relations for ~ with respect to ~o:u~{3u""
5.9. Ifwe consider semigroups only to within isomorphism, i.e., if we do not distinguish semigroups which are isomorphic to each other, then the construction of a free product defines an operation between semigroups. The result of this operation on the semigroups ~ and mis the semigroup IJ, decomposed into the free product of its subsemigroups~' and ~', which are isomorphic to the corresponding semigroups ~ and~. By 5.6 such a semigroup always exists and by 5.3 it is completely determined up to isomorphism. 5.10. Commutativity of the operation mentioned in 5.9 follows directly from Definition 5.2, and associativity also holds, as follows from 5.8. In fact, let 1J12 be the free product of the semigroups ~1 and ~2' and let IJ' be the free product of 1J12 and ~3' By 5.8, IJ' is the free product of the three subsemigroups ~l> ~r2' ~3' Similarly IJ", which is the free product of ~1 and 1J23' where 1J23 is the free product of~r2 and ~r3' turns out to be the free product of the three semigroups ~1' 1ll2' ~r3'
414
[CHAP. X
EMBEDDING OF SEMI GROUPS
5.11. By 5.9, 5.l0, starting from given semigroups, considered to within isomorphism, we can construct a commutative semigroup X whose elements are semigroups. The operation in X will be the construction of the free product of two semigroups. The initial semigroups will be the elements of the generating set for X. 5.12. Construction of a free product is related to the construction of a direct product. DEFINITION. A semigroup ~ is the DIRECT PRODUCT of its pairwise disjoint subsemigroups 58"" !B p, ••• if58 = 58", U 58 p U ... is a generating set ofm, where for any B~ E 58 Proc. Amer. Math. Soc. 7 (1956),729-734. [3] On the structure of semigroups on a compact manifold with boundary, Ann. of Math. (2) 65 (1957), 117-143.
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INDEX The numbers in parentheses are page numbers. Abstract characteristic I, 1. 15 (7) Adjacent sets IV, 2.8 (146) Algebraic operation I, 1.2 (1) Anti-automorphism I, 1.17 (7) Anti-isomorphism I, 1.17 (7) Anti-symmetry I, 5.6 (35) Associativity I, 1.18 (8) Automorphism 1,1.16 (7)
Defining set of relations IX, 1.9 (347); IX, 3.10 (356) Densely imbedded semigroup VII, 5.6 (290) Derivative relations 1,5.20 (41); 1,5.21 (41) Direct product X, 5.12 (414); X, 5.18 (415) Distributive lattice II, 4.13 (61) Divisibility 1,5.19 (39) Divisibility of elements II, 1.2 (43)
Band VIII, 1.6 (309) Basis classes VIII, 5.1 (336) Boolean algebra II, 4.16 (63) Bound I, 5.12 (37)
Endomorphism I, 3.18 (25); I, 3.19 (25); 1,3.20 (27) Envelope IV, 2.6 (146) Epimorphism VII, 1.1 (256) Equivalence I, 5.7 (36) External adjunction II, 2.12 (50)
Cancellation I, 1.18 (8); I, 3.2 (16) Cayley table I, 1.6 (2) Chain IV, 2.10 (147); IV, 4.1 (157) Character VII, 6.20 (305) Class regularity II, 6.11 (77) Commutative band VIII, 1.12 (312) Commutative semigroup of idempotents II, 4.1 (56) Commutativity I, 1.18 (8); I, 1.19 (8); I, 3.2 (16) Commutator condition IV, 6.7 (175) Complemented lattice II, 4.15 (62) Complete lattice I, 5.13 (37) Completely isolated subsemigroup IV, 6.1 (172) Completely regular element II, 6.1 (71) Completely regular semigroup II, 6.1 (72); VIII, 2.1 (314) Completely simple semigroup V, 3.9 (192); V, 4.2 (194) Congruence I, 5.18 (39) Corollary of relations IX, 1.7 (347); IX, 1.8 (347) Cyclic group III, 3.17 (112)
Decomposition I, 5.8 (36); VIII, 1.2 (307) Dedekind lattice II, 4.14 (61) Defining relation IX, 3.2 (354)
Factor-semigroup VII, 2.4 (266); VII, 4.15 (282) Fixed point II, 5.1 (66) Free product X, 5.2 (411); X, 5.18 (415) Free semigroup IX, 1.3 (346); IX, 5.1 (364) Free set IX, 5.1 (364) Galois theory I, 3.19 (27) Generalized aggregate II, 7.10 (83) Generating set III, 2.1 (98) Greatest common right divisor for homomorphisms VII, 1.15 (263) Group I, 3.2 (16); II, 1.3 (43); II, 2.15 (51) Groupoid I, 1.5 (2) Holoidal semigroup VIII, 4.8 (333) Homogroup V, 1.8 (182) Homomorphism VII, 1.1 (256) Ideal IV, 1.1 (138) Ideal chain IV, 4.1 (157) Ideal envelope IV, 3.3 (151) Ideal equivalence IV, 3.3 (151) Ideal factor IV, 4.5 (159) Ideal factor-group VII, 4.15 (282) Ideal layer IV, 3.3 (151) Idempotent II, 2.3 (48)
445
446
INDEX
Identity in a semi group IX, 4.1 (359) Identity transformation I, 3.13 (23) Inner automorphism VII, 6.14 (302); VII, 6.17 (303) Inverse ~lement II, 2.1 (47); II, 2.13 (50) Inverse isomorphism I, 1.17 (7) Inverse semigroup II, 7.2 (79); II, 7.4 (79) Invertibility of operations I, 1.18 (8); VI, 3.1 (229) Invertible element VI, 1.1 (218) Invertible transformation I, 3.13 (22); I, 3.14 (23); I, 3.16 (24) 'Involution I, 1.17 (7) Irreducible generating set III, 2.5 (99) Isolated subsemigroup IV, 6.1 (172) Isomorphic representation I, 1.15 (6) Isomorphism I, 1.8 (4); I, 1.10 (5); I, 1.11 (5); I, 1.13 (6) Lattice I, 5.13 (37) Lattice isomorphism III, 7.6 (129) Layer IV, 2.4 (146) Least common right multiple for homomorphisms VII, 1.15 (264) Left translation I, 3.9 (20) Linear ordering I, 5.11 (37) Linearly ordered semigroup X, 3.2 (395) Magnifying element III, 5.1 (118) Mapping, isomorphic I, 1..10 (5) Matrix band VIII, 1.14 (312) Matrix semigroup V, 5.1 (202); V, 6.3 (212) Maximal element II, 4.17 (63) Maximal set IV, 2.1 (145) Minimal ideal IV, 3.3 (151) Minimal set IV, 2.1 (145) Monogenic semigroup III, 3.1 (106) Monomorphism I, 1.13 (6) Multiplication of partial transformations 1,4.2 (29) Multiplication of relations I, 5.3 (34) Multiplication of subsets III, 1.1 (91) Multiplication table I, 1.6 (2) Multiplicative set I, 1.5 (2) Natural homomorphism VII, 2.5 (267) Nilpotent semigroup IX, 4.5 (361); IX, 4.6 (361) Nonzero ideal V, 3.13 (193) Normal complex VII, 4.1 (276) Normal subsemigroup VII, 4.9 (278)
Order of a semigroup I, 3.2 (16) Partial ordering I, 5.9 (36) Partial transformation I, 4.1 (28) Partially ordered semi group X, 3.2 (395) Partition I, 5.8 (36); VIII, 1.2 (307) Periodic semigroup III, 4.1 (113) Permutability I, 1.19 (8) Positive part of a partially ordered semigroup X, 3.5 (396) Potentially invertible element X, 4.1 (403); X, 4.13 (407) Primitive idempotent V, 3.10 (192) Principal ideal IV, 3.6 (152) Product of homomorphisms VII, 1.12 (261) Product of subsets III, 1.1 (91) Product of transformations 1,3.6 (18) Reflexivity I, 5.6 (35) Regular conjugate II, 6.6 (74) Regular element II, 6.1 (71) Regular semigroup II, 6.1 (72) Regular sUbsemigroup with respect to invertibility VI, 4.3 (238); VI, 5.5 (241) Regular unit II, 6.2 (72) Regularity I, 5.18 (39) Relation I, 5.1 (33) Relations in a semi group IX, 1.5 (346) Representation I, 1.15 (6); I, 3.10 (21); . 1,3.12 (22); VII, 1.6 (258) Right annihilating product VIII, 5.12 (341) Right translation I, 3.12 (22) Semigroup I, 3.1 (16) Semigroup of subsemigroups of a semigroup III, 7.2 (128) Semigroup of transformations I, 3.6 (18) Semigroup with separating group part VI, 6.3 (249) Semilattice I, 5.13 (37) Special product of elements IX, 6.2 (375) Stability I, 5.17 (39) Subsemigroup III, 1.5 (93) Subsemigroup characteristic III, 7.2 (128) Successively annihilating band VIII, 4.2 (330) Supersemigroup III, 1.5 (93) Symmetry I, 5.6 (35) Transformation I, 3.5 (17) Transitivity I, 5.6 (35)
INDEX Type of a monogenic semigroup III, 3.7 (109); III, 3.16 (112) Uniqueness of division I, 1.18 (8) Unit element II, 2.3 (48); II, 2.4 (48); II, 2.6 (48) Universal class III, 1.9 (94) Universally maximal set IV, 2.1 (145)
447
Universally minimal set IV, 2.1 (145) World in an alphabet IX, 1.2 (345); IX, 1.4 (346)
Zero II, 2.3 (48); II, 2.4 (48); II, 2.6 (48) Zeroid element II, 1.6 (45)
512.86 1696 Liapin, E. S" Semigroup6
512.8:6 L696
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