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tsemiendos and asking for its range: gap> phi := IsomorphismTransformationSemigroup(semiendos); MappingByFunction( <semigroup with 4 generators>. <semigroup with 4 generators>, function( a ) ... end) gap> tsemiendos := Range(phi); <semigroup with 4 generators> So = V suppose first that (13, S be determined by Xn S be the homomorphism determined by X l S defined by Xl
Exercise 8 Look at a few of the elements of tsemiendos and work out what the isomorphism is. Now create the 'V-classes gap> dcl := GreensDClasses( tsemiendos );;
Exercise 9 Find the size of each of the 'V-classes. To quickly apprehend the structure of tsemiendos type: gap> DisplayTransformationSemigroup(tsemiendos); Rank 120: *[H size = 120, 1 L classes, 1 R classes] Rank 2: *[H size = 1, 10 L classes, 1 R classes] Rank 2: [H size = 1, 15 L classes (15 image types), 1 R classes (1 kernel types)] Rank 1: *[H size = 1, 1 L classes, 1 R classes] The '*' indicates that the 'V-class is regular. Clearly, the rank 120 elements comprise the group of automorphsisIIlS of S5, and the rank 1 element is the 'zero' mapping everything to the identity. So, to completely determine the poset of 'V-classes, it only remains to decide the order relation between the two rank 2 'V-classes: gap> IsGreensLessThanOrEqual( dcl[3] , dcl[2] ); true
Exercise 10 Describe the action of the endomorphisms in the two middle 'V-classes
Hints 1. To get the 2nd element of the list of 'V-classes, type gap> d2 := dcl[2];;
14
2. To get a representative element of a V-class, (say d2) use gap> x :=
Representative(d2);
3. To see the preimage of an element x of tsemiendos under phi use:
gap> a := PreImageElm(phi, x)j MappingByFunction( Sym( [1 5 ] ), Sym( [ 1 .. 5 ] ), function( x ) ... end )
4. To see the image of an element a of semiendos under phi use
5. To see what the group element (1,2) maps to under the endomorphism a use: gap> (1,2) -aj (1,2) 5
An infinite exrunple - The Heisenberg group
The first and most important step in dealing with a finitely presented semigroup is to find a solution to the word problem. Most commonly this entails producing an algorithm which, given any word, returns a canonical word representing the same element of the semigroup. Then two words represent the same element precisely when they have the same canonical form. Under certain circumstances, the function CanonicalForm can be obtained by finding a finite confluent rewriting system for the semigroup. We give a brief informal explanation of this idea, referring the reader to Sims [15] for a detailed exposition. A reduction ordering on the free semigroup over an alphabet A is a total order with the descending chain condition, such that if u < v then for all words x and y we also have xuy < xvy. One example of a reduction word order is the ShortLex word order: one begins with a total order on the letters of A. The order is then defined by u < v if u is shorter than v, or if they are the same length and u would come before v in lexicographic order. Another reduction ordering which we use in this example is the Basic Wreath Product ordering [15], however its description is beyond the scope of this tutorial. A rewriting system is a presentation, together with a reduction ordering. A relation which would be written u = v in a presentation (with u > v)
15
is denoted by u --+ v in a rewriting system, and is known as a rule. Then, for any word w = xuy in the alphabet, we may rewrite w to w' = xvy, which represents the same element of the semigroup and is smaller under the reduction ordering. The word w' may then be rewritten and so on, until an irreducible word which cannot be rewritten, is obtained. A rewriting system is said to be confluent if every two words which represent the same element have the same irreducible form, and in this case, rewriting is a canonical form algorithm solving the word problem. The Knuth-Bendix procedure is an algorithm which, given a presentation and a reduction ordering, adds rules which are consequences of the original relations, attempting to produce a confluent rewriting system upon termination, although in many cases the procedure will not terminate since no finite confluent rewriting system exists. The Heisenberg group is an infinite nilpotent group with solvable word problem. It turns out that, while one can find a confluent rewriting system for the Heisenberg group, it cannot be done using the ShortLex word ordering, so it is a good example in which to experiment with: • infinite finitely presented groups and semigroups; • commutative finitely presented semigroups; • rewriting systems. A group presentation for the Heisenberg group is given [4] as follows.
(0:, {3, "I I [0:, {3h- 1 , [0:, "I], [(3, "I]) . 5.1
Solving the word problem
We use GAP to find a confluent rewriting system for this group allowing us to test equality of two words. We start by creating the group. First construct the free group on three generators. gap> f := FreeGroup( "gamma", "beta", "alpha");
16
gap> relators := [Comm(a,b)*g--l, Comm(a,g), Comm(b,g)] jj gap> h := f/relatorsj
17
5.2
The Heisenberg group is infinite
One can easily demonstrate this well known fact by considering the largest commutative quotient of the Heisenberg group. If this quotient is infinite, then the Heisenberg group is certainly infinite. Since finiteness of a commutative semigroup is decidable (see, for example Gilman's paper [8)) we are able to test the claim in GAP. In the transcript below, aq is the largest commutative quotient of the Heisenberg group s: gap> aq := Abelianization(s);
Acknowledgements and Further Information The material presented here represents the combined efforts of a number of mathematicians over the course of more than a decade. These people are collectively known as The GAP Group. The particular functionality in support of semigroups is principally due to Robert Arthur, Gotz Pfeiffer and the authors. To learn more about GAP, or to obtain GAP for your own use, you can: • visit the website http://www-gap.des.st-and.ae . ukr gap/ • email gap
18
References 1. R. Arthur and N. RuSkuc, 'Presentations for two extensions of the monoid of order-preserving mappings on a finite chain', Southeast Asian Bull. Math (to appear). 2. J. J. Cannon and C. Playoust, 'An introduction to Algebraic Programming in Magma', School of Mathematics and Statistics, The University of Sydney (1996) . 3. P. Catarino and P. M. Higgins, 'The monoid of orientation preserving mappings on a chain', Semigroup Forum (to appear). 4. D. B. A. Epstein et. al., Word Processing in groups, Jones and Bartlett, Boston, 1992. 5. The GAP Group, GAP - Groups, Algorithms and Programming, Version 4.1; http://www-gap.dcs.st-and.ac.uk/ gap (Aachen, St Andrews, 1999). 6. The GAP Group, The GAP Reference Manual, http://www-gap.dcs.stand.ac.uk/ gap (Aachen, St Andrews, 1999). 7. C. Nobauer, Glissando, GAP Share Package for Nearrings, (Technical University, Linz, 1997). 8. R. H. Gilman, 'Presentations of groups and monoids', Journal of Algebra, 57:544- 544, 1979. 9. T. G. Lavers, 'Fibonacci numbers, ordered partitions, and transformations of a finite set', Australasian Journal of Combinatorics, 10:147-151, 1994. 10. T. G. Lavers and A. Solomon. 'The endomorphisms of a finite chain form a rees congruence semigroup', Semigroup Forum, 59:1-4, 1999. 11. S. A. Linton, G. Pfeiffer, E. F. Robertson, and N. RuSkuc, 'Computing transformation semigroups', Journal of Symbolic Computation, 11, 1998. 12. S. Linton, G. Pfeiffer, E. Robertson and N. Ruskuc, Monoid, GAP Share Package for Transformation Monoids (University of St. Andrews, 1997). 13. D. Kozen 'Lower bounds for natural proof systems', Proc. 18th Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, Long Beach, CA , 254-266 (1977). 14. B. H. Neumann, 'Some Remarks on Semigroup Presentations', Can. J. Maths. 19 1018- 1026 (1967). Corrigendum and addendum, Can. J. Maths . 20, 511 (1968). 15. C. C. Sims. Computation with finitely presented groups, volume 48 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, New York, 1994.
THE SEMIGROUP EFFICIENCY OF DIRECT POWERS OF GROUPS H. AYIK Qukurova Universitesi, Matematik, Biiliimii, 01330 Adana, Turkey E-mail: [email protected]. edu.tr
C.M. CAMPBELL~ J.J. O'CONNOR AND N. RUSKUC Mathematical Institute, University of St Andrews, St Andrews KY16 988, Scotland E-mails:[email protected]@[email protected]. uk A finite semigroup 8 is said to be efficient if it can be defined by .. semigroup presentation (A I R) with IRI-IAI = rank(H2(8 1 )) where H2(8 1 ) is the second integral homology of the monoid 8 1 obtained from 8 by adjoining an identity. In this paper we show that certain classes of direct powers of finite groups are efficient as semigroups.
1
Introduction
The efficiency of semigroups was introduced in [1] and the first examples of efficient and inefficient semigroups were given in that paper. Moreover, in [2] and [3], the efficiency of monoids was considered, and it is shown that a (finite) group is efficient as a group if and only if it is efficient as a monoid. However, in [3], an example is given which shows that an efficient finite monoid (which is not a group) is not efficient as semigroup. In this paper we return to the semigroup efficiency of finite groups. Let A be an alphabet. Denote by A + the free semigroup on A consisting of all non-empty words over A. A semigroup presentation is an ordered pair ( A I R ) , where R ~ A+ X A+. A semigroup S is said to be defined by the semigroup presentation ( A I R) if S is isomorphic to the quotient semigroup A+ / p, where p is the congruence on A+ generated by R. The deficiency of a finite presentation P = ( A I R) is defined by IRI - IAI, and is denoted by def(P). The semigroup deficiency of a finitely presented semigroup S, defs(S), is given by defs(S) = min{def(P) I P is a finite semigroup presentation for S }. The group deficiency of a group is defined similarly, by considering group presentations, and is denoted defa . So a finitely presented group G has (at least) *The corresponding author
19
20
two deficiencies, namely defG (G) and defs (G). Since a semigroup presentation for a group G is also a group presentation, we have defs(G) 2: defG(G). It is well-known that, for a finite group G, defG(G) 2: O. A better bound for defG(G) is defG(G) 2: rank(H2(G)) where H2(G) is the second integral homology of G (see [8, Corollary 10.17)). (An analogous result for monoids and semigroups has been proved by S. J. Pride (unpublished)) . IT defG(G) = rank(H2(G)) we say that G is efficient as a group, and if defs(G) = rank(H2(G)) we say that G is efficient as a semigroup. In this paper we consider direct products of groups and prove that the direct powers of the dihedral groups D2n and the alternating group ~ are efficient as semigroups. 2
Presentations for direct products of groups
Proposition 1 Let (A I R) and (B I Q) be semigroup presentations for two groups G and H, respectively. Then the direct product G x H can be defined by the semigroup presentation p = ( AU B I R U Q U C U {e = J}) where C = {ab = ba I a E A, bE B}, e E A+ and f E B+ are any two words representing the identity elements of G and H, respectively. Proof. From the relation e = f, it is clear that e is an identity for the semigroup defined by P. Each a E A has an inverse in A+ C (A U B)+, and each b E B has an inverse in B+ C (A U B)+. It follows that P defines a 0 group and this group is obviously G x H. Notice that the deficiency of the semigroup presentation P is one greater than the deficiency of the standard group presentation ( AUB I RUQUC) for the direct product of groups. In some cases it is possible to find a semigroup presentation with the same deficiency as the standard group presentation for the direct product of groups.
Proposition 2 Let (A I R) and (B I Q) be two semigroup presentations for two groups G and H, respectively. If there is at least one relation of the form alual = al (al E A, u E A*) in R and at least one relation of the form bl vb l = bl (bi E B, v E B*) in Q, then the semigroup presentation Q
= (A UBI
R U Q' U C' U {ual
where Q' = Q\{bIvb l = bl } and C' product as a semigroup.
=
= bi v,
C\{bial
=
al bl u
= bl } )
albIl defines the direct
21
Proof. First we choose e == ual and f == vb l in the presentation P in Proposition 2. Then we apply Tietze transformations to show that P and Q define the same semigroup. (P ::::} Q) Since the relations al u = ual and bl v = vb l are consequences of R and Q, it follows from the relations bl al = al bl , ual = bl v and bl vb l = bl in P that
alblu = blalU = blual = blblv = blvb l = bl · (Q ::::} P) Since the relation al u = ual is a consequence of R, it follows from the relations al bl u = bl , ual = bl v and al ual = al in Q that
blvb l = blvalb l u = ualalbl u = al ualblu = alb l u = bl , and so the relation bl vb l = bl holds. Finally, observing that bl v = vb l holds in H and so is a consequence of Q = Q' U {bl Vbl = bd we have blal = (alblu)al = albl(blv) = alblvb l = alb l · Thus P and Q define isomorphic semigroups, as required. D Now we consider finite imperfect groups as in [6], but as semigroups. For a group G, we denote the minimum number of generators of G by d( G), and the n-th direct power of G by Gn. For a finite imperfect group G, it is known that there exists an integer no such that, for all n ;::: no, d(Gn) = nk where k is the rank of GIG' (k = d(GIG')), and G' denotes the commutator subgroup of G (see [9, Theorem B]. We shall refer to the least such no as the growth index of G. Moreover, we have
_ {(d(G) -l)/(k -1) if k ;::: 2 no 2d( G) + 1 if k = l. For a finite (imperfect) group G, let m be the rank of H 2 (G), let k be the rank of GIG', and let j be the number of cyclic factors of H2(G) whose orders are coprime to at least one of the orders of the cyclic factors of GIG'. If we denote the rank of H2(Gn) by mG(n), it is known that there exists an integer nl such that, for all n ;::: nl, mG(n) = n(m - j) + (k 2n(n - 1))/2 = (k 2n 2)/2 + an for some constant a (see [6]). We call the least such nl the multiplier index of G. Let n2 be the maximum of the growth index no and multiplier index nl of G. We say that G has quadratic efficiency at n if n ;::: n2 and Gn has a presentation ( A I R) such that IAI = d(Gn) and IRI -IAI = mG(n). The following theorem was proved in [6, Theorem 3.1.]. Theorem 3 With the above notation, if a finite imperfect group G has quadratic efficiency at n for every n, n2 ::; n ::; 2n2 - 1, then G has quadratic efficiency at n for all n ;::: n2·
22
The efficient group presentation for C n was obtained by using the standard presentation for the direct product of c[n/2] xc[(n+1)/2] where [x] denotes the integer part of x . Therefore, from Proposition 2 and Theorem 3 we have the following corollary: Corollary 4 If, for every n, n2 :::; n :::; 2n2 - 1, C n has a semigroup presentation (A I R) such that IAI = d(C n ), IRI-IAI = mc(n) and there is at least one relation of the form alual = al (al E A, u E A*) in R, then C n is efficient as a semigroup for every n 2: n2.
3
The semigroup efficiency of D;:' and
A:r'
It may be shown that the dihedral group D2n is efficient as a semigroup (see [1, Theorem 2.2] and [3, Theorem 3.1]). Efficient semigroup presentations of D2n are
(a,b I a 3 = a, bn = a 2, abn-1a = b) (a, b I bn a 2b = b, b(n-l)/2 a 3 b(n-l)/2
(n even)
= a)
(n odd).
(1) (2)
In this section we show that direct powers of D2n are efficient. Before doing this, we give the following technical lemma (for a proof see [7, Proposition 1.3] and [4, Theorem 3(iii)]).
Lemma 5 Let P = ( A I R) be a semigroup presentation. If there exists a word e E A+ such that, for each a E A, ea = a (ae = a) and uaa = a (au a = a) for some U a E A+, then P defines a group. In particular this group is isomorphic to the group which is defined by P when it is considered as a group presentation. Proposition 6 The semigroup presentation R2 = (a, b I a 2n b = b, (a n b)2a = a, (ab)2a 2n = b2n ) defines D~n with n odd. Proof. From the second and first relations of R2, a 2n+1 = a 2n (a n b)2a
== an (a 2n b)an ba =
(a n b)2a = a.
(3)
This, together with the first relation, implies that a 2n is a left identity for the semigroup S defined by R 2 . It is clear that a 2n - 1 is a left inverse of a. We show that anb2nanb2n-l is a left inverse of b. From the first and second relations of R2 ,
23 and so b2nanb2n = banb. It follows from the third relation, (3), and the first and second relations that (a n b2n )2
= anb2nan(ab)2a2n = anb2nan(ab)2a4n = anb2nanb2na2n = (a n b)2a2n = a 2n .
Thus, from Lemma 5, R2 defines a group. From [5, Lemma 4], this group is D~n with n odd. 0 Proposition 7 The semigroup presentation 2 x 2n - 1 ax - a , R3 = (a , x , z I zazaz = Z , (xzn)2n - a, 1 n n 1 x - Z = zx + , zn-lxzn+l = x, x2n = a 2 ) defines D~n with n odd. Proof. From the fifth and first relations of R3, we have x(az)2 = x. From this and the third relation of R 3 , we have a(az)2 = a. It now follows from the first relation that (az)2 is a right identity for the seroigroup S defined by R 3. It is clear that zaz, x 2n - 2axzaz and zn-2xzn+lx2n-2axzaz are right inverses of a, z and x, respectively. Thus, from Lemma 5, S is a group. Next we show that the relations x2n = a2 = (az)2(= 1) hold in S. Indeed, from the third and the sixth relations of R3, we have
and so x4n = a 4 = 1. It follows from the fourth relation of R3 that x n(n-l)zx n(3n-l) = z. Since n is odd, either n(n - 1) == 0 (mod 4n) and n(3n - 1) == 2n (mod 4n) or n(n - 1) == 2n (mod 4n) and n(3n - 1) == o (mod 4n) . In both cases, it follows from the equations x n(n-l)zx n(3n-l) = z and x4n = 1 that x2n = 1 holds in S. Therefore, if we add the relation x2n = 1 into R 3 , we obtain a presentation for D~n with n odd (see [5]), as required.
o Theorem 8 The direct power integers m and n. Proof.
D~
is an efficient semigroup for all positive
In [5] it was shown that rank(H (Dm)) = { ~(m2 - m) ~f n 2 2n 2m 2 - m if n
~s odd; IS
even.
(4)
Thus, for even n , the result follows from (4), (1), Proposition 2 and Corollary 4; and for odd n, the result follows from (4), (2), Propositions 2,6 and 7, and Corollary 4. 0
24
In (6) it is shown that all direct powers of the alternating group ~ of degree 4 are efficient groups. We complete this section by proving that Ar is also efficient as a semigroup. From Corollary 4, we require to prove the result just for m = 1, 2, 3, 4,5 as in [6, Theorem 4.1). Theorem 9 The following semigroup presentations 7i. = (a,b I a 3 = a, b2 = a 2, (ab)3b = b) , T2 = ( a,b I a4 = a, b3 = a 3, (ab)6b = b, (a 2b2ab)2 = a 3 ), 'Fa = (a,b , c I (a 4 b)6 a = a , a 3ba3 = b, (a 2b2a4 b)2c = c, (a 3c)3 = (a 4 b)6, a 2c = ca 2b3, be = cbc3 ), T4 = (a, b, c, d I a4 = a, (a 2b2ab)2c = c, (ab)6b = b, c3 = a 3, (cd)6d = d, be = cb, bd = db, ac = cab3, (c 2d 2cd)2 = d 3, dad 2 = a), 75 = ( a, b, c, d, e I a4 = a, (a 2b2ab)2c = c, (ab)6b = b, (c 2d 2c4 d)2e = e, c3dc 3 = d, (c 4 d)6 = a 3, (c 3e)3 = a 3, ec 2d 3 = c 2e, ede 2 = d, ea = aeb3, ac
= ca,
da
= ad,
be
= cb, db = bd,
be
= eb)
define Ar for m = 1,2,3,4,5, respectively. Therefore Ar is efficient as a semigroup for every positive integer m. Proof. By using Lemma 5, it is shown that all Tm (m = 1, ... , 5) define groups with identities a 2, a 3 , (a 4 b)6 , a 3 and a 3, respectively. It follows from [6, Theorem 4.1) that , for every m = 1, ... , 5, Ar is a homomorphic image of the group defined by Tm. Moreover, a coset enumeration program shows that the group defined by Tm has the same order as Ar. Thus we conclude that, for every m = 1, ... , 5, Tm defines Ar. Therefore, since if m = 1, if m = 2,
C2
H 2 (Ar)
=
C2 x C 6 { m (m-3) / 2 C2
x
em'f >3 6 1m _ .
where C 2 and C 6 are the cyclic groups of orders 2 and 6 respectively, it follows from Corollary 4 that Ar is efficient as a semigroup for every positive integer m. 0 References 1. H. Aylk, C. M. Campbell, J. J. O'Connor and N. RtiSkuc, M inimal presentations and efficiency of semigroups, Semigroup Forum, to appear. 2. H. AYlk, C. M. Campbell, J. J. O'Connor and N. Ruskuc, On the efficiency of wreath products of groups, Proc. Groups Korea '98, to appear. 3. H. AYlk, C . M. Campbell, J. J. O 'Connor and N. RtiSkuc, The semigroup efficiency of groups and monoids, Proc. Roy. Irish Acad. Sect. A , to appear.
25
4. C. M. Campbell, E. F. Robertson, N. Ruskuc and R. M. Thomas, Semigroup and group presentation, Bull. London Math. Soc. 27 (1995), 46-50. 5. C. M. Campbell, E. F. Robertson and P. D. Williams, Efficient presentations of some powers of groups, in "Groups Canberra 1989", Lecture Notes in Mathematics 1456, Springer-Verlag, New York (1990) , 106-113. 6. C. M. Campbell, E. F. Robertson and P. D. Williams, On the efficiency of presentations of direct powers of imperfect groups, Algebra Colloq. 4 (1997), 21-27. 7. T. W . Hungerford, "Algebra", Graduate Text in Math. 73, SpringerVerlag, New York , 19'74. 8. J. J. Rotman, "An Introduction to Homological Algebra", Academic Press, New York, 1979. 9. J. Wiegold, Growth sequences of finite groups II, J Austral. Math. Soc. 20 A (1974) , 225--229.
INVERSE TRANSVERSALS - A Guided Tour T . S. BLYTH
Mathematical Institute, University of St Andrews, Scotland
We offer an up-to-date survey of results concerning regular semigroups with inverse transversals, several of which are new. In Section 1 we derive the basic properties and in Section 2 we give a classification of the most important types arising. In Section 3 we list some interesting illustrative examples. Section 4 is devoted to various structure theorems, and Section 5 to congruences.
1
Basic Properties
If S is a regular semigroup then by an inverse transversal of S we mean an inverse subsemigroup T of S that contains precisely one inverse of every element of S. This notion has its roots in a 1978 paper by the author and D . B. McAlister on orthodox semigroups [13]' the term itself being introduced by the author and R. McFadden in 1982 [2] . If T is an inverse transversal of S then for every xES we shall denote by XO the unique element of Tn V(x) and write T as So = {XO ; xES}. Then in the inverse semigroup So we have (XO)-l = XOO and therefore XO = xOoo for every xES. In what follows, we shall assume that S is a regular semigroup with an inverse transversal So . Most of the following basic properties involving the unary operation x >-+ XV are drawn variously from [6 ,21,24 ,28]. For the reader's convenience, and in order to appreciate the ubiquitous calculus involved, we include all of the proofs in this section.
Theorem 1.1. (Vx, yES) {xy)O
=(XO xy)O XO = yO{xyyO)O = yO (XO xyyO)O XO
Proof. For example, as is readily seen, (XOxy)OXO E So
n V(xy) .
Two particularly important subsets of E(S) are I A
{x E S ; x {x E S; x
= XXO} = {x E S ; XO x = XO}; = XOx} = {x E S; XXV = XO}.
Equivalent definitions are 1= {XXO
A
xES},
= {XOx
; xES}.
For example, by Theorem l.1 we have XXO(XXO)O xxo Xoo XO = XXO, so xxo E I.
26
=
XXO(XVXXO)OXO
o
27 Theorem 1.2. I n A = E(5°). Proof. If x E I n A then XOx = XO and x = XOx. Thus x = XO E 5° n E(5) = E(5°). Conversely, if x E E(5°) then x 2 = x = XO gives x = xxo = J:ox. 0
Theorem 1.3. I and A are sub-bands of 5. Moreover, I is left regula r and A is right regulm o
•
Proof. Suppose that i,j E I and consider the sandwich element 9 E 5(i,j) given by 9 = j(ij)Oi. Using Theorem l.1 , we have 9 = j(ij)Oi = }(iOij)OiOi = j(iOij)"io = j(ij)O whence we see that ig = ij(ij)O E 1. We also have , by Theorem l.1 again, gO = [j0 j (ij)0 jO This, together with the fact that jO = jO j, gives gO = gO j. Using these facts, we see that 9 £ ig £ (ig)O whence (ig)O E 5° n V(g) and therefore gO = (ig)O Thus gO £g and so we have 9 = ggO = ggO j = gj which gives ij = igj = ig E 1. Hence I is a subsemigrou p which is clearly a sub-band. To see that I is left regular, observe from the above that ij = ig gives ij = iji. Dually, A is right regu lar. 0
no
Theore m 1.4. (Vx,yE5)
(xyO)o=yOOxO,
Proof. Since I is left regular , we have yO xOo . XOY yO xOo yO yyO xOo XOyyO xOo
=
(XVy)o=yOxOO
= yO yyO xOo XOxOo = yO xOo
and similarly XO y yO xOo . XO Y = XO y. Thus yO XOO E 5° n V (XV y) and so yO:roo = (XVy)O Dually, we have the second identity. 0
Theore m 1.5. Green's relations £ and R on 5 are given by (x , y) E £ <===> xOx = yOy , (x , y) E R <===> xx o = yyO Proof. Suppose, for example, t hat (x, y) E R . Then xS = y5 gives XXO = yz for some z E 5 whence XXO = yyO xxo Similarly, yyO = xxo yyO Since I is left regular we have XXO = yyO XXO = XXO yyO XXO = XXO yyO = yyO Conversely, if xxo = yy" then x = XXO x = yyO x E y5 and similarly y E x5 whence (x , Y)ER. 0
Other significant subsets of 5 are
= {x E 5 ; x = XXO XOO} = { x E 5 ; XO x = XOXOO }; R = {x E 5 ; x = XOO XO x} = {x E 5 ; XXO = XOO XO }. 1.6. L n R = 5° L
Theor e m
28 Proof. If x E L n R then x = xxo xoo conversely if x E 5° then x = xoo gives x xER .
= xoo Xo xxo xoo = xoo E 5°; and = xxo x = xxo xoo ELand similarly 0
Theorem 1.7. If x E Lor y E R then (xy)o
= yOxo.
Proof. If, for example, x E L then, using the basic formulae of Theorems 1.1 and 1.4, we see that (xy)O = yO(XOxyyO)OXO = yO(x"xOOyyO)OXO yO(yyO)O(XOXOO)OOXO = yOyOOyOxOxOOxo = yOx· 0 Theorem 1.8. L = U eEE(so)Le and R = U eEE(so)Re are subsemigroups of 5, with L left inverse (R-unipotent) and R right inverse (C-unipotent).
Proof. If x, y E L then by Theorem 1.7 we have (xy)O
= yOxO
It follows that
= yOxOxy = yOyOOyOxOxOOyyOyOO = yOxOxOOyOO E 5° and so (xy)Oxy = ((xy)Oxy)"o = (xy)O(xy)OO whence xy E L. Thus L is a subsemigroup. We now observe that E(L) = I. In fact, if i E I then iO E E( 5°) and so iO = iOO whence i = iio = iio iO = iio iOO E L; and conversely if g E E(L) then g"g = gOgOO E 5° and g"g E V(g) give g"g = gO (xy)Oxy
whence g = ggO E I. Since 5° ~ L it follows that L is orthodox with a left regular band of idem po tents and so is left inverse. 0 The situation can be summarised by the Venn diagram
R
L
E(S)
Note from the above that 5° is an inverse transversal of both Land R. Theorem 1.9. 5 is left [right] inverse if and only if A = E(5°) [I
= E(5°)] .
Proof. Suppose that 5 is left inverse. Then E(5) is a left regular band. If I E A we then have I WI lio 1° and so A E(5°). Conversely, if A E(5°) then for every x E 5 we have XOx x"xoo whence 5 L and so 5 is left inverse. 0
=
=
=
=
=
=
=
29 From the definition of R we clearly have R ~ So A. The reverse inclusion follows from the fact that R is a subsemigroup. Hence R = So A, and similarly L = ISo In view of this it is natural to consider also the set product IA . Theorem 1.10. IA
= {x E S;
XU
E E(SO)}.
Proof. If i E I and I E A we have (il)0 = l°(i°illO)OiO = l°(i°lO)OiO = laze E E(SO). Conversely, if XO E E(SO) then x = xxox = XXOXOx E IA . 0 Corollary. So n IA
= E(SO). o
Since iO E E(SO) for every i E I we have I
~
lA , and similarly A
~
IA .
Theorem 1.11. Th e following statements are equivalent : (1) IA is a subsemigroup of S; (2) IA = (E(S)); (3) x E E(S) => XO E E(SO) [i.e ., E(S) ~ IAl;
(4) (AI)O <;;: E(SO). Proof. (1) => (3) : If (1) holds then AI ~ lA, so x E E(S) gives XO = xOxxo XOxxx" E IA whence, by the Corollary of Theorem 1.10 , XO E E(SO). (3) => (4) : For i E I and I E A we have i(lWI E E(S). Observe that (i(litlf = 10(iOi(li)OUOriO = 1° (i°(li)OIOfiO = 10(litOiO
=
= (li)OO
Consequently, if (3) holds then (li)OO E E(SO) whence (li)O E E(SO). (4) => (1) : Suppose that (4) holds and that x, y E IA . Then (XOxyyO)O E E(SO) and xO, yO E E(SO). It follows that (xy)O = yO(xOxyyO)o :ru E E(SO) and therefore xy E IA. Hence IA is a subsemigroup. (3) => (2) : If (3) holds then E(S) ~ IA. But clearly IA <;;: (E(S)) and, since (3) => (1) from the above, IA is a subsemigroup . It follows that
IA
= (E(S)). (2)
=> (1) : This is clear.
0
We have the following important special case of Theorem 1.11. Theorem 1.12. The following statem en ts are equivalent: (1) S is orthodox;
(2) (Vx,yES) (xy)o=yOxO; (3) (Vi E I)(VI E A) (li)O = iO 1°; (4) x E E(S) ¢:::} XO E E(SO) [i .e., E(S)
= IAl·
30
Proof. (1) => (2) : If 5 is orthodox then yO XO E 5° n V (xy) whence (2) follows. (2) => (3) : This is clear. (3) => (2) : If (3) holds then we have the identity (yO yXXO)O = xOo XO yO yOO On pre- multiplying by XV and post-multiplying by yO, we obtain (y:r) ° = XO yV (2) => (4) : Suppose now that (2) holds. If x E E(5) then :1'0 = (xx)O = xOxo whence XO E E(5°); and conversely if XO E E(5°) then x = xxox = x(xO)2x = x(x 2)Ox = x(x 2 )ox 2(x 2)Ox = X(XO)2x 2(xo)2x = xxoxxxvx = xx whence x E E(5). (4) => (1) : If (4) holds then for all e,f E E(5) we have eo,r E E(5°) whence (eOefr)O = rO(ef)°eoo = r(ef)°eo = (ef)° But (4) of Theorem 1.11 also holds, and therefore (eOefr)O E E(5°). Thus (ef)° E E(5°) and so , by (4), ef E E(5). Hence 5 is orthodox. 0 2
A classification of inverse transversals
Historically, and in chronological order, the following basic types were first to emerge . An inverse transversal 5° is said to be
• multiplicative if AI ~ E(5°) [2]; • a quasi-ideal if 5° 55° ~ 5° or, equivalently, if AI ~ 5° [15]; • weakly multiplicative if (AW ~ E(5°) [21] . Note that each of these is characterised in terms of the set product AI. Observe also that the weakly multiplicative situation has been described III Theorem 1.11. As shown in [4], these conditions are naturally related: Theorem 2.1. The following statements are equivalent: (1) 5° is multiplicative; (2) 5° is weakly multiplicative and a quasi-ideal.
D
Now in Theorem 1.3 we have seen that an inherent property of the subband I is that it is left regular. It is natural to consider those inverse transversals for which I satisfies the stronger property of being left normal (in the sense that i)k = zk) for all i,) , k E I) , and simil arly those for which A is right normal. Surprisingly, there are many equivalent properties that surface in this investigation. From [7] and [10] we have the following list. Theorem 2.2. The following statements al'e equivalent: (1) I is left normal; (2) (Vi ,) E I) i) i)O; (3) E(5°) is a l'ight ideal of I; (4) (Vx , y, z E 5) xOyzo = xOyOOyOyzO; (5) 5° 15° ~ 5°;
=
31
(6) So is a right ideal of L; (7) (Vx E SO)(Vi E I) xixo E E(SO); (8) (Vx E S)(Vi,j E I) xi(xi)O and xj(xj)O commute; (9) AI ~ R; (10) (Vi E I)(VI E A) (Ii, (li)oo) En; (11) S is locally right inverse.
o
We say that So is
• left simplistic if anyone of the properties of Theorem 2.2 holds; • right simplistic if anyone of the dual properties holds. Theorem 2.3. The following statements are equivalent: (1) So is both left simplistic and right simplistic; (2) So is a quasi-ideal; (3) S is locally inverse.
o
The amalgamation of the above properties and their duals in the case of a quasi-ideal inverse transversal explains why such inverse transversals have enjoyed the greatest popularity. In addition to the above, we have the following result from [7]. Theorem 2.4. The following statements are equivalent : (1) So is weakly multiplicative and left simplistic;
(2) AI
~
o
A.
So we have the following semilattice: WID
Is
rs
(td)O ~E( S O)
AI~R
AI~L
J\I~A
In order to extend this diagram we highlight a natural property of inverse transversals that lies between being multiplicative and a quasi-ideal. For this purpose, we focus on the involvement of Green's relation n in Theorem 2.2. In general we have the egg-box picture
32
Ii
• li(li)O
Ii (lW (li) 00
• (li)Oli
(Ii) °
• (lW(li) OO
liliJoO (lit Ii
• (li)OO(li)O
(IWO
in which the elements marked. are idempotent. By Theorem 2.2( 10) we see that So is left simplistic if and only if, in each such egg-box, the top and bottom R-classes coincide . Simil arly, by Theorem 2.4 we see that So is weakly multiplicative and left simplistic if and only if the egg-box reduces to a single R-class and two .c-classes . Thus we see that natural collapses in the egg-box provide particular situations. In this connection , and fo cussing on symmetric situations , we shall be especially interested in the observations (a) (ii , (li)O) E 1i <==> li(li)O = (lWli ; (b) ((fi)O, (li)OO) E {R,.c, 1i} <==> (li) °(lWo = (li)o°(li)O We shall say that an inverse transversal So is • perfect if li( li) O= (li)Oli ; • weakly perfect if (li)°(li)OO = (li)o°(fi) O The following results are to be found in [10] . Here we denote by E(SO)( the centraliser in S of E(SO) , and by ;.t so the biggest idempotent-separating congruence on So Note that E(SO)( n So = ker ;.t so (see, for example [12]) . Theorem 2.5. The following statements are equivalent : (1) So is perfect; (2) (Vi E I)(VI E A) Ii = ZO WO; (3) AI ~ ker ;.t s o; (4) AI ~ E(SO)(.
D
Theor em 2.6. Th e following statements are equivalent : (1) So is weakly perfect; (2) (ViE I)(VIEA) (li)o=(iolil of; (3) (AW ~ ker ;.t so.
D
Theorem 2.7. Th e following statements are equivalen t : (1) So is weakly perf ect and left simplistic; (2) ( Ii, (ii)0) E R .
D
Observe that Theorem 2.5(2) can be expressed as the conjunction of th e identities Ii = iO Ii and Ii = lilo We shall say that So is
• left perfect if (Vi E I)(VI E A) Ii = iO li ; • weakly left perfect if (Vi E I)(VI E A) (fi)O
= (i°li)O ;
33
• right perfect if (Vi E 1)("1lEA) li = lilo; • weakly right perfect if (Vi E 1)("11 E A) (Ii) ° = (lil O)° Recall by Theorems l.11 and l.12 that if 5 is orthodox then 5° is weakly multiplicative whence, by Theorem 2.6, 5° is weakly perfect. Now there is a generalisation of the orthodox property that also implies 5° weakly perfect. A regular semigroup Q is said to be quasi-orthodox [32] (equivalently, E-solid [11]) if there is an inverse semigroup T and a surjective morphism f : Q --+ T such that 1 {e} is a completely simple subsemigroup of Q for every e E E( T). From [10] and [23] we have:
r
Theorem 2.8. The foll owing statements are equivalent: (1) 5 is quasi-orthodox; (2) ("Ix , y E 5) (xy)O(xy)OO = yOxOxOOyOO; (3) ("Ix, y E 5) (xy)OO(xy)O = xOOyOOyOxV; (4) ("Ix E 1)("11 E A) (li)°(li)"o = 10io; (5) (Vi E 1)("11 E A) (li)o°(li)O = 10io
o
As shown in [9]' various properties follow from this. For example : Theorem 2.9. The following statements are equivalent: (1) 5 is orthodox; (2) 5 is quasi-orthodox and 5° is weakly multiplicative.
o
Theorem 2.10. The following statements are equivalen t: (1) 5 is quasi-orthodox and 5° is left simplistic; (2) (Vi E 1)("11 E A) (li , 10iO) E R.
o
Combining all of the above results, we obtain the following semilattice: Is
ri
rs
34 Remarkably, in the case of monoids with inverse transversals the above diagram collapses considerably. In fact, we have from [7]: Theorem 2.11. If 5 is a monoid then 5° is left simplistic if and only if 5 is right inverse. 0 For monoids the above diagram therefore reduces to wlp
wrp p
Wffi
Is:o Ip:o ri
qo
Ii:o rp:o rs
i:Offi:O p:oqi
3
Some examples
For convenience in describing some illustrative examples, we shall make use of the following terminology. With reference to the general semilattice above , we shall say that the inverse transversal 5° is of type x if 5° satisfies the defining identities of x but not those of any y that lies below x in the diagram. Example 3.1. Let 5 be a naturally ordered regular semigroup (i.e. 5 has a compatible order which extends the natural order on the idempotents). Suppose also that 5 has a biggest idempotent a. Then 5° = a5a is an inverse transversal of 5 [3]. Since eO = aea E E(5°) for every e E E(5) we have by Theorem l.11 that 5° is weakly multiplicative. As shown in [14], 5 is also locally inverse so, by Theorem 2.3, 5° is a quasi-ideal. By Theorem 2.1, 5° is multiplicative. If 5 is not orthodox then 5° is of type m. Example 3.2. [4] If B is a boolean algebra then the semigroup B2 Mat2x2 B is regular and non-orthodox. Moreover, every
[~
!1
=
E B2 has
a biggest inverse, namely a [c
b]O _ [b'(a+ C)+c'(a+b)+d Q'(C+d)+d'(a+C)+b] d a'(b+d)+d'(a+b )+c b'(c+d)+c'(b+d)+a '
and the set B2 so described is an inverse transversal of B2 that is of type wm . Example 3.3. [6] Let 5 be the set of real singular 2 x 2 matrices having a non-zero element in the (1, I)-position; i.e. matrices of the form
[ac a _~ bc].
35
Let M consist of S with the 2 x 2 zero matrix adjoined. Then M is a regular semigroup and , relative to the definitions [:
a-~bC]O
[a~l~]
[°0 °0] ° -_ [0° °0] ' the set M ° _- {[x° °0] .,x i- °}U {[00]}. °°
and
an verse transversal of M and is of type p . If we adjoin the identity matrix with 12 defined to be h, then (Ml)O is of type wp. If Q is the subset M given by IS
.
111-
h
then QO = MO\{O} is an inverse (in fact, a group) transversal of Q and is of type qo 1\ p. Moreover, (Ql)O is of type qo.
Example 3.4. If S is a completely simple semigroup then every tl-class of S is an inverse transversal of S and is of type p. Example 3.5. Let M == M[S; 2, 2; P] be the regular Rees matrix semigroup over an inverse monoid S with sandwich matrix P
= [~
!]
where a E S is
to be specified. Consider the subset MO = {(I , x , 1) ; xES} . Clearly, this subsemigroup of M is isomorphic to S, so is inverse. Now V (m , x, n) n MO = {( 1, X -I , I)} so MO is an inverse transversal of M with (m, x, n)O = (1, X-I, 1). The elements of I are those of the form i == (m , x , n)(l , X-I, 1) = (m,xx- 1 ,1) and those of A are those of the form I == (l,y-l,l)(m,y , n) = (I, y-ly , n). It follows that Ii E MO and so MO is a quasi-ideal of M. Taking in particular i = (2,1,1) and 1= (1,1,2) , we have li(li)O = (1, aa- I , 1) and (li)Oli = (1, a-la , 1). Thus , if a is chosen such that aa- l i- a -I a then MO is not weakly perfect . Observe also that in general Ii iO/i li/ o
(1 , y-lypnmXx-l, 1); (1, xx-ly -IYPnmxx-l , 1);
= (1,
y-lYPnmxx-ly-ly, 1).
Now let S be the bicyclic semigroup (a, b ; ab = 1). If we choose a E {a, b} then aa - I i- a-lao Let Ma be the semi group obtained by choosing a = a. Since eae = ea for all idempotents e E S we see that in general iO Ii i- Ii = li/ o and so in this case M~ is right perfect but not left perfect. Thus M~ is of type qi 1\ rp. Now let Mb be the semigroup obtained by choosing a = b. Since ebe = be for all idempotents e E S we see that in general iO Ii = Ii i- li/ o and so in this case M; is left perfect but not right perfect. Thus M; is of type qi 1\ lp.
36
Example 3.6. If Ma and Mb are as in Example 3.5 then the cartesian product semigroup Ma x Mb has an inverse transversal M; x Mt of type qi. Examples of the other types in the above diagram can also be constructed. For instance, to obtain an inverse transversal of type wlp it suffices to consider the cartesian product of the semigroup Ml of Example 3.3 and the semigroup Mb of Example 3.5. 4
Structure theorems
A regular semi group may have several inverse transversals (see, for instance, Example 4 above) . How any two inverse transversals of S are related has been an outstanding problem. A partial solution to this can be found in [22] where it is shown that if S has a quasi-ideal inverse transversal then all inverse transversals of S are isomorphic. In fact, the following general result has been established only very recently [1]. Theorem 4.1. Inverse transversals of S are mutually isomorphic. More precisely, if SO, S* are inverse transversals of S then an isomorphism from S* 0 to SO is given by x >-+ (x*xxot. Structure theorems for regular semigroups with inverse transversals exist but are very complicated. The first to appear was the following characterisation for the case where So is a quasi-ideal [15]. Theorem 4.2. Let T be an inverse semigroup, let r be a left normal band, and let ~ be a right normal band. Suppose that r and ~ have a semilattice transversal that is (isomorphic to) E ( T) . Let [ , ] : ~ x r -+ T be stich that
(1) (Va, bE T)(Vi E r)(VI E~) a[l, i]b = [ai , ib]; (2) (Vi E r)(VI E~) [I, iO] = W, i] = 10io Consider the set W
= {(i,x,l) E r
x T x ~; iLx-lnl}
and the law of composition (i ,x, l)(j,y,m)=(iaa- l , a, a-lam)
where
a=x[l,j]y.
Then W is a regular semigroup with a quasi-ideal inverse transversal that is isomorphic to T Moreover, every regular semigroup with a quasi-ideal inverse transversal can be constructed in this manner. 0
More complicated results of this nature that describe the structure of regular semigroups having other types of inverse transversal can be found
37
in [2,19,24]. All are based on the coordinatisation x == (XXO,XOO,XOx) and algebraically we can assert that in general S is isomorphic to the set W
= {(i,x,l) E I x
So x A; i£x-1Rl}
under the multiplication (i, x, l)(j, y, m)
= (ixlj(xlj)O,
x(lj)OOy, (ljy)Oljym).
A corollary of this is the following result (of which Theorem 2.8 above is a consequence) [23] .
Theorem 4.3. S is completely simple if and only if So is a group.
Proof. If So is a group with identity element 1 then it is readily seen that we have I = {i E S ; iO = 1 and il = i} and A = {I E S ; 1° = 1 and 11 = I}. Now if i,j E I then ij = iljl = WHO = ijO = il = i and so I is a left zero semigroup. Similarly, A is a right zero semigroup. The above multiplication therefore reduces to (i , x, I) U, y, m) = (i, x(lJ)OO y, I). Hence W is the A x I Rees matrix semigroup over the group So with sandwich matrix P = [(lJ)OO]AxI. Conversely, suppose that S c:::: M[G;I,A;P]. If e = (i ,Pli1,1) and f = U, p;;;;, m) are idempotents of So then, since e and f are inverses of (i , p;;;~ , m) and So is an inverse transversal, we see that e f and so So has only one idempotent , whence it is a group. 0
=
The more general question of when S can be embedded in a Rees matrix semigroup over an inverse semi group has been addressed in [16] . This involves the existence of a quasi-ideal inverse transversal So which is rigid in the sense that the associated bands I and A have one-one structure mappings. There are equally complicated structure theorems that involve the subsemigroups Land R, a consequence of which is that algebraically S is a spined product of Land R. More precisely, S is isomorphic to the set
U
= {(x, a)
E L x R; XO
= aO}
with the multiplication
(x, a)(y, b)
= (xxOay(ay)"(ay)OO, (ay)"O(ay)OaybOb).
If in particular So is a quasi-ideal then we have ay = aOo aO ayy" yOO E So SSO So and so this product simplifies to
~
38
5
Congruences
In what follows we shall denote by Con S the set of congruences on S. As far as the unary operation x >-+ XO is concerned, the following result , which was first established in [29] , shows that every congruence on S is also a congruence on the algebra (S; 0). Theorem 5.1. If cp E Con S and (x, y) E cp then (xv , yO) E cp .
Proof. If (x, y) E cp then [XXO ]cp = [yxO ]cp = [yyOyx O]cp = [yy O]cp [yxO ]cp whence , since [yyO] cp E E(Sjcp), we have [yyOXXO]cp = [XXO ]cp. Similarly, [y OyxOx]cp = [yOy]cp. But , I being left regular, [XO]cp = [x"XXO]cp = [XOyyOxXO ]cp = [XOyyO ]cp; and, A being right regular , [yO]cp = [yOyyO]cp = [y"yxOxyO]cp = [XOxyO ]cp. The result now follows since (x, y) E cp. 0 Descriptions of the structure of congruences on S in terms of congruences on the building bricks I, So, A were given independently in [6] and [29]. Here we follow that of the former. We define a triple (L , 11" , >.) E Con I x Con So x Con A to be
(a) balanced if
LIE(SO)
= 1I"IE(SO) = >'IE(SO);
(b) linked if for all iI, i2 E I, all Xl, X2 E So, and all 11 , 12 E A,
(h id/l ido , 12 i2(12 i 2)0) (iI, i 2) E
(iI , i2) E (/ 1 ,/2 )
E
L,
(11,/ 2) E >.:::}
~, (Xl,
X2) E
>., (Xl, X2)
E
{
((11 ido, (l2i2)0) E
E
L
11"
[0'] [,B]
11" :::}
((ilidO/lil ' (/2i2)0/2i2) E>' (Xl ilx?, x2i2xn E ~
b] [6]
11" :::}
(xf/lXl' X2 /2 X2) E>'
[s]
Under the cartesian ordering, the set BLT(S) of balanced linked triples forms a lattice that is isomorphic to Con S. The structure of the congruences on S is then as follows.
>.) E BLT(S) then the relation Il1(L , 11" , >.) described by 1l1( L , 11", >.) ¢::=:} (aa ° , bb 0) E L , ( a °, b0) E 11" , ( a ° a, bOb) E >.
Theorem 5.2. If ( a , b) E
(~, 11" ,
is a congruence on S. Moreover, every congruence on S is of this form for some balanced linked triple. 0 As shown in [27], if So is left simplistic then every balanced triple satisfi es property [6] above . Moreover, in this case [,B] implies [0']. As a consequence, we have the following simplification in the case where So is a quasi-ideal.
39
Theorem 5.3. If SO is a quasi-ideal then every congruence on S is of the form \It( t, 7r, ).) where (L, 7r, ).) is a balanced triple such that (iI, i2) E
t,
(h,1 2) E). => (tli l , 12i2) E
7r.
0
As can readily be shown, it is not true in general that a congruence on the inverse transversal So can be extended to a congruence on S. We say that 7r E Con So is special if there exists 1'J E Con S such that 1'J IS0 = 7r. From [6] we have the following characterisation of such congruences. Theorem 5.4. The following statements are equivalent: (1) 7r E Con So is special; (2) (x, y) E 7r => (Vi E I)(VI E 1\) ((txW, (lyi)o) E 7r.
o
There are , of course, corresponding results concerning congruences on I and A that extend to congruences on S. In this connection, we say that So has the congruence extension property if every 7r E Con So is special. With reference to the semilattice of Section 2, we have the following from [10]: Theorem 5.5. If So is of type x then the following statements are equivalent: (1) So has the congruence extension property; o (2) x S Is or x S rs or x S o. The lattice of congruences on a regular semigroup has been the object of a deep study [17,18]. Important in this are the complete congruences (1) (1'J,IO) E Tr '¢::::::} 1'Jns.c= IOnS.c where e S.c f ¢> ef = e; (2) (1'J,IO)E Tl '¢::::::} 1'Jn'5:n=IO(')'5:n where eSnf¢>ef=f; (3) (1'J, 10) E U '¢::::::} 1'J n '5:= I" n S where '5:='5:.c n '5:n; (4) V=UnI(where(1'J,IO)EJ( '¢::::::} ker1'J=kerlO · These relations have a very natural interpretation in the presence of an inverse transversal. In fact , from [30,31] there can be extracted the following result, of which we offer a short and direct proof. Theorem 5.6. For X E {I , So, 1\} the mapping x : Con S --+ Con X given by x(1'J) = 1'Jlx is a complete lattice morphism with Ker x =
Tr V { Tl
if X if X if X
= I; = So; = 1\.
Proof. It is easy to see that x is a complete lattice morphism. Suppose now that (19 , 10) E Ker I and let (e, f) E 1'J n S,e. Then from ef = e we have e = erf; and from (e,f) E 1'J we have (ee°,fr) E 1'JI I = lOll. Multiplying on
40
the left bye, we obtain eeo
e '5:£ f => eO e '5:
r I·
In fact, if el = e then e = erl gives e"erl = eVe; and rleoe = = (ef)° el = eO e. Consequently, e '5: I gives eO e '5: I and likewise ee V '5: Ir Now (eO ,r) E
r I( ef)° el
r
eeo
= Ir eeo
and similarly we have (eO e,
e
leO eeo
= leO
Ir,
r f) E <po Hence
= eeoeooeoe
and therefore (e,t) E
7r
on 5°, the biggest extension of
(a, b) Err¢::::::> (Vi E I)(VI E A) ((lai)O, (lbi)O) E The smallest extension of it is given in [26).
7r
7r
to 5 is
7r.
to 5 is much more complicated. A description of
41
6
Other sites to visit
Since the notion of an inverse transversal arises in the study of naturally ordered regular semi groups with a biggest idempotent, several other results concerning these and related semigroups are worth noting.
Theorem 6.1. [15] A semigrotlp 5 contains a multiplicative inverse transversal if and only if 5 can be embedded as an ideal in a naturally O1'dered regular semigroup with a biggest idempotent. 0 Theorem 6.2. [3] If 5 is a naturally ordered regular semigroup with a biggest idempotent then the biggest idempotent separating congruence on 5 is gwen by (x, y) E J1.
¢::::}
(\ife E E(5))
xOex
= yOey,
xexo
= yeyO
0
Theorem 6.3. [25] Every regular semigroup with a multiplicative mverse tl'ansversal can be naturally ordered in such a way that XO is the biggest inverse of x and .c, n are regular. Conversely, if 5 is a naturally ordered regular semigroup in which every element has a biggest inverse and .c, n a1'e regu.lar the the set of biggest inverses forms a multiplicative inverse transversal. 0 Theorem 6.4. [5 ,8] Every orthodox semigroup with a multiplicative inverse transversal can be amenably ordered. Moreover, every amenable order on S° can be extended to a unique amenable order on 5. 0
References [1] T. S. Blyth and J. F. Chen, Inverse transversals are mutually isom01phic, submitted. [2] T. S. Blyth and R. McFadden, Regular semigroups with a multiplicative inverse transversal, Proc. Roy. Soc. Edin. 92 (1982), 253-270. [3] T. S. Blyth and M. H. Almeida Santos, On naturally ordered regular semigroups with biggest idempotents , Communications in Algebra 21 (1993), 1761-1771. [4] T. S. Blyth and M. H. Almeida Santos, On weakly multiplicative inverse transversals, Proc. Edin. Math. Soc. 37 (1994), 91-99. [5] T. S. Blyth and M. H . Almeida Santos, A menable orders on orthodox semigroups, J. Algebra 169, (1994),49-70. [6] T . S. Blyth and M. H. Almeida Santos, Congruences associated with inverse transversals, Collectanea Math. 46 (1995), 35-48. [7] T. S. Blyth and M. H. Almeida Santos, A simplistic approach to inverse transversals, Proc. Edin. Math. Soc. 39 (1996), 57-69.
42 [8] T. S. Blyth and M. H. Almeida Santos, Invariant subsemigroups of orthodox semigroups, Semigroup Forum 52 (1996), 163-180. [9] T. S. Blyth and M. H . Almeida Santos, On quasi-orthodox semigroups with inverse transversals, Proc. Edin. Math. Soc. 40 (1997), 505-514. [10] T. S. Blyth and M. H. Almeida Santos, A classification of inverse transversals, Communications in Algebra (to appear).
[11] T. E. Hall, Some properties of local subsemigroups inherited by la'-ger subsemigroups, Semigroup Forum 25 (1982), 35-48.
[12] J. M. Howie, Fundamentals of semigroup theory, Oxford University Press (1995). [13] D. B. McAlister and T. S. Blyth, Split orthodox semigroups, Journal of Algebra 51 (1978), 491-525 . [14] D. B. McAlister and R. McFadden, Maximum idempotents in naturally ordered regular semigroups, Proc. Edin. Math. Soc. 26 (1983), 213-220. [15] D. B. McAlister and R. McFadden, Regular semigroups with inverse transversals, Quart. J. Math. Oxford 34 (1983),459-474. [16] D. B. McAlister and R. McFadden, Semigroups with inverse transversals as matrix semigroups, Quart. J. Math. Oxford 35 (1984), 455-474. [17] F. Pastijn and M. Petrich, Congruences on regular semigroups, Trans. Amer. Math.Soc. 295 (1986), 607-633. [18] F. Pastijn and M. Petrich, The cong,·uence lattice of a regula,· semigroup, J. Pure and App!. Algebra 53 (1988), 92-123. [19] Tatsuhiko Saito, Construction of a class of regular semigroups with an inverse transversal, Proc . Semigroup Coru. Greifswald (1984), 108-112. [20] Tatsuhiko Saito, Structure of regular semigroups with a quasi-ideal inverse transversal, Semigroup Forum 31 (1985), 305-309. [21] Tatsuhiko Saito, Regular semigroups with a weakly multiplicative inverse transversal, Proc . 8th Symposium on Semigroups, Shimane University (1985), 22-25. [22] Tatsuhiko Saito, Relationship between the inverse transversals of a regular semigroup, Semigroup Forum 33 (1986), 245-250. [23] Tatsuhiko Saito, Quasi-orthodox semigroups with inverse transversals, Semigroup Forum 36 (1987), 47-54. [24] Tatsuhiko Saito, Construction of regular semigroups with inverse transversals, Proc. Edin. Math . Soc. 32 (1989), 41-5l. [25] Tatsuhiko Saito, Naturally ordered semigroups with maximum inverses, Proc. Edin. Math. Soc. 32 (1989) , 33-39. [26] M. H. Almeida Santos, Inverse transversal congruence extensions, Communications in Algebra 26 (1998), 889-898. [27) M . H. Almeida Santos, Stable congruences associated with inverse transversals , Trabalho de Investigac;ao, F.C .T., Universidade Nova de Lisboa (1999).
43 [28] Xi Lin Tang, Regular semigroups with inverse transversals, Semi group Forum 55 (1997), 24-32 . [29] Xi Lin Tang and Li Min Wang , Congruences on regular semigroups with inverse transversals , Communications in Algebra 23 (1995), 4157-417l. [30] Xi Lin Tang and Li Min Wang, Congruence lattices of regular semigroups with inverse transversals, Communications in Algebra 26 (1998), 1243-1255. [31] Li Min Wang, On congruence lattices of regular semigroups with Q-inverse transversals, Semigroup Forum 50 (1995), 141-160. [32] M. Yamada, Structure of quasi-orthodox semigroups, Mem. Fac. Sc. Shimane University 14 (1980), 1-18.
SEMIGROUPS SATISFYING SOME VARIABLE IDENTITIES MIROSLAV caRlC AND TATJANA PETKOVIC University of NiS, Faculty of Philosophy, Cirila i Metodija 2, 18000 NiS, Yugoslavia E-mail: [email protected].{mciric.tanjapet}@archimed.filfok.ni.ac.yu
STOJAN BOGDANOVIC University of NiS, Faculty of Economics, Try VJ 11, 18000 NiS, Yugoslavia E-mail: [email protected]
Putcha and Weissglass in (13) and (14) used variable identities to characterize periodic semigroups which are nilpotent extensions of unions of groups and semilattices of groups. In this paper they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases: The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell [2) concerning rings satisfying variable semigroup identities and the results of Ciric and Bogdanovic [7) concerning semigroups satisfying some ordinary identities.
1
Introduction and Preliminaries
An identity over an alphabet A is a pair of words from the free semigroup A + which is usually written as a formal equality of these words. A semigroup S is said to satisfy a set of identities ~ over A if the kernel of each homomorphism from A+ into S contains~. But if the kernel of each homomorphism from A+ into S contains a non-trivial identity from ~ , then we say that S satisfies variabily ~ , or that it satisfies ~ as a variable identity. This is the same concept which was introduced by Putcha and Weissglass in [13] and [14] , but the definition given here is closer to the definition of ordinary identities than the one of Putcha and Weissglass. In a way, this concept traces one's origin to the concept of pseudo identities and pseudo varieties, introduced by Schein in the 1960's (or disjunctive identities and varieties, as they were called in [12]) . The related concepts, the s~called inclusive identities and collective identities, were studied in [1], [10], [11] and [12] . Putcha and Weissglass in [13] and [14] used variable identities to characterize periodic semigroups which are nilpotent extensions of unions of groups Supported by Grant 04M03B of RFNS through Math. lnst. SANU.
44
45 and semilattices of groups. In this paper they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases. The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell [2] concerning rings satisfying variable semigroup identities and the results of Cirie and Bogdanovie [7] concerning semigroups satisfying some ordinary identities. Note also that Cirie and Bogdanovie in [8] used variable identities to describe semigroups having some properties as hereditary ones. For more information about variable identities and various related concepts we refer to the survey paper by Bogdanovie, Cirie and Petkovie [6]. Throughout the paper, N denotes the set of all positive integers and for m, n EN, m :S n, we set [m, nJ = {i EN Im :S i :S n}. For a semigroup S, E(S) denotes the set of all idempotents of S, and Gr(S) is the set of all group (completely regular) elements of S. For e E E(S), G e denotes the maximal subgroup of S with e as its identity, and Te = {x E S I (3n EN) xn E G e }. A semigroup S with zero 0 is a nil-semigroup if for any a E S there exists n E N such that an = 0, and a nilpotent semigroup if there exists n E N such that = {O}, and then S is also called n-nilpotent. Let a semigroup S be an ideal extension of a semigroup T. If the Rees factor semigroup SIT is a nilsemigroup (nilpotent, n-nilpotent) then S is called a nil-extension (nilpotent extension, n-nilpotent extension) of T, and if there exists a homomorphism cp : S ---> T such that acp = a, for each a E T, then cp is called a retraction and S is a retractive extension of T. We shall use the following notations for classes of semigroups:
sn
notation
class of semigroups
U9
unions of groups
1:-9
left groups
R9
right groups
9
notation
S
N Nn
class of semigroups semilattices nil-semigroups n-nilpotent semigroups
groups
Let Xl and X2 be two classes of semigroups. By Xl OX2 we denote the Mal'cev product of the classes Xl and X 2 , i.e. the class of all semigroups S on which there exists a congruence {2 such that S I {2 E X 2 and any g-class which is a subsemigroup of S belongs to Xl. It is clear that X 0 S is the class of all semilattices of semigroups from the class X. If X2 is a subclass of the class N of nil-semigroups, then Xl 0 X 2 is the class of all semigroups which are ideal (nil) extensions of semigroups from Xl by semigroups from X2. In this case by Xl ® X 2 we denote the class of all semigroups which are retractive extensions of semigroups from Xl by semigroups from X2.
46
The free semigroup over an alphabet A is denoted by A+ , and for n EN, An = {Xl,X2 , '" ,xn}. For a word wE A+ , Iwl denotes the length of w , Ixlw the number of appearances of the letter x in w, h(w) (t(w)) the first (last) letter of w (head and tail of w), and c(w) denotes the set of all letters which appear in w. To emphasize the fact that Xl , X2, .. . ,Xn are all letters that appear in w we write W(Xl' X2, .. . ,X n ) instead of w . For w E A+ such that Iwl ~ 2, h(2)(w) (t(2l(w)) denotes the prefix (suffix) of w of the length 2. If wE A+ , X E A , such that w = xv (w = vx) for v E A+ and x¢. c(v) , then we write x II w (x II w). Otherwise we write x.ftw (x.ftw). I
I
r
r
The next lemma, taken from [4], will be used in the further work. Lemma 1 Let S be nil-extension of a semigroup K which is a union of groups. If there exists a retraction cp of S onto K, then it is unique and has the following representation: xcp = xe,
where e E E(S) such that x E Te.
According to the well-known Munn's lemma, there exists at most one e E E(S) such that x E T e , and then we also have xe = ex , so we can write ex instead of xe in the above representation for cpo For undefined notions and notations we refer to the book by Howie [9] .
2
The Main Results
Variable identities that are studied here consist of some particular kinds of identities. The first kind are the identities over An of the form
(1) with n
~
3, having some of the following properties:
(AI) for a fixed i E [1, n], twice on another side; (Bl) Iwl =I lui + 2;
(Cl.l) xl.ftw ;
Xi
appears once on one side of (1) and at least
(Cl.2) h(2l(w) = XI;
(Cl.3) h(w) =I Xl ;
(Dl.2) t(2l (w) = x; ;
(Dl.3) t( w) =I x n .
I
(Dl.l)
We also deal with identities over An of the form
(2) with n
~
3, having some of the following properties:
47
(A2) for a fixed i E [1, n), twice on another side; (B2) lui =F Ivl;
Xi
appears once on one side of (2) and at least
(C2.2) h(2l(v) = x~;
(C2.3) h(v) =F Xl;
(D2.2) t(2l(u) = X~;
(D2.3) t(u) =F Xn.
The third considered kind of identities are the ones over An of the form (3)
with n
~
2, having some of the following properties:
(A3) for a fixed i E [1 , n), twice on another side; (B3) Iwl ~ n + 1; (C3.1) xl.fl'w; I
(D3.I) xn.fl'w;
Xi
appears once on one side of (3) and at least
(C3.2) h(2l(w) -- x 2l'.
(C3.3) h(w) =F Xl;
(D1.2) t(2l(w) = X~;
(D1.3) t(w) =FXn.
T
The main result of the paper is the following theorem. Theorem 1 A semigroup S satisfies a variable identity consisting of all identities of a form F having properties P if and only if S is a periodic semigroup from a class C, where F, P and C are given by the following table:
F 1.1.1. 1.1.3. 1.3.1. 1.2.2. 2.2.2.
(1) (1) (1) (1) (2)
1.2.3. 2.2.3.
(1)
1.3.2. 2.3.2.
(1)
1.3.3. 2.3.3.
(AI), (AI), (AI),
(2) (2)
(1) (2)
(AI), (A2),
P (BI), (CLl), (BI), (Cl.I), (BI), (Cl.3), (Bl), (C1.2), (B2), (C2.2), (Bl), (C1.2), (B2), (C2.2), (Bl), (C1.3), (B2), (C2.3), (Bl), (C1.3), (B2), (C2.3),
(DLl) (D1.3) (DLl) (D1.2) (D2.2) (D1.3) (D2.3) (D1.2) (D2.2) (D1.3) (D2.3)
C ugoN (I:-g oS) oN (7(.goS) oN U9®N (I:-g oS) ®N (7(.goS) ®N (goS) ®N
48
:F
3.1.1. 3.1.3. 3.3.1. 3.2.2. 3.2.3. 3.3.2. 3.3.3.
(3) (3) (3) (3) (3) (3) (3)
P (A3) , (B3), (C3.1), (A3), (B3), (C3.1), (A3), (B3), (C3.3), (B3), (C3.2), (B3) , (C3.2), (B3), (C3.3), (B3), (C3.3),
C
(D3.1) (D3.3) (D3.1) (D3.2) (D3.3) (D3.2) (D3.3)
UQoNn (CQ 0 S) oNn (RQ 0 S) oNn UQ®Nn (CQ 0 S) ®Nn (RQoS) ®Nn (Q oS) ®Nn
Proof of 1.1.1 Let S satisfy the variable identity consisting of all identities of the form (1) having the properties (AI), (B1), (C1.1) and (D1.1). By (B1) it follows that S is periodic. Let XES, e E E(S). First we have that some of the mentioned identities lies in the kernel of the homomorphism C{J : A;t -+ S determined by XIC{J = xe
and
XjC{J = e, for j E [2, n].
By this it follows that if hew) = if hew) #
Xl Xl
In the first case, if h( w) = Xl, by (CLl) it follows IXllw 2: 2, and then xe E Cr(S), which was to be proved. Consider the second case: hew) # Xl. In this case we have that xe = e(xe)IX1I w = ee(xe)IX1I w = exe. Next we have that an identity of the form (1) with the properties (AI), (B1), (C1.1) and (D1.1) lies in the kernel of the homomorphism 'I/J : A;t -+ S determined by xi'I/J = xe
and
Xj'I/J = e, for j E [1, n], j
# i,
where i E [1, n] is the one fixed in (AI). By this it follows that erxe = eS(xe)t, for some integers r, s 2: 0 and t 2: 2, and since xe = exe, then xe = (xe)t, so xe E Cr(S) . This proves that Cr(S) is a left ideal of S. In the same way we prove that it is a right ideal, and since S is periodic, then S is a nil-extension of Cr(S) E UQ. Conversely, let S be periodic and a nil-extension of a semigroup K E UQ, and let x,a,y E S. Then a k E K, for some kEN, whence xaky E K. Thus, xaky belongs to some periodic subgroup of K, so xaky = (xaky)rn+l, for some mEN. This means that S Fv {XIX~X3 = (XIX~X3)rn I k, mEN}. •
49
Proof of 1.1.3 Let S satisfy the variable identity consisting of all identities of the form (1) having the properties (AI) , (BI), (C1.I) and (D1.3). Since (D1.3) implies (D1.I), then by 1.1 it follows that S is a nil-extension of a semigroup K E Ug . To prove that K E .cg 0 S, it is enough to prove that E(K) is a left regular band. Let e,j E E(K) and let the homomorphism
f
and
Xj
e, for j E [1, n - 1] .
Then the kernel of
• The proof of 1.3.1 will be omitted , because it is dual to the proof of 1.1.3. Proof of 1.2.2 and 2.2.2. Let S satisfy the variable identity consisting of all identities of the form (1) with (BI), (C1.2) and (D1.2). By (BI) we have that S is periodic. Let x, a, yES, and let
Xn
and
Xj
for j E [2, n - 1].
Then the kernel of
= X
and
Xj
a, for j E [2, n],
50
we have that xa k = x 2s, for some kEN and s E S, and considering the homomorphism 'I/J : A;t ---> S defined by xn'I/J = y
and
Xjtp = a, for j E [1, n - 1],
we obtain that amy = ty2, for some mEN and t E S. Therefore, xak+my = x 2sty2, and as in the previous case we conclude that S E Ug ® N . Conversely, let S be periodic and a retractive nil-extension of K E Ug , let tp be the retraction of S onto K, and let x, a, yES. Then there exists n EN such that xk,yk E E(S) and a k E K. Moreover, by Lemma 1 it follows that xtp = xxk = xk+1 and ytp = yyk = yk+1, whence
xaky
= (xaky)tp =
(xtp)ak(ytp) = xk+ 1a k yk+1 (xtp)aky = xk+ 1 a k y { xak(ytp) = xa k y k+l
Therefore, S Fv {XIX~X3 = x~+1x~x~+1 IkE N} and S x~+1x~x31 kEN}.
Fv
{XIX~x~+1 = •
Proof of 1.2.3 and 2.2.3. Let S satisfy the variable identity consisting of all identities of the form (1) with (B1), (C1.2) and (D1.3). Since (C1.2) implies (AI) and (C1.1) , then by 1.1.3 we have that S is a nil-extension of a semigroup K E .cg 0 S. We prove that for each xES and e E E(S) the following conditions hold xe E xm Se , ex E eSx m ,
for each mEN, for each mEN.
(4) (5)
It is clear that (4) holds for m = 1. Let mEN such that xe = xmse , for some s E S and consider the homomorphism tp : A;t ---> S determined by Xl = xm
and
Xj = se, for j E [2, n].
Since the kernel of contains an identity of the form (1) with (B1) , (C1.2) and (D1.3) , we have that xm(se)k = x 2mte, for some kEN and t E S. We also have that se E K = Gr(S), so se'H.(se)k, where 'H. is the Green's relation on S , whence se = (se)kp, for some pES. Now we have that xe
= xmse = xmsee = xm(se)kpe = x 2m tepe E xm+ISe.
Therefore, by induction we conclude that (4) holds for each mEN. On the other hand, considering the homomorphism 'I/J : A;t ---> S determined by x n'I/J
= ex
and
Xj'I/J
= e,
for j E [1 , n - 1],
51
we obtain that ex = (ex)k e, for some kEN, whence ex = exe. Let mEN. Then ex'H(ex)'ffi, so ex = s(ex)'ffi, for some s E S, whence it follows that ex = eex = es(ex)'ffi = esex'ffi E eSx'ffi,
since ex = exe. Therefore, we have proved (5). Using (4) and (5) we prove that the mapping cp : S -+ K defined by xcp = xe, where e E E(S) such that x E Te, is a retraction of S onto K. It is enough to show that is is a homomorphism. Let x, yES and let x E Te, y E Tf and xy E T g , for some e,j,g E E(S). By (4) and (5) it follows that yg = Jyg , xJ = exJ , exy = exyg and ey = eyJ, whence (xy)cp = xyg = xJyg = exJyg = exyg = exy = xey = xeyJ = (xcp)(ycp).
FUrther, let S satisfy the variable identity consisting of all identities of the form (2) with (B2), (C2.2) and (D2.3), and let xES and e E E(S). Using the same methodology we obtain that xe = (xe)k, for some kEN, k 2: 2, and ex = exe, whence it follows that ex = exe = e(xe)k = (ex)ke = (ex)k. Thus, Gr(S) is an ideal of S and S is a nil-extension of Gr(S). In the same way as in the previous case we prove that Gr(S) E .cQ 08 and S is a retractive extension of Gr(S). Conversely, let S be periodic and a retractive nil-extension of a semigroup K E .cQ 0 8, let cp be the retraction of S onto K, and let x, a, yES. As in 1.1.3 we obtain that there exists kEN such that xk, a k E E( S) and xaky = xakyx k = xakya k , whence xaky _ (xaky)(() _ { (xakyxk)cp = (xcp)akyx T (xcp)aky = xk+laky
k
= xk+lakyxk
Therefore, S Fv {X1X~X3 = X~+lX~x3xt IkE N} and S x~+lx~x3 1 kEN}.
Fv
{X1X~X3X~ = •
The proofs of 1.3.2 and 2.3.2 will be omitted, because they are dual to the proofs of 1.2.3 and 2.2.3. Proof of 1.3.3 and 2.3.3. Let S satisfy the variable identity consisting of all identities of the form (1) with (AI), (BI), (C1.3) and (D1.3) . Since (C1.3) implies (C1.I) and (D1.3) implies (D1.I), then by 1.1.3 and 1.3.1 it follows that S is a nil-extension of K E .cQ 0 8 n RQ 0 8 = Q 0 8. By Theorem 3 of [5], (Q 0 8) oN = (Q 0 8) ®N. Let S satisfy the variable identity consisting of all identities of the form (2) with (A2) , (B2), (C2.3) and (D2.3). As in the previous proofs, by (B2) we have that S is periodic, using (C2.3) and (D2.3) we prove that the idempotents of S are central, i.e. ex = xe , for all e E E(S), XES, and by (A2) we obtain
52
that ex = (ex)k = (xe)k = xe, for some kEN, k ~ 2, and conclude that Gr(S) is an ideal of S. Therefore, S E (90S) 0 N = (9 0 S) ® N . Conversely, let S be periodic and a retractive nil-extension of a semigroup K E 90S, let ep be the retraction of S onto K, and let x, a, yES. Then there exists kEN such that ak,xk,yk E E(S) = E(K), and then xaky E K, xep = xk+1, yep = yk+l. Since the idempotents of S are central, then xaky
= (xaky)ep =
(xep)ak(yep) = xkxakyyk = ykxakyxk . xak(yep) = xa 2k y k+1 = xa ky k+1a k { (xep)aky =xk+la2k y=a k xk+la ky
Hence, S Fv {XIX~X3 = X~XIX~X3X~ IkE N} and S x~x~+lx~X31 kEN} .
Fv
{XIX~x~+IX~ -
The proofs of 3.1.1-3.3.3 will be omitted because they are immediate consequences of the previous theorems and the following one: Theorem 2 Let n E N, n ~ 2, and let S be a nil-semigroup satisfying the variable identity of the form (3) having the property (B3). Then S is nnilpotent. Proof Let us prove that any nilpotent subsemigroup Q of Sis n-nilpotent. Let Q be a k-nilpotent semigroup. By the hypothesis, for arbitrary aI, a2, .. . ,an there exists a word WI = WI (Xl, ... ,xn ) such that IWII ~ n + 1 and
This proves that Qn = Qn+l, whence we obtain that Qn = Qm, for any mEN, m ~ n , and hence, Qn = Qk = {O}. Therefore, the set of indices of nilpotency of all nilpotent subsemigroup of S is bounded, so by Theorem 3 of [15] we have that S is nilpotent, and hence, it is n-nilpotent. The theorems characterizing nilpotent and nil-extensions of bands, left regular bands and semilattices, and their retractive analogues, are very similar to the previous ones, so they will be omitted. We only note that the variable identities describing these semigroups consist of the corresponding identities from the above theorems, having an additional property:
(Al- 3)* for a fixed i E [1, n), Xi appears once on one side of (1) (resp. (2), (3) ), and exactly twice on another side. This condition forces all subgroups of a semigroup satisfying it to be oneelement.
53
Remark that the conditions (Ci.j) and (Di.k) in the claim i .j.k are necessary. Namely, adding to the variable identity from the claim i.j.k an identity which does not satisfy (Ci.j) or (Di.k) we leave the class of semigroups from i.j.k. This is an immediate consequence of the results given by Ciric and Bogdanovic in [7] (see also [6]). References
1. J. Almeida, On power varieties of semigroups, J. Algebra 120 (1989), 1- 17. 2. H. E. Bell, A commutativity study for periodic rings, Pacific J. Math. 70 (1977), 29- 36. 3. S. Bogdanovic, Semigroups of Galbiati- Veronesi, Algebra and Logic, Zagreb, 1984, Novi Sad, 1985, 9-20. 4. S. Bogdanovic and M. Ciric, Retractive nil-extensions of regular semigroups, Proc. Japan Acad. Ser. A, 68 (1992), no. 6, 126-130. 5. S. Bogdanovic and M. Ciric, Semigroups of Galbiati- Veronesi IV (Bands of nil-extensions of groups, Facta Univ. (Nis), Ser. Math. Inform. 7 (1992) , 23- 35. 6. S. Bogdanovic, M. Ciric and T. Petkovic, Uniformly 7r-regular rings and semigroups: A survey, Topics from Contemporary Mathematics, Zborn. Rad. Mat. Inst. SANU 9 (17) (1999), 1- 79. 7. M. Ciric and S. Bogdanovic, Nil-extensions of unions of groups induced by identities, Semigroup Forum 48 (1994), 303- 311. 8. M. Ciric and S. Bogdanovic, The five-element Brandt semigroup as a forbidden divisor, Semigroup Forum (to appear). 9. J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, Oxford: Clarendon Press, 1995. 10. E. S. Lyapin, Atoms of the lattice of inclusive varieties of semigroups , Sibir. Mat. Zhurn. 16 (1975), no. 6, 1224- 1230 (in Russian). 11. E. S. Lyapin, Identities valid globally in semigroups, Semigroup Forum 24 (1982) , 263-269 12. G. Mashevitzky, On a finite basis theorem for universal positive formulas , Algebra Universalis 35, (1996), 124- 140. 13. M. S. Putcha and J . Weissglass, Semigroups satisfying variable identities, Semigroup Forum 3 (1971),64-67. 14. M. S. Putcha and J. Weissglass, Semigroups satisfying variable identities II, Trans. Amer. Math. Soc. 168 (1972), 113-119. 15. L. N. Shevrin, Semigroups all of whose subsemigroups are nilpotent, Sibir. Mat. Zhurn. 11 (1961), no. 6, 936-942 (in Russian).
SOME VARIATIONS ON THE NOTION OF LOCALLY TESTABLE LANGUAGE JOSE CARLOS COSTA Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal E-mail: [email protected] The aim of this paper is to complete the characterization of the languages that are Boolean combinations (of a subset) of languages of the form wA·, A·w or L(w,r,t,n), where A is an alphabet, w E A+, r,t ~ 0, n ~ 1 and L(w,r,t,n) denotes the set of all words u in A + such that the number of occurrences of the factor w in u is congruent to r threshold t mod n. For each class C of languages such that A+C is a Boolean algebra generated by some of the following types of languages: wA·, A·w, A·wA* = L(w, 1, 1, 1) or L(w , r, t, 1), and such that C does not constitute a variety of languages, we compute the smallest variety of languages containing C and the largest variety of languages contained in C.
1
Introduction
In this paper we are interested in classes of languages C such that, for each alphabet A, the Boolean algebra A +C is generated by some of the following types of languages: wA* , A*w , A*wA* (= L(w, 1, 1, 1)), L(w, r, t, 1) or L(w, r, t, n), where w E A+ , r, t ~ 0 and n ~ 1. As an example we have the well-known class of locally testable languages, denoted Ct, which is such that A+ Ct is the Boolean algebra generated by the languages of the form wA*, A*w and A*wA*, where w E A+. The locally testable languages were characterized independently by Brzozowski and Simon [3] and McNaughton [6] as being those languages whose syntactic semigroup lies in LSI, the pseudovariety of all locally idempotent and locally commutative semigroups. Recall also that a language L is locally testable if one can decide the membership of a given word u in L by considering the factors of a fixed length k of u and its prefixes and suffixes of length < k. In [2] , Beauquier and Pin considered three variations on this last definition of locally testable languages and obtained three different classes of languages. First, they dropped the conditions about the prefixes and the suffixes and defined strongly locally testable (Sit) languages to be those languages whose elements are determined by factors of a fixed length. The class of all such languages in A+ is the Boolean algebra generated by the languages of the form A*wA* with wE A+. This class is not a variety of languages but it is decidable and characterized by a nice algebraic property. In this paper we 54
55
consider a class of languages intermediate between locally testable languages and strongly locally testable languages, which we call locally testable by prefi:xes (Ct-p). Membership of a word u in this type of language is determined by the factors of u of a fixed length k and by the prefixes of u of length < k. Thus a language in A+ is locally testable by prefixes if it is a Boolean combination of languages of the form wA* and A*wA* where w E A+. This class of languages is characterized by an algebraic property similar to that of Beauquier and Pin. Secondly, Beauquier and Pin characterized the languages in A + that are Boolean combinations of languages of the form wA*, A*w or L(w, r, t, 1), which they called threshold locally testable (Tlt). Membership of a word u in such a language is determined by the factors of u of a fixed length k, but taking in account their number of occurrences up to a certain "threshold", and by the prefixes and suffixes of u of length < k. Finally, by dropping the conditions about the prefixes and the suffixes on this last condition, Beauquier and Pin introduced another class of languages whose elements, called strongly threshold locally testable (Stlt) languages, are Boolean combinations of languages of the form L(w, r, t, 1). However, the syntactic characterization of these languages only recently was obtained by Pin [8]. Once again we describe a "lateralized" version of this work, by dropping only the condition about the suffixes. One obtains a class of languages whose elements are Boolean combinations of languages of the form wA* or L(w, r, t, 1), which we call threshold locally testable by prefixes (Tlt-p) languages. If one replaces wA* by A*w on the generators of the languages "by prefixes" above, one obtains dually the classes of locally testable by suffixes (Ct-s) and of threshold locally testable by suffixes (Tlt-s) languages. We complete our study by considering the languages that are Boolean combinations of languages of the form wA*, A*w or L(w,r,t,n). These languages, which we call counting locally testable (Clt), were also characterized in [2]. Here, we show that they can also be obtained using only Boolean combinations of languages of the form L( w, r, t, n), i.e., they coincide with its "strongly" version. Now, we recall that the class C of all languages such that A+C is the Boolean algebra generated by the set {wA* : w E A+} (resp. {A*w: w E A+}, {wA*,A*w : w E A+}) is already characterized (see [7], for instance). It is the class of languages associated, via Eilenberg's Theorem, with the pseudovariety K (resp. D, LI), consisting of all finite semigroups S such that eS = e (resp. Se = e, eSe = e) for each idempotent e of S. This means that the characterization of the languages that are Boolean combinations (of a subset) of languages of the form wA*, A*w, A*wA* or L(w, r, t, n) is now
56
complete. In the last part of this paper we compute the smallest (resp. largest) variety of languages containing (resp. contained in) the classes of languages mentioned above and that are not varieties of languages. For instance, we show that the class of all Ct (resp. Ct and .J-trivial) languages is the smallest (resp. largest) variety of languages containing (resp. contained in) the class of all Slt languages. In other words, Ct is generated (as a variety of languages) by the languages of the form A*wA* with w E A+. We remark the analogy of this result with the well known characterization of the variety of languages Sl, associated with the pseudovariety 81 of semilattices, as being the Boolean algebra generated by the languages of the form A * aA * with a E A. 2
Preliminaries
We begin by presenting basic definitions and notation concerning words and finite semigroups. Next we recall the notion of pseudovariety of semigroups and define the pseudovarieties mentioned in this paper. We then present the main definitions about recognizable languages and their relations with pseudovarieties. For omitted proofs and missing definitions, the reader is referred to the book of Pin [7]. 2.1
Words
Let A be a finite alphabet. We denote by A+ the set of non-empty words over A and by A* the set A+ U {I}. If u = al'" ak (ai E A) is a word, the number k is said the length of u and is denoted by lui. For each word u of length ~ k, we denote by Pk(U) (resp. Sk(U)) the prefix (resp. suffix) of u of length k. For each word u, we denote by ik(U) (resp. tk(U)) the word U if Iwl < k, and Pk(U) (resp. Sk(U)) otherwise. We will denote by Fk(U) the set of all factors of length k of u. For each pair of words w and u, we denote by [,\,:] the number of occurrences . W. D 2sillce ' . two o f the f ac t or U III ror 'illSt ance [abaabaaa] abaa =, a baa occurs III different places in abaabaaa: abaabaaa, abaabaaa. Let us now introduce a congruence on the set of non-negative integers, which is crucial in what follows. Let x, y, t ~ 0 and n ~ 1 be integers. We say that, x is congruent to y threshold t mod n, denoted x =t,n y, if either x = y or x, y ~ t and x is congruent to y mod n. For instance, the classes of =2,3 are {O}, {I}, {2, 5, 8, ... }, {3, 6, 9, ... } and {4, 7, lO, ... }. For an alphabet A, a word wE A+, integers r, t ~ 0 and n ~ 1 set
L(w,r,t,n) = {u E A+: [~l
=t,n
r}.
57 For instance, L(w, 1,1,1) = A*wA* and L(a, 1,0,2) = B*aB*(aB*aB*)*, with B = A \ {a}, is the set of all words in A+ containing an odd number of occurrences of the letter a E A. 2.2
Pseudovarieties of semigroups
Let S be a finite semigroup and let s E S. We denote by SW the unique idempotent of the subsemigroup of S generated by s. We say that S is aperiodic if sw+l = SW for all s E S. Recall that a pseudovariety of semigroups is a class of finite semigroups closed under taking subsemigroups, homomorphic images and finite direct products. We denote by A, Com, Acorn, SI and J, respectively, the pseudovarieties of all finite aperiodic, commutative, aperiodic and commutative, idempotent and commutative (or semilattices) and .J-trivial semigroups. Particularly important in this paper is the pseudovariety LSI of all finite semigroups S such that eSe E SI for every idempotent e E S. It is well mown, by Reiterman's Theorem [9), that every pseudovariety V is defined by a family I: of pseudo identities, written V = [I:]. We refer the reader to Almeida [1) for background on pseudovarieties and pseudoidentities. We have, for instance, the following equalities:
A = [xw+l = XW],
= SI =
Com
[xy [x 2
= J =
Acorn
= yx], = X, xy = yx].
= x W, xy = yx] [(xy)W = (yx)W,xw+1 = XW] [xw+l
Now, we recall three calculations of semidirect product of pseudovarieties of semigroups which will be used later. The first was obtained by Brzozowski and Simon [3) and McNaughton [6) and the two last ones by Therien and Weiss [10). SI * D
=
[XWyxWyx W = xWyxW, xWyxW ZXW
= XW zxWyxW] = LSI
Com * D = [XWryW sxWtyW = xWtyW sxWryW] Acorn * D = (Com * D) nA. 2.3
Recognizable languages
Let A be an alphabet and let V be a pseudovariety. A subset L of A+ is called a language. It is said to be recognizable (resp. V-recognizable) if there exists a finite semi group S (resp. in V) and a morphism f..L : A+ ---. S such
58
that L = f-L-l(f-L(L)). In that case, we say that S recognizes L. The syntactic congruence of a language L is the congruence "'L over A+ given by
u
"'L
v
if and only if
xuy E L {:} xvy E L for all x,y E A*.
The syntactic semigroup of L, denoted by S(L), is the quotient of A+ by We know that L is recognizable (resp. V-recognizable) if and only if S(L) is finite (resp. S(L) E V). The natural morphism rJ : A+ ~ S(L) is called the syntactic morphism of Land P = rJ(L) is its syntactic image. For more details on recognizable languages, the reader is referred to [7,5]. A class of (recognizable) languages is a correspondence C associating with each alphabet A a set A+C of (recognizable) languages of A+. A variety of languages is a class V of recognizable languages such that '" L.
(1) for every alphabet A, A+V is closed under finite union, finite intersection and complement;
(2) for every morphism cp: A+
~ B+, L
E B+V implies cp-l(L) E A+V;
(3) if L E A+V and a E A, then a-1L = {u E A+ : au E L} and La- 1 = {u E A+ : ua E L} are in A+V. Let V be a pseudovariety and let V be the class of recognizable languages which associates with each alphabet A the set A+V of V-recognizable languages of A+. One can show that V is a variety of languages. Moreover, Eilenberg [5] proved the following fundamental result. Theorem 2.1 The correspondence V f-+ V defines a bijective correspondence 0 between pseudovarieties of semigroups and varieties of languages. 3
Languages defined by factors of words
In this section, we begin by presenting some equivalence relations which will be used to describe the languages we are interested in. We then present the characterizations of the languages. 3.1
Some equivalence relations
Let k, n ~ 1 and t ~ 0 be integers. We define an equivalence =k,t,n of finite index on A + by setting u =k,t,n V if and only if, for every word x of length:::; k, [~] =t,n [~]. For instance, if u = a 3 bababa 2 and v = a 2babababa 3 , we have u =32 1 v but U ¢:3,2,2 v since laba] = 3 ¢:2,2 4 = [a~a]. However u =3,2,2 a2bab~bababa3
59 We note that =k,t,n is not a congruence in general. For instance, consider A = {a, b} , u = aba and v = abab. One has u =2,1,1 v, but ua ¢;2,1,1 va. Indeed a 2 is a factor of length 2 of ua but it is not a factor of va. Let now "'k ,t,n be the congruence of finite index on A+ given by U "'k,t,n V if and only if ik-1(U) = ik-1(V), tk-1(U) = tk-1(V) and u =k,t,n V.
If on the definition of "'k,t,n we drop the condition about the suffixes we obtain a new equivalence on A+, which we denote by ~k,t,n. That is, ~k,t,n is given by U ~k,t , n V if and only if i k- 1(U) = i k- 1(V) and u =k,t,n V. This equivalence is not a congruence in general. We say that an equivalence relation = on A+ saturates a language L ~ A+ if L is a union of =-classes. Definition 3.1 Let A be an alphabet. We say that a language of A+ is • locally testable (resp. strongly locally testable, locally testable by prefixes) if it is saturated by "'k,l,l (resp. =k,l,l, ~k,l,l) for some k; • threshold locally testable (resp. strongly threshold locally testable, threshold locally testable by prefixes) if it is saturated by "'k,t,l (resp. =k,t,l, ~k ,t,1) for some k and t; • counting locally testable (resp. strongly counting locally testable, counting locally testable by prefixes) if it is saturated by "'k,t,n (resp . =k,t,n, ~k , t,n) for some k, nand t. • The notions of locally testable by suffixes, threshold locally testable by suffixes and counting locally testable by suffixes can be defined dually by dropping the condition about the prefixes, instead of the suffixes, on the definition of "'k,t,n. We will use, respectively, the notations Ct , Tlt, Clt, Slt, Ct-p, etc, either for the classes of all locally testable, threshold locally testable, counting locally testable, strongly locally testable, locally testable by prefixes, etc, languages, or for the languages themselves. The next proposition describes these classes as Boolean algebras. For a set of languages C we denote by B(C) the Boolean algebra generated by C. Proposition 3.2 Let A be an alphabet. Then A+Clt = B{wA*,A*w,L(w,r,t,n) I wE A+, r,t ~ 0, n ~ I}, A+Clt-p = B{wA*,L(w,r,t,n) I w E A+, r,t ~ 0, n ~ I}, A+Clt-s = B{A*w,L(w,r,t,n) I w E A+, r,t ~ 0, n ~ I}, A+Sclt = B{L(w,r,t,n) I w E A+, r,t ~ 0, n ~ I}.
60 Similar results are valid for the four classes of "locally testable" languages and the four classes of "threshold locally testable" languages. We only need to substitute L(w, r, t, n) by L(w, 1, 1, 1) and by L(w, r, t, 1), respectively. 0 The next result shows that the generators of the form wA* and A*w are superfluous for Cit. That is, we can restrict the generators of Cit to the languages of the form L(w, r, t, n) . Proposition 3.3 We have the equalities Cit = Cit-p = Cit-s = Sclt. Proof. It is clear that it suffices to prove the inclusion Cit ~ Sclt. For that, we will show that for each alphabet A and each w E A+, the languages wA* and A*w are Boolean combinations of languages of the form L(u, r, t, n) with u E A+, r, t 2: 0 and n 2: 1. To be more precise, we show that wA* (for A*w is similar) is the (disjoint) union of all languages of the form L( w, Q, 1,2) n
n
L( aw, (3a, 0, 2) aEA where Q E {1,2}, {3a E {O, I}, "£aEA{3a is even if Q = 1 and it is odd if Q = 2. We begin by observing that a word u E A+ lies in wA* if and only if
L
[a:J [:J -
u E
n
(1) = 1. aEA Let u E wA*. Then, either [~l is odd, or it is even and not null. In the first case u E L(w, 1, 1,2). Furthermore, we deduce from (1) that "£aEA[a':ul is even. This implies that L(aw,{3a,0,2) aEA for some family ({3a)aEA, {3a E {O, I}, such that "£aEA (3a is even. Analogously, one can show that in the second case
n
n
L(aw, (3a, 0, 2) aEA for some family ({3a)aEA, {3a E {O, I}, such that "£aEA (3a is odd. This shows one of the inclusions. Conversely, if u E L(w, Q, 1,2) n naEA L(aw, (3a, 0, 2) where Q and {3a are in the conditions of the statement, we have that Q (and so also l~]) is even if and only if "£aEA {3a is odd, that is, if and only if "£aEA la':ul is odd. Consequently l~l and "£aEA la':ul are different whence w is a prefix of u. So u lies in wA*. 0 u E L(w, 2,1,2)
We shall see that the other classes of languages are all distinct between themselves and from Cit . The inclusion relation between them is as shown in the next figure.
61
Clt
I
Tlt
/ /Tlt-s " "Ct ""'Y--. ~Ct-s Stlt Ct-p ""'// Slt
Tlt-p
We shall also see that from these classes only Ct, Tlt and Clt constitute varieties of languages. Furthermore we will prove that Ct-p n Ct-s = Sit, that Tlt-p n Tlt-s i:- Stlt and that Ct (resp. Tit) is the smallest variety of languages containing Slt (resp. Stlt) . Example 3.4 Let A = {a, b} . The language L = ba* ba* is threshold locally testable by prefixes since
L = bA* n L(b, 2, 3,1). Less obvious is that L is also strongly threshold locally testable. Indeed, L = L(b, 2, 3, 1) \ [L(ab, 2, 2,1) U L(abb, 1, 1, 1)] .• 3.2
Syntactic characterizations
In this section we describe effective characterizations of the classes of languages presented in the previous section. These characterizations are all given in terms of an algebraic property of the syntactic morphisms of the languages. The classes Ct, Tlt and Cit are characterized by a property of the syntactic semigroups of their languages. For the other classes it is also necessary to consider the syntactic images of the languages. This means by Eilenberg's Theorem that the first three classes are varieties of languages while the others are not. We begin by presenting the characterizations of Ct, Tlt and Clt. The first is due to Brwzowski and Simon [3] and McNaughton [6]. The others are due to Beauquier and Pin [2]. Theorem 3.5 Let L be a recognizable language.
(1) L is Ct if and only if S(L) E Sh D. (2) Lis Tlt if and only if S(L) E Acorn * D.
(3) L is Cit if and only if S(L) E Com * D.
o
62
Thus, it is decidable whether a given language is Ct, Tlt or Clt. We now proceed to describe characterizations of the remaining classes of languages. Let 8 be a finite semigroup. Define =: to be the smallest equivalence relation on 8 containing the relation .J and satisfying the condition: \;Ie = e 2,f = f2 E 8\;1r,s E 8, erfse =: fserf.
Beauquier and Pin [2] and Pin [8] gave, respectively, the characterizations of the classes Slt and Stlt. Theorem 3.6 Let L be a recognizable language of A+ , let 8 be the syntactic morphism of L and let P be its syntactic image.
(1) L is Slt if and only if 8 E LSI and P is a union of .J-classes of 8. (2) L is Stlt if and only if 8 E Acorn * D and P is a union of =:-classes of 8. 0 We now present a "lateralized" version of this last theorem. Theorem 3.7 Let L be a recognizable language of A+ , let 8 be the syntactic morphism of L and let P be its syntactic image. (1) Lis Ct -p if and only if 8 E LSI and P is a union ofR-classes of 8 .
(I') L is Ct-s if and only if 8 E LSI and P is a union of C-classes of 8 . (2) L is Tit-p if and only if 8 E Acorn * D and P is a union of R-classes of 8 . (2') L is Tlt-s if and only if 8 E Acorn * D and P is a union of C-classes of 8. Proof. The proofs are adapted without difficulty from the corresponding proofs of Theorem 3.6. We only recall the proof of (2). Suppose first that Lis a Tlt-p language. Then, L is saturated by :::::ik,t , l for some k and t . Since Tltp ~ Tit , L is also Tlt and Theorem 3.5 shows that 8(L) E Acorn * D. Since the syntactic morphism TJ : A + -- 8 is onto, one can fix , for each element s E 8 1 a word S E A* such that TJ(s) = s (if s = 1, we take S = 1). To prove that P is a union of R-classes of 8 , let us consider two R-equivalent elements r and s of .') and suppose that rEP. We want to show that s E P . Since r R s there exist x, y E 8 1 such that rx = sand sy = r. Now, since 8 is finite, there exists an integer n such that , for any s E 8, sn is idempotent. Choosing one such n 2 kt , we have r(xy)n :::::ik,t,1 r(xy)n x. But TJ(r(xy)n) = rEP and thus r(xy)n E L. This implies r(xy)nx E L , whence TJ(r(xy)nx ) = s E P
63
Conversely, since S E Acorn * D it follows from Theorem 3.5 that L is saturated by "'k,t,l for some k and t. We will show that L is saturated by ~k,T,l for some T sufficiently large (more precisely, one can take T ~ (1 + t· (IAlk)!)(l + IAI)). To each word w we associate a labeled graph N(w) where the set of vertices is Fk-l(W) and if U E Fk(W), there exists an edge of label [~] threshold t from Pk-l(U) to Sk-l(U). The vertex Pk-l(W) (resp. Sk-l (w)) is called the initial (resp. final) vertex of N( w). Let w and w' be two words such that w ~k,T,l w' and w E L. We want to show that w' E L. If Iwl < k (or Iw'l < k), then w = w' SO, we may suppose Iwl, Iw'l ~ k. Suppose now that Iwl < T. We claim that w "'k,T,l w'. Since w ~k,T,I w', it remains to prove that Sk-l(W) = Sk-l(W ' ). If k = 1 this is clear. Consider now k ~ 2 and put S = Sk-l(W). Since Iwl < T we have [';'] = ['J.n < T. Put n = [';'] and suppose that Sk-l(W' ) i= s. Then
L [:] =n-1 aEA
and
L ~~ =n.
aEA
But this contradicts the assumption that w ~k,T,l w', since in this case [~] = [~~] for all a E A. Then Sk-l(W) = Sk-I(W ' ) and the claim is proved. It follows that w "'k ,t,I w', since t < T, and we may conclude that w' E L. Thus, we may assume that Iwl, Iw'l ~ T . Since [~] =t,I [~'] for any word U of length k and since Pk-I(W) = Pk-I(W ' ), the labeled graphs N(w) and N(w ' ) are equal, except possibly for the final vertices. We denote by f and 1', respectively, the final vertices of N(w) and N(w' ). We say that two vertices VI and V2 are in the same strongly t-component, if there are two oriented paths from VI to V2 and from V2 to VI using only edges of label t. Since N(w) and N(w ' ) have the same initial vertex, one has (see the proof of [8, Theorem 3.3]) that f and f' are in the same t-component. Now, one can show that 7](w) R7](w' ). Since P is union of R-classes, we deduce that 7]( w') E P and thus that w' E L, which concludes the proof. 0 Since each .:J-class of a finite semigroup is a union of R-classes and a union of .c-classes, we have the following consequences of the last theorem. Corollary 3.8 Let L be a recognizable language of A+, let S be the syntactic morphism of L and let P be its syntactic image. (1) L is both .ct-p and .ct-s if and only if L is Slt. (2) L is both Tlt-p and Tlt-s if and only if S E Acorn * D and P is a union of .:J -classes of S. 0
We remark that a language L being both Tlt-p and Tlt-s does not imply
64
that L is Stlt, that is, the class Tlt-p n Tlt-s strictly contains the class Stlt, as it is shown in the next example. Example 3.9 Let A = {a,b} and let L = a*b+a· Then L is recognized by the following automaton. a
a
b
The syntactic semigroup of L is defined by the relations a 2 = a, b2 = b and bab = o. Its.J -class structure is represented in the following diagram, where the grey boxes represent the syntactic image P of L.
o
_
--o II
Thus P is a union of.J -classes of S(L) and L is Tlt-p and Tit-s. Indeed, we have L
= b+a· u a+b+a" = [bA* \ = a*b+ U a*b+a+ = [A*b \
L(ab, 1, 1, 1)] U [aA*
n L(ab, 1, 1,2)]
L(ba, 1, 1, 1)] U [A*a n L(ba, 1, 1,2)].
Let us now verify that L is not Stlt . We prove that P is not a union of ::=-classes. Indeed, by definition of::= and since a and bare idempotents, we have aabaa ::= baaab, that is, aba ::= O. Thus P is not a union of ::=-classes since aba E P and 0 f/. P Note also that L is not Lt , since S(L) f/. LSI. Indeed, for instance, a is idempotent and aba is not. _ Example 3.10 Let A = {a, b, c}, and let L L is recognized by the following automaton.
= (ab)+ U a(ba)* U {e2 }.
Then,
The syntactic semigroup S (L) has seven elements and it is defined by the relations a 2 = ae = b2 = be = ea = cb = c3 = O. Its.J -class structure is
65
represented in the following figure.
[I]
o As one can show, S(L) E LSI. On the other hand, the syntactic image of L is the set P = {ab,a,c 2 }. Since it is a union ofR-classes of S(L), L is Lt-p. Indeed, we can write
Note that P is not a union of L-classes of S(L). So L is not Lt-s.
4
•
The varieties of languages generated
In this section \\'e compute the smallest (resp. largest) variety of languages containing (resp . contained in) each one of the classes of languages considered in the last section. As we have seen in Theorem 3.5, the classes Lt of locally testable languages and Tit of threshold locally testable languages are varieties of languages . Let us prove the following result. Proposition 4.1 The class Lt (resp. Tit) is the smallest variety of languages containing the languages of the form A*wA* (resp. L(w, T, t, 1)) for any alphabet A and wE A+ (resp. and T, t 2 0). Proof. We only give the proof for Lt . The proof for Tit is a consequence of this one since A*wA* = L(w, 1, 1, 1). Let V be the smallest variety of languages containing the languages of the form A *wA *, where A is any alphabet and w E --t + First , it is clear that V is contained in Lt since for every alphabet A and It: E A+, the language A*wA* is locally testable. Let now A be a fixed alphabet and let L E A+ Lt. Then L is a Boolean combination of languages of the form wA*, A*wA* or A*w, where wE A+. Thus, to prove that L E A+V it suffices to show that each one of these languages lies in A,+V. This is clear for every language of the form A,*wA* , by definition of V. Consider now a language of the form w A * Let B be the alphabet obtained from A by the addition of a new letter b, i.e ., B = AU {b} .
66
The language B*bwB* lies in B+V. Then the language b- 1 (B*bwB*) = B*bwB* U wB* is also in B+V, since B+V is closed under cancellation. Now, (B*bwB* U wB*) \ B*bB* = wA* is also a language of B+V, since B*bB* E B+V and B+V is closed under complementation. Consider now the morphism cp: A+ _ B+ given by cp(a) = a for all a E A. We have cp-l(wA*) = wA*, whence wA* is a language of A+V, since V is closed under inverse image of morphisms between free semigroups. That every language of the form A*w, with w E A+, lies in A+V can be proved analogously. So, we deduce that L E A+V and, consequently, that A+Ct ~ A+V. Since this holds for all alphabets A, we conclude that Ct ~ V .
o Corollary 4.2 The class Ct (resp . Tlt) is the variety of languages generated by each of the classes Slt, Ct-p and Ct-s (resp. Stlt, Tlt-p and Tlt-s). 0 This result and Proposition 3.3 imply the following corollary. Corollary 4.3 The pseudo variety LSI (resp. Acorn * D, Com * D) is generated by the syntactic semigroups of the languages of the form A*wA* (resp. L(w, T, t, 1), L(w, T, t, n)), where A is any alphabet and w E A+ (resp. and T, t ;::: 0 and n ;::: 1). 0 Now we consider varieties of languages contained in the classes of languages we are studying. Let us begin by considering the equivalence relation == defined immediately before Theorem 3.6 and show the following observation. Lemma 4.4 Let S be a finite semigToup. Then, S is ==-trivial if and only if S lies in the pseudovariety W = J n [x'" zy"'tx'" = y"'tx'" zy"'] . Proof. By definition of the equivalence ==, S is ==-trivial if and only if S is .J-trivial (since .J is contained in ==) and, for all idempotents e, f E S and all T, S E S , eT f se = f seT f. It follows that S is ==- trivial if and only if S E J and S satisfies the pseudoidentity x'" zy"'tx'" = y"'tx'" zy"', that is, if and only 0 if SEW. Now we can prove our last result. Proposition 4.5 (1) The class Ct n .J (resp . Ct n R , Ct n C) of Ct and.Jtrivial (resp. Ct and R-trivial, Ct and C-trivial) languages is the largest variety of languages contained in the class Slt (resp . Ct-p, Ct-s) . (2) The class Tlt n W (resp . Tlt n R, Tlt n C) ofTlt and ==-trivial (resp. Tlt and R-trivial, Tlt and C -trivial) languages is the largest variety of languages contained in the class Stlt (resp. Tlt-p, Tlt-s).
67
Proof. We only give the proof of (2) for Tlt n W. The other cases are similar. Let V be the largest variety of languages contained in the class of all strongly threshold locally testable languages and let £ E Tlt n W. Then S(£) E (Acorn * D) n W. In particular, S(£) E Acorn * D and the syntactic image of £ is a union of =-classes of S(£), since they are trivial by Lemma 4.4. Hence, by Theorem 3.6, £ is strongly threshold locally testable. So, by definition of V, we have Tlt n W <;;; V. Let now A be an alphabet and let L E A+V. Then L is strongly threshold locally testable, whence £ is also threshold locally testable. Now, we prove that £ is =-trivial. By Lemma 4.4, we have to show that £ is .J-trivial and that S(£) satisfies the pseudoidentity x"'zy"'tx'" = y"'tx"'zy"'. Suppose first, by way of contradiction, that £ is not .J-trivial, that is, suppose that S(£) does not verify the pseudoidentity (xy)'" = (yx)"'. Then, there exist u, v E A + such that (uv) n rf L (vu) n for all n 2:: 1. Hence, for each n 2:: 1, there exist rn, Sn E A* such that either rn(Uv)n Sn E £ and rn(vu)n sn fI£ , or rn(uv)n sn fI- £ and rn(vu)n sn E L. Then, either (uv)n E r;;1Ls;;1 and (vu)n fI- r;;1 £s;;1, or (uv)n fI- r;;1 £s;;1 and (vu)n E r;;1 £s;;1. Let k, t 2:: 1 and let n 2:: kt . Then, we have (uv)n =k,t ,1 (vu)n So, for all k,t 2:: 1, r;;1£s;;1 is not saturated by the equivalence =k,t,1. This implies that r;;1£s;;1 is not strongly threshold locally testable. But this is absurd since r;;1 £s;;1 E A+V since £ E A+V and A+V is closed under cancellation. So L must be .J-trivial. Let us now show that S(£) satisfies the pseudoidentity x"'zy"'tx'" = y"'tx"'zy"'. Since £ is Stlt, S(£) is aperiodic by Theorem 3.5. So, there is an integer m such that , for all S E S(£) , sm = sm+! . Suppose that S(£) does not satisfy x"'zy"'tx'" = y"'tx"'zy'" , that is, suppose that there are u, v, p, q E A+ such that unpv"qu n rf L vnqu"pv" for all n 2:: m. Then, without loss of generality, we may suppose that there are rn, Sn E A* such that rnunpvnquns n E £ and rnvnqunpv"sn fI- £. Hence, unpvnqu n E r;;1£s;;1 and vnqu"pvn fI- r;;1Ls;;1. Let k,t 2:: 1 and let n 2:: max{kt,m}. We have unpvnqun =k,t,1 vnqunpvn and, consequently, r;;1 £S;;1 is not strongly threshold locally testable. But this is a contradiction by the same reasons as above and so S(£) must satisfy the pseudoidentity x"'zy"'tx'" = y"'tx"'zy"' . By Lemma 4.4 we deduce that L is =-trivial, which shows that £ E A+(TltnW). We have proved that A+V <;;; A+(TltnW) and since this holds 0 for every alphabet A we conclude that V <;;; Tlt n W. We summarize in the next diagram the inclusion relations stated in the results of this section. The emboldened classes are the varieties of languages and we denote by Tlt-ps the class Tlt-p n Tlt-s . '
68
Tlt n "R.
.ct n.c
.ct n:r References 1. J . Almeida, Finite Semigroups and Universal Algebra, World Scientific, Singapore, 1994. 2. D. Beauquier and J.-E. Pin, Languages and scanners, Theoretical Computer Science 84 (1991) 3-21. 3. J. Brzozowski and I. Simon, Characterization of locally testable events, Discrete Mathematics 4 (1973) 243-271. 4. J. Costa, Quelques intersections de varieUs de semigroupes finis et de varieUs de langages, operations implicites, Ph.D. Thesis, Universite Paris 6,1998. 5. S. Eilenberg, Automata, languages and machines, vol. B, Academic Press, New York, 1976. 6. R. McNaughton, Algebraic decision procedures for local testability, Math. Systems and Theory 8 (1974) 60-76. 7. J.-E. Pin, Varieties of formal languages, Plenum, New York and North Oxford, London, 1986. 8. J .-E. Pin, The expressive power of existential first order sentences of Biichi's sequential calculus, in Proc. 23rd ICALP, Lecture Notes in Computer Science 1099, Springer, Berlin (1996) 300-311. 9. J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982) 1-10. 10. D. Therien and A. Weiss, Graph congruences and wreath products, J. Pure Appl. Algebra 36 (1985) 205-215.
SOLID VARIETIES OF SEMIRINGS KLAUS DENECKE University of Potsdam, Institute of Mathematics, PF 601553, D-14415 Potsdam, Germany E-mail:[email protected] HIPPOLYTE HOUNNON University of Pot-sdam, Institute of Mathematics, PF 601553, D-14415 Potsdam, Germany E-mail: [email protected] In [5] the concept of an ID-semiring (S;+ , ·) as an algebra of type (2,2) where (S; +) and (S; ·) are idempotent semigroups (bands) and where four distributive laws: xy + z ~ (x + z)(y + z), x + yz ~ (x + y)(x + z) , x(y + z) ~ xy + xz , (x + y)z ~ xz + x z are satisfied, was introduced. An ID-semiring (S; +, .) is normal if x + y + u + v ~ x + u + y + v and xyuv ~ xuyv are identities. In this paper we want to determine the least and the greatest solid variety of normal I D-semirings, i.e. varieties in which every identity is satisfied as a hyperidentity. The result is that the variety of all normal I D-semirings is solid. The variety RA(2 .2) of rectangular algebras of type (2,2), i.e. the variety generated by all projection algebras of type (2, 2) is the least non-trivial solid variety of normal I D-semirings. Ke1J words: Normal ID-semiring, hyperidentity, solid variety, macro systems
1991 AMS Mathematics Subject Classification: 08B05, 08B15, 08A50, 16Y60
Introduction
Hyperidentities in the sense of this paper were introduced by W. Taylor [7] with the aim to extend the concepts of an identity and of a variety. Hyperidentities are equations consisting of variables and operation symbols in which one can substitute for the variables concrete elements of the appropriate algebraic structure and for the operation symbols concrete term operations of this structure. Therefore hyperidentities are formulas in a second order language where quantification is allowed as well for individual variables as for operation symbols. For instance, one can consider the " hyperassociative law" as formula of the form:
'Vx'Vy'Vz'VF(F(x, F(y , z))
~
F(F(x, y), z)),
where for the binary operation symbol F any binary term of the considered language can be substituted. If we request that all identities of a variety are satisfied as hyperidentities we call this variety solid [2]. Solid varieties are second order varieties. The 69
70
important fact that all solid varieties of a given type T form a complete suIT lattice of the lattice of all varieties of type T is a helpful tool to study lattices of varieties. In our paper we will apply the general theory of hyperidentities to the concrete case of semirings, i.e, algebraic structures with two binary asssociative operations where the distributive laws are valid. In the first section we will mention the most important basic concepts which are needed in subsequent sections. In the second section we give necessary conditions for solid varieties of semirings. After this we characterize normal forms of binary terms in the variety of all normal I D-semirings. In the last section we prove that this variety is solid. The main result is the determination of the greatest and the least nontrivial solid variety of normal semirings and is an important step on the way to find out all solid varieties of semirings. 1
Basic Concepts
Since semirings are algebras of type T = (2,2) two binary operation symbols F and G are needed. To define an appropriate language for algebras of type T = (2,2) we choose a set X of variables and then we can define the set W C2 ,2) (X) of all terms of type (2,2) in the usual way. Using the n-element alphabet Xn = {Xl, .. . Xn } we define the set WC2,2)(X n ) of all n-ary terms of type (2,2). To define hyperidentities we need the notion of a hypersubstitution. Hypersubstitutions of type T = (2,2) are mappings a: {F,G} -; W C2 ,2) (X2). That means, each of the operation symbols F and G is mapped to a binary term s and t, respectively. By a s,t we denote the hypersubstitution defined by F t-+ S, G t-+ t and by Hyp(2, 2) we denote the set of all hypersubstitutions of type T = (2,2). A hypersubstitution a of type (2,2) can be extended to a mapping a defined on the set W C2 ,2) (X) of all terms in the usual inductive way: a[x] := X if X E X is a variable and a[J(tl,t2)]:= a(J)(a[td,a[t2]),J E {F,G}. If we define an operation: Oh : Hyp(2,2) x Hyp(2,2) -; Hyp(2, 2) by (aloha2)(J) := al[a2(J)] , J E {F, G} and if we define:
aid(J)
:=
J(XI , X2), J E {F, G}
then we obtain a monoid. An identity s >::! t in a variety V of type T = (2,2) is called hyperidentity of V if a[s] >::! a[t] is an identity for all a E Hyp(2,2) . In this case we will
71
also say that a preserves the identity s
~
t.
The idea of hypersubstitutions and hyperidentities in classes of semirings is basically used and applied in the definition of macro systems in connection with tree automata in the Theory of Automata [3]. Clearly, it is not necessary to apply all hypersubstitutions on the identity s ~ t if we want to check if s ~ t is satisfied as a hyperidentity in V . In [4] a relation "'v given on Hyp(2,2) was introduced which is defined by:
where IdV is the set of all identities valid in V . It is clear that "'v is an equivalence relation on Hyp(2,2). It is useful to form the quotient set Hyp(2, 2)/ ~v and to select from each equivalence class using a choice function
Necessary conditions for solid varieties of semirings
If s is a term of type (2,2) then we define a term shd in the following inductive way: (i) if s = x is a variable then shd = s = x ,
72
(ii) if F(tl' t2) or G(tt, t2) are composed terms and assume that tfd, i = 1,2 are already defined, then F(tt, t2)hd = G(t?d, t~) and G(tt, t2)hd =
F(t?d, t~d). Definition. 2.1 A variety V of type (2,2) is called hyperdualizable iffor every identity s ~ t in V the equation shd ~ t hd is also an identity in V. If aG(x,y) ,F(x,y) is the hypersubstitution mapping F to G(x, y) and G to F(x,y) then aG(x,y),F(x,y)[t] = t hd . Therefore every solid variety of type (2,2) is hyperdualizable. Proposition. 2.2 If V is a solid variety of semirings then V is a variety of I D-semirings. Proof. Every variety of semirings satisfies the distributive laws G(x,F(y,z)) ~ F(G(x,y), G(x, z)) and G(F(x,y),z) ~ F(G(x,z),G(y,z)). Since V is solid, it is hyperdualizable and therefore
F(G(x, y), z) = aG(x,y),F(x,y) [G(F(x, y), z)] ~ aG(x,y),F(x,y)[F(G(x,z),G(y,z))] = G(F(x,z), F(y, z)). This means that x· y + z ~ (x + y)(y + z). The other identity can be proved in a similar way. If we apply the hypersubstitution aF(x,y),x to the identity G(x,F(y,z)) ~ F(G(x,y),G(x,z)) we obtain x ~ F(x,x) and by the hyperdualizability also x ~ G(x,x). • We remember the following denotations for particular varieties of semigroups: RB = Mod{(xy)z ~ x(yz) ~ xz,x 2 ~ x}, the variety of rectangular bands, NB = Mod{(xy)z ~ x(yz),x 2 ~ x,xyzu ~ xzyu}, the variety of normal bands, Reg = Mod{(xy)z ~ x(yz), x 2 ~ x, xyzx ~ xyxzx}, the variety of regular bands. For a variety V of semirings by V+:= ModId{(S;+) I (S;+,·) E V} and by
V
:=
ModJd{(S;·) I (S; + , .)
E
V}
we denote the additive and the multiplicative reduct, respectively. Then we have:
Proposition. 2.3 Let V be a solid variety of semirings. Then for V+ and V there are exactly the following possibilities: 1. V+ and V are both varieties of rectangular bands. 2. V+ and V are both varieties of normal bands. 3. V+ and V are both varieties of regular bands.
73
Proof. If V is solid then by [1] each reduct is also solid. Therefore the reducts V+ and V are solid. Moreover by Proposition 2.2 they are varieties of bands. By [1] there are exactly three solid varieties of bands, namely RB, Reg and N B. Since V is hyperdualizable, both reducts are RB or Reg orNB . •
3
Solid varieties of rectangular semirings.
A semiring is called rectangular if both reducts are rectangular bands, i.e in the case of Proposition 2.3,l. In [6] the variety RA,. of algebras of type T was defined as variety which is generated by all projection algebras of type T. It turns out that RA,. is the least non-trivial solid variety of type T. Moreover in [6] an equational basis for the variety RA,. was given. In the case of type T = (2,2) the variety RA,. is defined by : RA,. = M od{ (x + y) + z R! X + (y + z) R! X + z, x + X R! x, (x. y) . Z R! X· (y. z) R! X· Z, X· X R! x, (x + y)(u + v) R! xu + yv}. The identity: (x + y)(u + v) R! xu + yv is also called entropic law. Theorem 3.1 The variety RA(2,2) is the least non-trivial solid variety of semirings. Proof. Since RA(2,2) is the least non-trivial solid variety of type T = (2,2) we have only to show that it is a variety of semirings, i.e we have to show that the distributive laws are satisfied. Infact, if we substitute in the entropic law for v the variable u we obtain (x + y)(u + u) R! (x +y)u R! xu + yu and if we substitute for y the variable x we obtain (x + x)(u + v) R! x(u + v) R! xu + xv.
•
Now we show: Lemma. 3.2 If V is a variety of semirings and assume that both reducts are rectangular bands then V ~ RA(2,2) ' Proof. To show that V ~ RA(2,2) we have to show that every identity in RA(2,2) is satisfied in V. For the associative and idempotent laws and the identities: xyz R! XZ, X + y + Z R! X + Z this is clear. We show the entropic law. Then we have: (x + y)(u + v) R! xu+xv+yu+yv R! xu+yv by the distributive laws and by: x+y+z R! X+Z .• Lemma 3.2 makes it clear that in the first case of Proposition 2.3 there is exactly one solid variety of semirings, namely RA(2,2) and this is the greatest solid variety with the property that both reducts are rectangular bands.
74
Binary terms in normal I D-semirings
4
Now we consider the case that both reducts are normal bands. Such varieties of semirings are called normal [5]. To determine the set of all normal form hypersubstitutions we need normal forms of binary terms with respect to such varieties. Consider the following set of binary terms:
BinI := {x,y,xy,yx , xyx,yxy}. Lemma. 4.1 Let V be the variety of all normal ID-semirings. Then every binary term over V can be written in the form: t = tl +t2+t3 +t4 +t5 +t6 +t7, where tj E BinI, j = 1 . . . 7. Proof. We give a proof by induction on the complexity of the definition of the binary term t. If t is a variable then t = tl + t2 + t3 + t4 + t5 + t6 + t7, tj = t using the idempotent law. 7
We assume that ti
=L
tIj,tIj E Binl,j
= 1 ... 7,i = 1, 2.
j=1
For t F(tl' t2) it is clear that t can be written in the form t = t~ + t; + t3 + t~ + t~ + t~ + t~,tj E Binl,j = 1. .. 7 because of the idempotency and the medial law. 7
For t = G(tl, t2), we have t =
7
0::: tIj)( L
t2j). After the development j=! we see that t is a sum of products, each consisting of two elements from Bin!. Since Bin! is the set of all binary terms constructed only by multiplication, the product of two elements from Bin! belongs also to Bin!. Using the idempotent and medial laws t can be written in the form: t = t~ + t; + t3 + t~ + t~ + t6 + t~, tj E Bin!,j = 1. .. 7. • j=1
The following lemmas show some possibilities to reduce the number of summands. Lemma. 4.2 Let V be the variety of all normal I D-semirings. Then the following identities are satisfied:
(i) x
+ xy + y ~ x + y
+ yx + y ~ x + y x + xyx + xy ~ x + xy
(ii) x (iii)
(iv) x
+ xyx + yx ~ x + yx
(x) xy + yxy
+ yx ~ xy + yx
(xi) xy + yxy + y (xii) x
~
xy + y
+ xyx + y ~ x + y
(xiii) x+yxy+y
~x+y
75
(v) x
(xiv) xyx + yx + y
(vi)
(xv) xyx + xy + y
(vii) (viii)
(ix)
+ yx + yxy ~ x + yxy x + xy + yxy ~ x + yxy x + xyx + yxy ~ x + yxy xy + xyx + x ~ xy + x xy + xyx + yx ~ xy + yx
xyx + y
~ ~
xyx + y
(xvi) xyx + xy + yxy
~
xyx + yxy
(xvii) xyx + yx + yxy
~
xyx + yxy
(xviii) xyx + yxy + y
~
xyx
Proof. The proofs can be given by easy calculations.
+ y.
•
Now we consider binary terms written as sums which start and end with the same term of BinI. We can restrict ourselves to the cases when the first and the last term is x, xy or xyx. The other cases are obtained by permuting x and y. Lemma. 4.3 Let V be the variety of all normal ID-semirings. Then (i) x + xyx + yx + x ~ x + yx + x (vi) x + y + xyx + x ~ x + y + x (ii) x + xyx + xy + x ~ x + xy + x (vii) x + y + yxy + x ~ x + y + x (viii) x+xyx+yxy+x ~ x+yxy+x (iii) x+xy+y+x ~ x+y+x (iv) x+yx+y+x ~ x+y+x (ix) x+xy+yxy+x ~ x+yxy+x (v) x+yxy+yx+x ~ x+yxy+x (x) x+xy+yx+x ~ x+yxy+x. Proof. All identities can be proved applying Lemma 4.2. •
Lemma. 4.4 Let V be the variety of all normal I D-semirings. following identities are satisfied:
Then the
(i) xyx + y (ii) (iii) (iv) (v) (vi)
+ xy + xyx ~ xyx + y + xyx xyx + xy + yxy + xyx ~ xyx + yxy + xyx xyx + y + yxy + xyx ~ xyx + y + xyx xyx + yxy + yx + xyx ~ xyx + yxy + xyx xyx + y + yx + xyx ~ xyx + y + xyx xyx + xy + yx + xyx ~ xyx + yxy + xyx.
Proof. All these identities can be proved if we substitute for x the term xyx in the appropriate identities of Lemma 4.3. • Lemma. 4.5 Let V be the variety of all normal I D-semirings. following identities are satisfied:
Then the
76
(i) xy + y + yxy + xy
~
(ii) xy + xyx + yxy + xy (iii) xy + xyx + x (iv) xy + yx
~
xy + yxy + xy
+ xy ~ xy + x + xy
+ yxy + xy ~ xy + yx + xy
(v) xy + yx + xyx + xy
(vi) xy
xy + y + xy
~
xy + yx + xy
+ x + yxy + xy ~ xy + x + yx + xy
(vii) xy + xyx + y
+ xy ~ xy + yx + y + xy .
Proof: All proofs can be given applying 4.2 and 4.3.
•
Finally we have: Corollary. 4.6 Every binary term over the variety of all normal IDsemirings can be represented as a sum t = tl +t2+t3+t4, tj E Binl, consisting of at most four summands. Proof. By Lemma 4.1 t can be represented as a sum consisting of at most seven summands from Binl. If tl = x then t ~ x + t2 + t3 + X + t4 + ts + t6 + t7 ~ X + t' + t4 + ts + t6 + t7 ~ . .. ~ x + s + t7, s E Binl using Lemma 4.3. If tl = xyx and assume that t consists of more than four summands then by Lemma 4.4 if x does not occur under the summands we can reduce the term to three summands using the same method as before. If x occurs under the summands by the medial law and the idempotency we have t = xYX+X+t2+t3+t4+tS+t6+t7 ~ xYX+X+t2+t3+X+t4+t5+t6+t7 ~ xyx + x + t' + t4 + t5 + t6 + t7 ~ . .. ~ xyx + x + til + t7 by Lemma 4.3 and we obtain a term consisting of four summands. Now we consider the case that the term begins with xy. Assume that x occurs or that y occurs under the summands. If x occurs we can write t ~ xy + x + t2 + t3 + X + t4 + ... + t7 ~ ... ~ xy + x + t' + t7 (see above) and we obtain a term which consists of at most four summands. If y occurs we conclude in the same way using the identities which are obtained by permuting x and y in Lemma 4.3. If t starts with y or yx or yxy we can use the identities which can be derived from the identities above changing x andy. • Using the previous lemmas we will give a full description of all normal forms of binary terms over the variety of all I D-semirings. All these terms are
77
represented as sums of elements from BinI. At first we determine all binary terms represented as sums consisting of three elements of BinI. Lemma. 4.7 There are exactly the following terms over the variety of all normal I D-semirings which are represented as sums of elements from BinI consisting of three summands:
(i) x+xy+t,t E {x, xyx, yx} (ii) x+yH, t E BinI \{y} (iii) x
+ yx + t, t
E {x, xyx, xy}
(iv) x+yxy+t, t E {x,xyx} (v) x +xyx +x (vi) xyx +x +t,t E BinI\{x} (vii) xyx + y (viii)
+ t, t E BinI \{y} xyx + xy + t, t E {x, yx, xyx}.
(ix) xyx +yxy + t, t E {x,xyx} (x) xyx+yx+t, t E {x, xy, xyx} (xi) xy +x + t, t E BinI\{x} (xii) xy +yx + t, t E BinI \ {yx} (xiii) xy + y
+ t, t E BinI \ {y} (xiv) xy + xyx + xy (xv) xy + yxy + xy
and all terms which arise from the given ones by exchanging x and y. Proof. (i) For t we have to consider only three possibilities since x+xy+t {y, yxy} and by Lemma 4.2 (i) and (vi). (iii) For t we have only three possibilities since x + yx + t by Lemma 4.2 (ii) and (v).
~
~
x+t, t E
x + t, t E {y, yxy}
(iv) is clear since x+yxy+y ~ x+y by Lemma 4.2 (xiii) and x+yxy+xy ~ x + yx + yxy + xy ~ x + yx + xy by Lemma 4.2 (v), (ix), respectively, and x + yxy + yx ~ x + xy + yx by Lemma 4.2 (vi) and (x). (v) We have x + xyx + t ~ x + t, t E {yx, xy} by Lemma 4.2 (iv), (iii) and x + xyx + t ~ x + t, t E {y , yxy} by Lemma 4.2 (vii), (xii). (viii) follows from xyx + xy + t ~ xyx + t, t E {y, yxy} by Lemma 4.2 (xv) and (xvi).
(ix) follows from xyx + yxy + y ~ xyx + y by Lemma 4.2 (xviii) and further we have xyx+yxy+yx ~ xyx+xy+yxy+yx ~ xyx+xy+yx by Lemma 4.2 (xvi) and (x). Finally we get xyx + yxy + xy ~ xyx + yx + xy , by Lemma 4.2 (xvii), and the identity arises from (ix) by permuting x and y.
78 (x) We have xyx (xvii).
+ yx + t
~ xyx
+ t,t
E {y,yxy} by Lemma 4.2 (xiv) and
(xiv) Using Lemma 4.2 we have xy + xyx + t ~ xy + t if t belongs to {x,yx} and xy + xyx + t ~ xy + yx + t if t belongs to {y,yxy}. (xiv) In this case using Lemma 4.2 we have xy + yxy + t ~ xy t E {x, xyx} and xy + yxy + t ~ xy + t if t E {y, yx} . For (ii) , (vi) , (vii), (xi) (xii) (xiii) we have nothing to prove.
+ yx + t
if
•
Now we will determine all binary terms which can be represented as sums of length 4. Lemma. 4.8 There are exactly the following binary terms over the variety of all normal I D-semirings which are represented as sums of elements from BinI consisting of four summands.
(i) xyx+x+y+t,t E Binl\{X , y} (ii) xyx +x +xy + t, t E {xyx,yx} (iii) xyx+x+yx+t,t E {xyx,xy} (iv) xyx + x
(v) xy+x+yx+t,t E {xy,xyx} (vi) xy+yx+y+t, t E {xy,yxy} (vii) xy+x+y+t, t E BinI \{x,y}
+ yxy + xyx
and all terms which arise from the given ones by exchanging x and y. Proof. (ii) xyx + x (iii)
+ xy + t ~ xyx + x + t , t E {y,yxy} by Lemma 4.2 (i) , (vi). Here we have also xyx + x + yx + t ~ xyx + x + t, t E {y, yxy} by Lemma 4.2 (ii) ,(v).
(iv) xyx+x+yxy+y ~ xyx+x+y by Lemma 4.2 (xiii), further xyx+x+yxy+ xy ~ xyx+x+yx+xy by Lemma 4.5 (vi) and the last term was already considered in (iii). Finally xyx+x+yxy+yx ~ xyx+yx+x+yxy+yx ~ xyx + x + xy + yx by the identity arising from 4.5 (vi) by permuting x and y. The last term occurred already in (ii). (v) xy
+ x + yx + t ~ xy + x + t, t
E {y, yxy} by Lemma 4.2 (ii) ,(v).
(vi) xy + yx + y + t ~ xy + y + t, t E {x,xyx} by Lemma 4.2 (ii) and the identity arising from 4.2 (vi) by permuting x and y. In the case of (i) and (vii) we have nothing to prove. •
79
Using all these results we can give a full list of all binary terms over the variety of all normal I D-semirings: Xj xYj XYXj x + t, t E BinI \ {x}j xY + t, t E Binl\{XY}jxyx+t,t E Binl\{XYX}jx+xy+t,t E {x,xyx,yx}jx+y+t,t E BinI \{Y}jX + yx + t, t E {x,xyx,xy}jX + yxy + t,t E {x,xyx}jX + xyx + Xjxyx+x+t,t E Binl\{X}jxyx+y+t,t E Binl\{Y}jxyx+xy+t,t E {x,yx+xyx}jxyx+yxy+t,t E {x,xyx}jxyx+yx+t,t E {x,xy,xyx}jxy+ x+t,t E Binl\{X}jxy+yx+t,t E Binl\{YX};xy+y+t,t E Binl\{Y};XY+ xyx+xy;xy+yxy+xYjxyx+x+y+t,t E \{x,y};xyx+x+xy+t,t E {xyx,yx};xyx+x+yx+t,t E {xy,xyx};xyx+x+yxy+xYXjxy+x+y+t,t E Binl\{X,y};xy+yx+y+t,t E {xy,yxy};xy+x+yx+t,t E {xy,xyx}; and all terms which are obtained by exchanging x and y in the given terms.
5
Hyperidentities in the variety of all normal I D-semirings
Obviously, the idempotent laws are satisfied as hyperidentities. We want to check the associative, the distributive and the medial laws. We need all normal forms of binary hypersubstitutions and use the results of section 4. But we can reduce our checking using some lemmas. Let Bin be the set of all binary terms over the variety of all I D-semirings. Lemma. 5.1 For arbitrary terms t, t', t" belonging to Bin the following identities are satisfied in the variety V of all normal I D-semirings:
+ t")(y, z))
~
t(x, t'(y, z))
+ t(x, t"(y, z)).
+ t")(x, y), z)
~
t(t'(x, y), z)
+ t(t"(x, y), z).
(i) t(x, (t' (ii) teet'
Proof. (i) We give the proof in two steps: 1~ step: Let t E BinI and t', t" E Bin. We substitute for t step by step all elements from the set BinI: For t = x or t = Y we have equality because of the idempotent laws. For t = xy or t = yx we can use the distributive laws and get identities. For t = xyx we have x(t'(y,z) + t"(y,z))x ~ xt'(y,z)x + xt"(y,z)x by the distributive laws. For t = yxy we obtain on the left hand side, (t' + t")(y, z)x(t' + t")(y, z) ~ t' (y, z )xt' (y, z) + t' (y, z )xt" (y, z) + t" (y, z)xt' (y, z) + t" (y, z )xt" (y, z) by the distributive laws. Using the identity x + xy + yx + y ~ x + y, the distributive and the idempotent laws we can see that this side is equal to t'(y, z)xt'(y, z) + t"(y,z)xt"(Y,z) which agrees with the right hand side. 2nd step: Let t, t', t" E Bin.
80 4
Since t E Bin then we can write t =
l: tj, tj
E Bin1 by Corollary 4.6. and
i=1
obtain t(x, (t'
+ t")(y, z))
4
=
(l: tj )(x, (t' + t")(y, z) j=1
~
4
l: (tj(x, (t' + t")(y, z))
j=1 ~
4
l: (tj(x, t' (y, z)) + tj(X, til (y, z)))
j=1 ~ ~
4
using the 1sf step
4
(l: tj )(x, t'(y , z)) + (l: tj )(x, t"(y, z)) i=1
t(x, t'(y , z)) (ii) can be proved similarly.
i=1
+ t(x, til (y, z)). •
Lemma. 5.2 Let V be the variety of all normal ID-semirings. Assume that S1 + S2 + .. . Sn-1 + Sn ~ S1 + s; + ... + S~_1 + s~ E IdV with S1 ~ S1 E IdV and Sn ~ s~ E IdV, then Sn +Sn-1 + .. .+S2+S1 ~ s~ +s~_1 +s;+s1 E IdV-
Proof. As in V holds S1 +S2+ ... Sn-1 +sn ~ S1 +s;+ ... +S~_1 +s~, we have also the following identities s~ + S1 + S2 + ... Sn-1 + Sn + s~ ~ s~ + s~ + s~ + ... + s~_1 + S~+S1 and Sn+S1+S2+· · · Sn-1 +Sn+S1 ~ S~+S1 +s;+ .. . +S~_1 +s~+s~ (by S1 ~ S1 and Sn ~ s~ E IdV). Using the idempotent and the medial laws we obtain Sn + Sn-1 + ... + S2 + S1 ~ s~ + s~_1 + s; + s1 E IdV • Let t be a term of Bin, we define t i by t i ~ t4 + t3 + t2 + t1 if t ~ t1 + t2 + t3 + t4 where tj E Bin1. FUrther we define t C in the following inductive way: If t = x then t C = y, if t = Y then t C = x. If t = t1 + t2 then t C = t1 C + t2 C and if t = t1 t2 then t C = t1 Ct2" assumed that t 1c, t2 C are already defined. Then we have: Lemma. 5.3 If at preserves the associative law in the variety V of all I Dsemirings, then ati and ate preserve also the associative law. Here at denotes the hypersubstitution mapping the operation symbol F to the binary term t and G to an arbitrary term or conversely. Remarks: 1. Clearly, it is enough to map one of the operation symbols to a term since the associative law contains only one operation symbol. 2. It easy to see that all elements from Bin1 preserve the associative law. Proof. Let us consider the associative law: F(x , F(y, z )) ~ F(F(x,y),z) .
81
Then we have o-tc [F(x, F(y, z)] RJ tCW(x, y), z) RJ tC(t(y, x), z) RJ t(z, t(y, x) RJ o-t [F(z, F(y, x))] and o-tc [F(F(x, y), z)] RJ t(t(z, y), x) RJ o-t[F(F(z, y), x)]. But by assumption we have o-t[F(z,F(y,x))] RJo-t[F(F(z,y),x)] EldV. Soweobtain o-tc[F(x,F(y,z))] RJ o-tc[F(F(x,y),z)] E IdV. By assumption we have t(t(x, y), z) RJ t(x, t(y, z)) E IdV, and by Corollary 4
4.6 we obtain
4
4
(2: tj)(( 2: tj )(x, y), z) j=l j=l 4
RJ
4
(2: tj )(x, (2: tj )(y, z)) E IdV. j=l j=l
4
4
4
2: 2: tj(tk(X,y),z) RJ 2: 2: tj(X,tk(Y'Z)) EldV, j=lk=l j=lk=1 i.e. tl (ti (x, y), z) + ... + tl(t4(X, y), z) + ... + t4(t4(X, y), z) RJ tl (x, tl (y, z)) + ... + tl (x, t4(Y, z)) + ... + t4(X, t4(Y, z)) E IdV. (*) Using Lemma 5.1 we get:
Because of the previous remark and using Lemma 5.2, from the identity (*) we obtain:
t4(t4(X,y)Z) +t4(t3(X,y),z) + ... +t4(tl(X,y),z) + ... tl(t4(X,y),z) + ... + tl (t4(X, y), z) + .. . + tl (tl (x, y) , z) RJ t4(X, t4(Y, z)) + ... + t 4(x, tl (y, z)) + . .. + tl (x, t 4 (y, z)) + ... + tl (x, tl (y, z)) E I dV. Using Lemma 5.1 one has: 4
4
t4((2: t5_j)(X,y),z) + ... +tl((2: t5_j)(X,y),z)
j=l
RJ
j=l
4
4
t4(X, (2: t5-j )(y, z)) + ... + tl (x, (2: t5-j )(y, z)) E IdV, i.e.
j=1
4
j=l 4
4
(2:t5-k)(2:t5-j)(X,y),z)
k=1
j=l
RJ
4
(2:t5-k)(X,(2:t5-j)(y,z)) E IdV, and
k=1
j=1
therefore ti(ti(x, y), z) RJ ti(x, ti(y, z)) E IdV. • To check the associative hyperidentity we prove at first some more identities. Lemma. 5.4 Let V be the variety of all normal I D-semirings. The following
identities are satisfied: (i) x + xy + xyz
RJ
X + xy + xz + xyz,
(ii) x + yx + zyx
RJ
X + yx + zx + zyx,
(iii) x + xyx + xyzx
RJ
X + xyx + xzx + xyzx,
(iv) x + yxy + zxyz
RJ X
+ yxy + zxz + zxyz,
(v) xyz + yxz + zxy + zyx
RJ
xyz + xzy + yzx + zyx .
Proof. All these identities can be proved by calculation. The next identities are satisfied for all terms from BinI.
•
82
Lemma. 5.5 For arbitrary terms tt, t2 belonging to BinI, the following identity is satisfied in the variety V of all normal ID-semirings: 2
2
2
2
L: L: ti(X, tj(Y, z)) ~ L: L: ti(tj(X,y),z)).
i=lj=l
i=lj=l
Proof. For h and t2 we substitute all elements from Binl and apply Lemma 5.2. • To prove that the distributive law is a hyperidentity in the variety of all normal I D-semirings we need the following identities. Lemma. 5.6 For all t E Binl in the variety of all normal I D-semirings there holds: t(X,yz)
~
t(x,y)t(x,z).
Proof. This becomes clear after substitution using the idempotent and distributive laws. •
Lemma. 5.7 For arbitrary terms t, t' belonging to Binl the following identities are valid in the variety V of all normal I D-semirings: t(x, yz) + t'(x, yz) ~ t(x, y)t(x, z) + t(x, y)t'(x, z) + t'(x, y)t(x, z) + t'(x,y)t'(x,z) ~ (t(x,y) +t'(x,y))(t(x,z) +t'(x,z)). Proof. We substitute for t and t' all elements from Binl using lemmas 5.7 and 5.2. • Further we have the following lemma Lemma. 5.8 For arbitrary t belonging to Bin and for arbitrary t' belonging to Binl, the following identity is satisfied in the variety V of all normal IDsemirings: t(x, t'(y, z)) ~ t'(t(x, y), t(x, z)) .
Proof. The identity is satisfied for t'(x, y) = x and for t'(x, y) = y if t is an arbitrary binary term. Consider the case t'(x,y) = xy. By Corollary 4.6 the 4
term t can be written as a sum
L: tj, tj E Binl . j=l
Then we have: 4
4
j=l
j=l
(L: tj)(x,yz) = L: tj(x,yz) ~ tl(X,yZ) +t2(X,yZ) +tl(X,yZ) +t3(X,yZ) +tl(X,yZ) +t4(X,Z) +t2(X,yZ) +t3(X,yZ) +t2(X,yZ) +t4(X,yZ) +t3(X,yZ)+ t 4 (x, yz) ~ tl(X,y)tl(X,Z) +tl(X,y)t2(X, Z) +t2(X,y)tl(X,y) +t2(X,y)t2(X,Z)+
83
+ t1 (x, y)t3(X, z) + t3(X, y )t1 (x, z) + t3(X, y)t3(X, z)+ + t1 (x, y)t4(X, z) + t4(X, y )t1 (x, z) + t4(X, y)t4(X, z)+ t2(X, y)t2(X, z) + t2(X, y)t3(X, z) + t3(X, y )t2(X, z) + t3(X, y)t3(X, z)+ t2(X, y )t2(X, z) + t2(X, y )t4(X, z) + t4(X, y)t2(X, z) + t4(X, y)t4(X, z)+ t1 (x, Y )t1 (x, z) t1 (x, Y )t1 (x, z)
t3(X,y)t3(X,Z) +t3(X,y)t4(X,Z) +t4(X,y)t3(X,Z) +t4(X,y)t4(X,Z) by Lemma 5.8, the medial and the idempotent laws. This sum can be written as 4
(2: tj)(x,yz) j=l
~
4
t1(X,y)((2: tj)(x,z)) j=l
4
4
4
+ (t2(X,y)(2: tj)(x,z)) + j=l 4
4
(t3(X,y)(2: tj)(X,Z)) + (t 4(X,y)(2: tj)(X,Z)) ~ (2: ti)(X,y)(2: tj)(X,Z)) ~ j=l j=l i=l j=l t(x, y)t(x, z). This proves t(x,yz) ~ t(x,y)t(x,z) (***) . If we replace y by z and z by y in (***) we obtain t(x,zy) ~ t(x,z)t(x,y), i.e., t(x, t' (y, z)) ~ t' (t(x, y), t(x, z)) for t' = yX. For t' = xyx we substitute in equation (***) for y the product yz and for z the variable y and obtain t(x,yzy) ~ t(x,y)t(x,z)t(x,y) and this gives t(x, t'(y, z)) ~ t'(t(x, y), t(x, z)). For t = yxy we get our result substituting in (***) for y the product zy . • The previous lemma means that the distributive law G(x,F(y,z)) ~ F(G(x, y), G(x, z)) is hypersatisfied if we substitute for G arbitrary binary terms over V and for F arbitrary terms from BinI. Now we can prove: Theorem. 5.9 Let V be the variety of all normal ID-semirings. Then the
distributive law: G(x, F(y, z)) tity.
~
F(G(x,y), G(x, z)) is satisfied as hyperiden-
Proof. For G we substitute an arbitrary binary term and give a proof by induction on the complexity of the definition of a binary term t' which we substitute for F. If t'(x,y) = x then we have t(x,y) ~ t(x,y) and for t'(x, y) = y we get t(x, z) ~ t(x, z). Now we assume that for tl, t2 E Bin the distributive law is satisfied and have only to consider the case t' = G(tl, t2) since if G is the outermost operation symbol by the distributive law we can construct an equivalent term of the variety V where F is the outermost operation symbol. By assumption we have: t(X,t1(Y,Z)) ~ t1(t(X,y),t(x,z)) E IdV and t(x, t2(Y, z)) ~
t2(t(X,y),t(x,z)) E IdV.
By addition it follows : t(X,t1(Y,Z)) + t(X,t2(Y,Z)) ~ t1(t(X,y),t(x,z)) + t2(t(X,y),t(x,z)) E IdV, i.e. t(X,t1(Y,Z)) + t(X,t2(Y,Z)) ~ (t1 + t2)(t(X, y), t(x, z)) E IdV, i.e. t(x, (t1 + t2)(Y, z)) ~ (t1 + t2)(t(X, y), t(x, z)) E IdV by Lemma 5.1 and then t(X,F(t1,t2)(Y,Z)) ~ F(t1,t2)(t(X,y),t(x,z)).
84
•
Thus the distributive law is a hyperidentity.
The last step is to prove that the medial law is satisfied as a hyperidentity. Lemma. 5.10 Let V be the variety of all normal ID-semirings. For arbitrary
tl, t2, t3 belonging to BinI, assume that the binary term tl + t2 + t3 preserves the medial law in V. Then the medial law is satisfied as hyperidentity in V. 4
Proof. Let t E Bin i.e. t =
L: tj, tj
j=1
E BinI by Corollary 4.6. Then we have:
4 4 4
t(t(x, y), t(u, v)) =
(L: tj)(( L: tj)(X, y), (L: tj)(u, v)) j=1 ~
j=1 j=1 (ti + t2 + t3)((t1 + h + t3)(X, y), (ti + t2 + t3)(U, v)) +(ti + t2 + t4)((tl + t2 + t4)(X, y), (ti + t2 + t4)(U, v)) +(ti + t3 + t4)((t1 + t3 + t4)(X, y), (ti + t3 + t4)(U, v)) +(t2 + t3 + t4)((t2 + t3 + t4)(X, y), (t2 + t3 + t4)(U, v)) by Lemma 5.1, the medial and the idempotent laws
~
(ti + t2 + t3)((t1 + t2 + t3)(X, u), (ti + t2 + t3)(Y, v)) +(ti + t2 + t4)((tl + t2 + t4)(X, u), (ti + t2 + t4)(Y, v)) +(ti + t3 + t4)((tl + t3 + t 4)(x, u), (ti + t3 + t 4)(y, v)) +(t2 + t3 + t4)( (t2 + t3 + t4)(X, u), (t2 + t3 + t4)(Y, v)) by assumption
~
4
4
4
j=l
j=1
j=1
(L: tj)(( L: tj)(X, u), (L: tj)(Y, v)) by Lemma 5.1, the medial and the idempotent laws
~
t(t(x, u), t(y, v).
•
Lemma. 5.11 Assume that the binary term t = tl + t2 + t3, tj E BinI pre-
serves the medial law in the variety of all normal ID-semirings. Then all binary terms t' arising from t by permuting the summands, preserve also the medial law. Proof. It is enough to prove that t' = tl + t3 + t2 and til = t2 + tl + t3 preserve the medial law. Using Lemma 5.1, the medial and the idempotent laws we obtain:
t' (t'(x, y), t' (u, v))
~
3 3 3
L: L: L: ti(tj(X, Y), tk( u, v)) + t2(t2(X, y), t2( u, v))
i=1 j=1 k=1 333 ~ (L: tj)(( L: tj )(X, y), (L: tj)(u, v)) + t2(t2(X, y), t2(U, v)) j=1 j=1 j=1 3 3 3 ~ (L: tj))( (L: tj )(X, u), (L: tj )(y, v)) + t2(t2(X, u), t2(Y, v)) j=l j=l j=1 by assumption and by the fact that both reduets of the
85
variety of all normal I D-semirings are solid + t3 + t2)((tl + t3 + t2)(X, u), (tl + t3 + t2)(Y, v)) by Lemma 5.1, the medial and the idempotent laws c::::! t'(t'(x, u), t'(y, v)). Using the same idea we can prove that til preserves also the medial law. • c::::!
(tl
Lemma. 5.12 Let t E Bin and assume that t preserves the medial law in the variety of all normal I D-semirings. Then the binary term t C arising from t by permuting x and y has the same property. Proof. similar to 5.3. • To check that the medial law is a hyperidentity we need more identities. So we have: Lemma. 5.13 Let V be the variety of all normal ID-semirings. The following identities are satified: (i) x + y + u + uv + yuv c::::! x + y + u + yv + yuv, (ii) x + y + u + vu + vuy c::::! x + y + u + vy + vuy, (iii) x + yxy + uxu + vxuv + vuxyv c::::! x + yxy + uxu + vyxv + vuyxv, (iv) x + xy + yx + xu + ux + xuv + uvx + vux + vuyx c::::! x + xy + yx + xu + ux + xyv + yvx + vyx + vyux. Proof. The proofs are straightforward. • Now we can prove: Lemma. 5.14 Let V be the variety of all normal ID-semirings. Then all binary terms t = tl + t2, where tj E BinI preserve the medial laws in V. Proof. This can be proved using the previous lemmas. • Now we can prove that the medial law is a hyperidentity: Theorem 5.15 The medial law is a hyperidentity in the variety V of all normal I D-semirings. Proof. Because of Lemma 5.11 and of Lemma 5.13 it is enough to prove that all binary terms over V represented by t = tl + t2 +t3 with tl E {x,xy,xyx} and t2, t3 E BinI preserve the medial law. We consider two cases: 1st case: t can be represented by t' + til, t' and til E BinI. In this case we have the result by Lemma 5.15. 2 nd case: t cannot be represented by t' + til, t' and til E BinI. In this case using lemmas 5.12 , 4.2 and 5.13 we can see that we have to consider only t = x + xy + yx. • As a consequence, we have our main result: Theorem 5.16 The variety of all normal ID-semirings is solid and is the greatest solid variety of normal semirings. Every nontrivial solid variety of normal semirings is in the interval between RA(2,2) and the variety of all
86
normal I D-semirings. Proof. We have to prove that every identity of an identity basis of the variety of all normal I D-semirings is satisfied as a hyperidentity. We remarked already that the idempotent laws are satisfied as hyperidentities. By Theorem 5.6 and Theorem 5.16 the associative and the medial laws are satisfied as hyperidentities in the variety of all normal I D-semirings. By Theorem 5.10 the distributive law: G(x,F(y,z)) ~ F(G(x,y), G(x, z)) is a hyperidentity in the variety of all normal I D-semirings. That means, for all hypersubstitutions as,t : F 1-+ s,G 1-+ t the equation as,t[G(x,F(y,z))] ~ as,t[F(G(x,y),G(x,z))] is an identity. Consider the hypersubstitution aF(x,y) ,G(y,x). Clearly aF(x,y) ,G(y,x)[G(x,F(y,z))] = G(F(y,z),x) and aF(x,y) ,G(y,xJfF(G(x,y),G(x,z))] = F(G(y,x),G(z,x)) and then (as,t 0h aF(x,y) ,G(y ,x») " [G(x,F(y,z)] ~ (as,t 0h aF(x,y) ,G(y ,x») " [F(G(x,y),G(x,z)] is also an identity for arbitrary hypersubstitutions as,t. The other distributive laws are clearly also satisfied as hyperidentities. • References 1. K. Denecke, S. L. Wismath, Hyperidentities and Clones, to appear in
Gordon and Breach Publishers. 2. E. Graczyllska, D. Schweigert, Hypervarieties of a given type, Algebra
Universalis 27, 305-318 (1990) . 3. W. Kuich, Pushdown tree automata, algebraic tree systems, and algebraic tree series, preprint. 4. J. Plonka, Proper and inner hypersubstitutions of varieties, in Proceedings of the International Conference: Summer School on General Algebra and Ordered Sets, Palacky University Olomouc 1994, pp. 106-115. 5. F. Pastijn, A. Romanoswka, Idempotent distributive semirings I., Acta Sci. Math. 44, 239-253 (1982). 6. R. P6schel, M. Reichel, Projection Algebras and Rectangular Algebras and Applications, Research and Exposition in Mathematics 20, 180-195 (1993), Heldermann-Verlag Berlin. 7. W. Taylor, Hyperidentities and hypervarieties Aequationes Mathematicae 23, 111-127 (1981).
DECIDING SOME EMBEDDABILITY PROBLEMS FOR SEMIGROUPS OF MAPPINGS DAVID H. FREMLIN AND PETER M. HIGGINS Department of Mathematics, University of Essex, U.K.
We give a simple way of showing that the problem of embeddability into finite inverse semigroups and finite semigroups of order-preserving mappings is decidable
1
Introduction
This note, based on a part on the talk of the second author at the Braga Conference, gives a method allowing solution of decidability problems including the ones mentioned in the abstract. These are not open problems but we believe that the simple technique used here is worthy of attention. The problem of embeddability into an inverse semigroup was solved by Schein nearly forty years ago [1]. Schein showed that a necessary and sufficient condition for embeddability into an inverse semigroup was that the so called strong quasi-order relation on the semigroup be an order relation. This can be reformulated as an infinite system of quasi-identites (equational implications) which, although containing redundancies, cannot be replaced by any finite system. It does also furnish an effective procedure for deciding the embeddability question for any finite semigroup. The corresponding question for semigroups of isotone mappings has more recently received attention as semigroups related to semigroups of orderpreserving mappings have been intensively studied. Here the related papers [2] of Vernitskil and [3] of Volkov in the Proceedings of St Andrews Conference on Semigroups in 1997 are relevant. In particular Lemma 5.1 of [3] shows that the question of whether a finite semigroup can be faithfully represented as a semigroup of partial order-preserving mappings on a chain is decidable. Volkov presented further results along these lines at the GAP Conference in Lisbon in 1997 which included the case of embeddability into the semigroups On of total order-preserving mappings on a finite chain of length n and there is continuing joint work of RepnitskiY Volkov and VernitskiY on bases for quasivarieties generated by isotone mappings.
87
88
2
Main Theorem
We denote by PT (X) the semigroup of all partial mappings on X under composition. A subsemigroup T of PT (X) is closed under restrictions if whenever a E T, Y ~ X and y. a ~ Y then the restricted mapping alY is also in T. The semigroup of all partial one-to-one mappings on X, known as the symmetric inverse semigroup, is written I (X). A mapping a (partial or total) on a chain X is order-preserving or isotone if whenever x ~ y (x, y E X) then, x . a ~ y . a if both sides are defined. Without further comment S and T will denote semigroups and we write S ~ T to indicate that S is a subsemigroup ofT. We write IAI to denote the cardinality of a set A. Let A ~ PT (X) and Y ~ X . We say that Y separates A if, for each pair of distinct mappings G, (3 E A there exist x E Y such that XG i- x(3. (Note that this allows for the possibility that one of XG, x(3 is defined while the other is not.) In these circumstances we say that x separates G and (3. Separation Lemma Let A ~ PT (X) with IAI = n. Then there exists Y ~ X with no more than n - 1 elements that separates A. Proof We proceed by induction, there being nothing to prove in the base case where n = 1. Hence we take n ~ 2 and a subset B of A containing n - 1 mappings. By induction there is a subset W of X with IWI ~ n - 2 that separates B. Let G denote the unique mapping in A\B. Either W separates A, in which case we are finished, or G is not separated from some (3 E B by any member of W, that is to say GIW = (3IW In the latter case consider any I E B\(3. Since W separates (3 there exists x E W such that x, i- x(3 = XG, and so it follows that W separates G from every member of B\(3. Since G i- (3 there exists a E X such that aG i- a(3. We thus obtain a required separating set Y by taking Y = W U {a} . • Remark We note that the bound of n - 1 is in general best possible. To see this let X = {O, 1" . " n - I} and consider the n mappings Gj,j = 0, 1, " ', n-1 given by the following rule:
. ZQj
=
{Oi
ifi<j othe~ise.
The n mappings Gj are then pairwise distinct but are not separated by any subset of A of X of size n - 2, as for such a set A there exists i rt- A with 1 ~ i ~ n - 1 but then Gi-liA = GilA. Indeed this set of n mappings forms a semigroup under composition isomorphic to the chain X under the operation of taking the maximum.
89
Embedding Theorem Let S S; T , where T S; PT (X) is closed under restrictions and S is a semigroup of finite cardinal n. Then S embeds in the subsemigroup TnPT(Y) for some subset Y of X with IYI S; (~+ 1)2. Proof It is sufficient to prove that there is a separating set Y ~ X of size no more than (~ + 1) 2 for S such that Y . S ~ Y for then the mapping which sends each mamber a E S to its restriction alY is an embedding into TnP (Y) as alY E T because T is closed under restrictions: the mapping is faithful as Y separates S and is a homomorphism as Y . S ~ Y. To this end put r = maxxEX Ix . SI and fix Xo E X such that Ix. SI = r. List the members of Xo . S as {Xl, X2, . .. , x r } . Put Sj = {s E S : Xo· s = Xj} for j S; r and write nj for ISjl , noting that nl + . .. + nr = n . For each j S; r we have, by the Separation Lemma, some Yj ~ X such that IYjI S; nj - 1 and slYj =I=- tlYj for distinct s, t E Sj. Let us take Y = {xo, xl," .. , x r } U
U(Yj U Yj . S) . j$.r
Then r
IYI
S; (r+1)+
L(r+1)IYjI j=l
:0; (r
=
(r
+ 1)
(1+
t,(n;
+ 1)(1 + n -
r) S;
-1)) f.
(~ + 1
To see that Y separates S take any s, t E S with s E Si , t E Sj say. If i =I=- j then Xo . s = Xi =I=- x j = Xo . t so that Xo separates sand t. If on the other hand i = j then s, t E Sj and there exists some x E Yj such that x . s =I=- x . t, as required. Finally, the fact that y. S ~ Y follows from the general observation that for any subset A of X of the form Z U Z . S we have A . SeA rut A . S = Z . S U Z . S2 ~ Z . S U Z . S = Z . S ~ A.
The required result follows as Y = Z U z· S where Z = {xo} U Uj=l Yj.
•
Corollary The question of whether or not a finite semigroup S can be embedded in a finite inverse semigroup, or a finite semigroup of partial orderpreserving mappings, or a semigroup of total order-preserving mappings, is decidable.
90
Proof In the first case recall that any (finite) inverse semigroup can be embedded in some (finite) I (X) so we need only decide whether or not Scan be faithfully represented by injective one-to-one partial mappings on some finite set X . We test whether or not S is isomorphic to a subsemigroup of I (X), where IXI is the integer part of (~ + 1)2 and n = lSI. If the answer is YES then S has been embedded in a finite inverse semigroup. If not, then by the Embedding Theorem S cannot be embedded in any symmetric inverse semigroup I (X) (whether finite or not) and so the question is thus decided in the negative. (We take T = I (X) in applying the theorem.) The argument for the case of partial order-preserving mappings is the same, being on this occasion based on the semigroup PO (X) of all partial order-preserving mappings on a chain X. Finally suppose that S is a finite semigroup that can be embedded in o (X), the semigroup of all order-preserving mappings on a chain X. The argument of Theorem 3.2 then shows that S can be embedded in some semigroup of total mappings on a subchain Y of X of order (~ + 1) 2 and these mapping inherit the order-preserving property with respect to the chain Y. n the same fashion as before we conclude that the question of embedding into some finite semigroup of order-preserving mappings is decidable. 3
Optimum Bounds
The final line of calculation of the cardinality of the set Y gives as bound the integer part of (~ + 1)2 = (n + 2)2. Indeed this is the maximum value of the expression (r + 1) (n + 1 - r) : in the case where n is even the maximum is (~ + 1)2 while for odd n the integer part has the form (n + 1) (n + 3). We construct examples of left zero semigroups of order n suitable embedded in certain partial transformations semigroups PT (X) for which this bound is best possible. The general idea of the construction is as follows. Let IT be a collection of partitions of the set X of order n . For each 7r, 7r' etc. in IT let p, p' etc. denote the corresponding equivalence relation on X. Suppose further that IT has the following properties. For each 7r E IT :
i
i
3x,yEX:(x,y)~pbut
(x,Y)Ep' 'v'7r'EIT\{7r}
(1)
That is to say each partition 7r separates a pair of elements of X that is separated by no other partition in IT. Our second property is that each pair (x, y) of distinct elements of X is separated by some (not necessarily unique) 7rEIT:
37r E IT: (x,y)
~p.
(2)
91
Let m be the number of partitions in the collection II and let 0 be a symbol not in X. For our base set we take
z=
U {i}X(7fiU{O}).
(3)
l:5i:5m
Consider the left zero semigroup S on X, identifying it with the image of the following action on Z : (i,O) . s = (i, Q), where sEQ, a class of 7fi; (i, P) . s = (i, P) , otherwise. This does define a (right) action of S on Z as for any pair s, t E S the effect of s followed by t is the same as that of s = st and moreover the action is faithful as by Property (2) there exists 7fi E IIsuch that (s, t) rJ. Pi so that (i,O) . s = (i, Q) , (i, 0) . t = (i, P) say, and the construction ensures that P =J Q. Moreover we show that Z contains no proper separating set Y for S. Suppose that (i, 0) rJ. Y ~ Z. By Property (1) there exists s, t E S such that (s, t) rJ. Pi but (s, t) E Pi for all7fi E II with j =J i. Then (j,O)· s = (j,O)· tfor all j =J i and all other points of Yare fixed by both sand t. It follows that such a proper subset Y of Z cannot distinguish between s and t. The order of the base set Z is evidently given by m
IZI =m+ LI 7fil.
(4)
i=l
We next construct particular examples of collections II which gives rise to sets Z for which the upper bounds for closed separating sets Y given in the first paragraph of this section are attained. First let us suppose that n 2:: 6 is even so that n = 2k with k 2:: 3. Let p be the partition of X with two classes that are triples and the rest pairs:
(5) Our collection II consists of k + 1 partitions which are refinements of p. We let 7fi, 3 S; i S; k - 1 be formed from p by replacing the single class {ai, bi } of p by the two singleton classes {aJ and {bJ . The partitions 7f1 and 7f2 are formed by replacing the classes {al,bl,cd and {a2,b 2,c2} by {ad, {bl,Cd and {a2}, {b 2, C2} respectively. Finally 7fk and 7fk+1 are formed by respectively replacing the two triple p-classes by {aI, bd, {cd and {a2, b2}, {C2}. Consider 7fi with 1 S; is; k -1. Here we have (ai, bi ) rJ. Pi but (ai, bi ) E Pi for all j =J i. For 7fk we have (bl, CI) rJ. Pk yet (b l , CI) E Pi for all j =J k with
92
similar comment applying to 7rk+l, thus showing that the collection II satifies Property (1). Property (2) is satisfied by p (and thus by each 7ri) with respect to all pairs taken from distinct sets in (5), whereas a pair formed by the one set in (5) is separated by some 7ri as described in the previous paragraph; (noting also that (a I , C!) rf. PI, (a2, C2) rf. P2). Take our base set Z as defined by (3). We now apply the formula (4) :
2k-6
m
n
= IIII = 2·2 + - 2 - = k + 1 = "2 + 1, 2k-6
n
l7ril=1+lpl=1+2+-2-=k="2' thus IZI =
m
+ m(l + Ipl) = m(2 + p) =
(% + If,
thereby showing that the bound of (~ + 1) 2 is the best possible in the case where n is even and at least 6. Next let us suppose that n is odd and at least 3 so that n = 2k -1 say with k ~ 2. We take p as before except there is now only one triple class {a 1, b1, CI } and so there is no element named C2. The collection II again satisfies Properties (1) and (2) . Again we take Z as defined by (3) and apply equation (4). The (2k - 4) /2 = k - 2 doublet classes each contribute one partition to II while the triplet {aI, b1, Cl} contributes two to the count. Hence we obtain n m = IIII = (k - 2) + 2 = k = "2 + 1,
2k l7ril=l+lpl=l+ ( 1 + -2
IZI =
m
4)
=k; thus 1
+ m (1 + Ipl) = k + k 2 = k (k + 1) = "4 (n + l)(n + 3).
Therefore the upper bound provided by the theorem is obtained in this instance also. Finally we address the three remaining values of n = 1,2 and 4. (n = 1) In this case the best bound is WI = 0 obtained by taking Y =;. The empty mapping is contained in T as T is closed under restrictions. 2 (n = 2) The standard bound in this case is b = (~+ 1) = 4 but this is only obtainable with a value of r = ~ = 1. However to distinguish between the two members sand t of S requires that r ~ 2 and so we obtain a lower bound of 3 for the cardinal of Y. This is achievable through our standard
93
construction by taking X = {I, 2}, II = {7r}, where 7r = {{I}, {2}} which yields IY I = 1 + 2 = 3. (n = 4) The standard bound in this case is b = (~ + 1) 2 = 9 which may be achived only with a value of r = ~ = 2. This is obtained using our standard construction on X = {I,2,3,4} as follows. We put II = {7rl,7r2,7r3} where 7rl
= {{I, 2}, {3, 4}}, 7r2 = {{I}, {2, 3, 4}}, 7r3 = {{4}, {I, 2, 3}}.
One easily checks that Properties (1) and (2) are satisfied by II and then IZI = 3 + (2 + 2 + 2) = 9, as required. The quadratic bound obtained in the Embedding Theorem for arbitrary semigroups might be bettered if we restrict ourselves to some standard classes of interest. For a group G of order n we may always find a suitable set Y of cardinal not exceeding n as we now show. Let S be a finite semigroup of order n acting faithfully on the right on a set X by partial mappings: x· st = (x. s) . t
(6)
\::Ix E X, s, t E S
for s, t E S 3x E X such that
X·
s
=1=
x . t.
(7)
That is to say either both sides of (6) are undefined or both are defined and equal. We may, and henceforth do, treat 'undefined' as an additional new symbol, 00, which is a fixed point of all actions. The action of s E S on Xu {oo} is then that of a (total) mapping fixing 00. Let us suppose that S = G is a group with neutral element e. Let Xl = X . G : then G acts on Xl (that is to say that the restricted action satisfies (6)) and for any Xl = X· g (x E X, g E G) we have Xl'
e
= (x. g) . e = X·
and so the action of G on X* =
{x
Xl
(ge)
= X· g = Xl,
is agroup action (by permutations). Next write
E Xl : 3g E G such that X· g =1=
x}.
This definition excludes any point of Xl that is fixed by every member of G (including the point 00). It follows that G has a group action on X* which is faithful as for any two distinct gl, g2 E G there is some x E X such that X· gl =1= X· g2 whence Xl = X· e E Xl and
Moreover
Xl E
X* as
Xl
separates two members of G.
94
Henceforth we may take X = X*, as a suitable subset Y of X* is also suitable as subset of the original base set X . Moreover we may assume that X is finite as by the argument of the Separation Lemma there is a finite separating set A ~ X for the action of G on X, and then without loss we may take X to be the finite set A· G. In summary we may assume that X is a finite set on which G acts faithfully by permutations and that every point of X is shifted by at least one of these permutations. Theorem With X and G as above there exists a subset Y of X of cardinal no more than n such that Y . G = Y and the restriction of the action of G to Y is faithful.
Proof We may assume that IXI ~ n. For any orbit Z of G we have IZI ~ n and if we have equality here we can take Y = Z for then Z = x . G for some x E X could act as a required separating set. If no such orbit exists we may write X = ZI U Z2, a distinct union of non-empty sets each of which is a union of orbits of G. For i = 1,2 define Hi = {g E G : x · 9 = x "Ix E Zi} .
Clearly Hi
for this action is well-defined by definition of Hi. By induction there exist disjoint subsets Yi (i = 1,2) such that Yi ~ Zi and
with Yi separating the points of G / Hi. Put Y = Y 1 UY2 and note that WI :::; n. By the definition of the actions of the quot ients we see that y. G = Y. Finally we verify that the action of G on Y is faithful as if y . gl = Y . g2 for all y E Y then 919"2 1 E HI n H2 = {e} whence gl = g2 , as required. _
95
References 1. Schein, B. .' A system of axioms for embedding of a semigroup into a
generalized group' , Doklady Akad. Nauk SSSR 134 (1960) , 1030-1033 (Russian) ; English trans!. Soviet Math . Doklady 1 (1961), 1180-1183. 2. Vernitskil A.S. 'The Semigroup of order-preserving mappings: quest for quasiidentities', in Semigroups and Applications, J.M. Howie and N. RuSkuc (eds) , World Scientific, Singapore (1998) , pp 229-38. 3. Volkov, M.V. 'The finite basis problem for the pseudovariety PO', in Semigroups and Applications, J.M. Howie and N. Ruskuc (eds) , World Scientific, Singapore (1998) , pp 239-57.
ON THE SEMIGROUPS WITH VERY GOOD MAGNIFIERS M . GUTAN Laboratoire de Mathematiques Pures, Universite Blaise Pascal, 63177 Aubiere Cedex, FRANCE E-mail: gutan@ucfma .univ-bpclermont.fr In this paper we characterize some remarkable elements in semigroups containing very good left magnifiers. We generalize a result of N. Jacobson concerning a sufficient condition under which a semigroup does not contain very good magnifiers . A more general construction than that used for semigroups under a sandwich operation enables us to give a new meaning of semigroups with very good magnifiers . Furthermore we present several general methods for obtaining semi groups containing such elements. We also characterize the minimal right ideals associated with very good left magnifiers. Finally we give conditions under which two semigroups with very good left magnifiers are isomorphic.
1
Introduction
An element a of a semigroup 5 is a left (resp. right) magnifier if Aa (resp . Pa), the inner left (resp . right) translation associated to a in 5 , is surjective and is not injective. This notion has been introduced by E . S. Ljapin [15] in a general context and by N. Jacobson [13] in the particular case of underlying multiplicative monoids of unitary rings . The bicyclic monoid B = B(p, q) =< p, q I pq = 1 > possesses both left and right magnifiers and is very important for the study of left (resp . right) magnifiers (see E. S. Ljapin [15] and R. Desq [4]). Thus, a semigroup 5 with left (resp. right) identities has left (resp . right) magnifiers if and only if it contains a subsemigroup B isomorphic to the bicyclic monoid such that the identity of B be a left (resp. right) identity of 5. This means that 5 has left magnifiers if and only if there exist e, u, v in 5 such that Ae = Is, uv = e, vu #- e, ue = u and ve = v . If the semigroup 5 does not contain left identities the study of left magnifiers is more complicated and the characterization of semigroups with left magnifiers is an open problem . Throughout this paper we denote by £9)1(5) (resp. 919)1(5)) the set of left (resp. right) magnifiers of 5 and by 913(5) (resp. £3(5)) the set of right (resp. left) invertible elements of 5 , that is the elements for which Aa (resp. Pa) is surjective. Generally, £9)1(5) C 913(5) and if 5 has no left identity then £9)1(5) = 913(5) ([5]). If a is a left magnifier in a semigroup 5 one can find a proper subset M of 5 such that, for every 8 belonging to 5 , the set M n A;;1(8) is a singleton. 96
97
In this case M is a minimal subset for the left magnifier a . If such a set is a subsemigroup (resp. right ideal) it will be called a minimal subsemigroup (resp.right ideal) for the left magnifier a and a will be called good (resp. very good) left magnifier of 5. These notions have been introduced by F . Migliorini ([17], [18]). Some properties of semigroups which contain good and very good left magnifiers have been established in [14] and [19]. A characterization of semigroups with good (resp. very good) left magnifiers has been obtained in [7] ; it has been proved that these semigroups are extensions of semigroups M with left identities and left magnifiers by endomorphisms (resp . right translations) satisfying some conditions. Notice that if the semigroup 5 contains left identities then all the left magnifiers of 5 are very good . 2
Characterizations and fundamental properties of semigroups with very good left magnifiers
The semigroups with very good left magnifiers can be characterized as being extensions of semigroups M having left identities and left magnifiers by right translations of M. This construction can also be understood as follows. Let p be a right translation of a semigroup (M,). We use p to manufacture another semigroup structure on M, denoted (M, p) , obtained for an operation "0" defined by : mom' = p(m)m', for every m, m' in M. Related to this method, we establish in this section that every semigroup with very good left magnifiers can be obtained in this manner. Throughout this section we suppose that (M,) is a semigroup containing three elements e, u, v such that:
uv = e, vu
i: e, ue =
u, ve = v and em = m, for every mE M
(1)
and p : M -+ M is a map satisfying, for every m, m' in M , the conditions:
p(mm') = mp(m'), p(m)e = p(m), p(m)v = me, p(e) = u.
(2)
Consider A a disjoint set from M, having the same cardinal as M \ vM, and r : M\ vM -+ A a bijective map . Let S = AUM and ¢ : M -+ 5, where: um ifm E vM ¢(m) = { r(m) otherwise. Denote ¢(e) = a and define on 5 an operation" "by:
¢(m) . ¢(m')
= ¢(p(m)m') ,
for every m and m'
III
M.
(3)
Then (5, .) is a semigroup denoted 5 = .5(M, u, v, p) for which a is a very good left magnifier (M is a minimal right ideal associated with a in 5, M = e· 5
98 and 'a 1M) . It is the extension of M (containing the elements e, u, v which satisfy (1)) by the right translation p of M (such that (2) hold). Using
mom'
= p(m)m',
for every m and m' in M.
(4)
Denote by (M, p) the semigroup obtained in this way. We also use the notations ((M,); u, v, p) in order to emphasize the elements u, v and the initial operation " "on M which occurs in conditions (1) and (2). For this semigroup e is a very good left magnifier and vM is a minimal right ideal associated with e. Also, (M,) -+ (vM, 0) is a semigroup isomorphism. We have established in [7) that every semigroup containing very good left magnifiers is of type S(M , u, v, p), whence of type ((M , ); u , v, p) . To see this is so , it suffices to notice that if a is a very good left magnifier of a semigroup Sand M is a minimal right ideal associated with a then :
>.: :
i) There exist u, v, e in M, uniquely determined by : ae and these elements satisfy conditions (1) .
ii) The map p : M -+ M, where p(m)
= a, av = e, au = a2
= ma, for every m E M, fulfils (2).
iii) If we choose A = S \ M and T : M \ vM -+ A the map defined by T(m) = am , for every mE M \ vM, then 'a 1M. iv) The semigroups (S, ) and (S(M, u, v, p) ,, ) coincide . Let Ide(S) (resp. Idr(S)) be the set of left (resp . right) identities of the semigroup S. For e E Ide(S) and s E S we denote by ry{(S; s, e) the set {s' E S Is s' = e} of right inverses of s with respect to e. Then in S (M, u, v, p) the following characterizations of left (right) identities, left (right) invertible elements, left (right) magnifiers and idempotents hold . 2.1. Proposition. ([7), [14)) .
= a{m EM I p(m) = e} Idr(S(M,u ,v, p)) = a{m EM I p(m')m =
i) Ide(S(M,u ,v, p)) ii)
m' , for every m' E M} C
a[£J(M)nry{(M, u , e) )
= a{m E M I p(m) E ry{J(M)} C ary{J(M) £J(S(M, u, v, p)) = a{m E M I p(M)m = M}
iii) ry{J(S(M, u,v , p)) iv)
v) £9J1(S(M, u, v, p))
= a{m E M I p(m)
E £9J1(M)}
99
= a{m E M I PmOP is surjective and is not injective} Idemp(.5(M , u , v , p)) = a{m E M I p(m)m = m} . •
vi) 919J1(.5(M, u, v , p)) vii)
The result below is frequently used in §3 . 2.2. Proposition. i) p(M) is a left ideal of M;
ii) Mu C p(M) C Me;
iii) mM C p(m)M, for every mE M.
•
In the next section we establish that if p(M)
=Mu
or p(M)
= Me
then
p is uniquely determined and we give a general method to obtain semigroups
for which the foregoing inclusions 2.2. ii) and 2.2 . iii) are proper . We now characterize the left (right) ideals and the subsemigroups of the semigroups 5 = .5(M, u, v , p). 2 .3. Proposition. Let M' C M, M' =1= 0. Then : i) aM' is a subsemigroup of 5 if and only if p(M')M' C M' ii) aM' is a right ideal in 5 if and only if M' is a right ideal of M and p(M') eM' . iii) aM' is a left ideal of 5 if and only if p(M)M' C M'
•
Using Theorem 9.5 ([14), page 56) and Proposition 2.1.iii) we obtain : 2.4. Proposition.
i) Mo
=
M\91J(M) is a proper right ideal of M which contains all the proper right ideals of M .
ii) 50 = ap-l(Mo) is a proper right ideal of 5 which contains all the proper right ideals of 5. • This result is partially contained at (3) and (7) of Theorem 10. 9 in [14] but the proof given there has a gap. From 2.2. iii) it follows that p-l(Mo) C Mo and generally, Mo C M C 50 c aMo C 5,Mo =1= M, aMo =1= 5. In Section 3, examples illustrating situations when the equalities 50 = M and 50 = aMo occur are presented. In the next Proposition we present a sufficient condition under which a semigroup does not contain very good left magnifiers . Thus we extend a result that N. Jacobson ([13), Theorem 1) has established for the underlying multiplicative semigroups of unitary rings.
100
Notice that if we put Mk = Vk M, for every k E IN-, then : p(Mk) C Mk, Mk:J M k+1 and v k E Mk \ Mk+l' Hence (aMk)kEIN. is a strictly descending chain of right ideals of S. Moreover, avk M = v k - 1M = Vk -1uk-1S. It follows that we have : 2.5. Proposition. Let S be a semigroup satisfying th e descending chain condition for principal right (left) ideals generated by idempotents. Then S does not contain very good left (right) magnifiers. • 3
Examples of semigroups with very good magnifiers
We begin this section by constructing M and p for which conditions (1), (2) and p(M) Me or p(M) Mu hold. Finally we present a general method for obtaining M and p such that the inclusions M u C p( M) C Me be proper. 3.1. We have already noticed in §1 that if a semigroup S contains left identities then all its left magnifiers are very good. Let e: E Idl(S) and a E'c9J1(S) . Hence there exists b E Se: such that ab = e: and ba f:. e:. Remark that M = bS is a minimal right ideal associated with a and its elements e = ba, u = ba 2 and v = b2 a satisfy conditions (1). More than that , the map p : M -+ M defined by p(bs) = bsa , for every s E S, satisfies conditions (2). In this context, if A = S \ M and r: M\vM -+ A, r(bs) = s, for every s E S , then ¢ = AalM (where ¢ is defined as in §2) and the semigroup (.5(M, u, v , p) ,, ) coincides with the semigroup (S, ). Conversely, every semigroup .5 (M , u , v, p) containing left identities (that is, according to 2.1.i), one has e E p(M)) can be obtained by this previous construction (see 3.3 and 3.4 of [7]). From Propositions 2.1. i) and 2.2 yield :
=
3.1.1. Proposition.
=
The following statem ents are equivalent:
i) .5(M, u ,v, p) has left identities..
ii) e E p(M) ;
iii) p(M) = M e.
=
•
3.1.2. Remarks. According to Proposition 2.4, the semigroup M bS possesses a greatest proper right ideal Mo {bs I sbS f:. S} and S possesses S\~J(S). As a ae E aMo and a >t. So a greatest proper right ideal So (notice that in this example, aMo is not a right ideal of S) it follows that the inclusion So C aMo can be proper. Furthermore, the inclusions in Proposition 2.2. iii) can also be proper because if we choose m be we get that mM b2 S, p(m)M bS, whence b E p(m)M \ mM . We also have that the sets ~J(M) and {m EM I p(m)M M} do not coincide.
=
=
=
=
=
=
=
3.2. It is easily seen that Pu, the inner right translation of M associated with u, fulfils conditions (2) . For the semigroup S .5 (M, u, v, Pu) we have :
=
101
3.2.1. Proposition. i) Idl.(S)
= 0.
ii) Idr(S)
= { a 91(M, u, e)
0
iii) ~9J1(S) =91J(S)
if if
Me Me
f.
M
= M.
= a 91J(M).
iv) Idemp(S) = a{m E M
I mum =
m}.
v) (am)n S (ami) if and only if mn M mi .
vi) If Me
=
M then Pu is the unique right translation of M for which conditions (2) hold. •
3.2.2. Proposition.
The following statements are equivalent :
=
=
i) P Pu; ii) p(M) Mu. Proof: It is obvious that i) implies ii) . Suppose that ii) holds, that is p(M) = Mu, and consider m E M. Then there exists m ' E M such that p(m) = m'u. It follows that me = p(m)v = m'e, whence m'u = mu = p(m) . Therefore p = Pu, as required. • 3.2.3. According to 2.3, the lattice of right ideals of .5(M, u, v, Pu) is isomorphic to that of right ideals of M. Hence, as Mo = {m E M I mM f. M} is the greatest proper right ideal of M, we deduce that So = aMo is the greatest proper right ideal of S. 3.2.4. The semigroups which appear in 3.1 and 3.2 are particular cases of a more general construction used by J. B. Hickey ([8] and [9]), K. Chase ([3]), T. S. Blyth and J. B. Hickey ([1]) , K . D . Magill Jr . ([16]), W. C. Huang ([12]) and others. We briefly recall it here: for any element a of a semigroup (S,) one may define a sandwich operation on S by x 0 y = xay, where x, y belong to S. Under this operation the set S is again a semigroup called a variant of S and denoted (S, a). We remark that (S, a) = (S, Pa). In this context, the semigroups considered at 3.2 are isomorphic to (M, u) and those at 3.1 are isomorphic to (S, c:). 3.3. Using the constructions given at 3.1 and 3.2 we now give a method to obtain semigroups for which the inclusions mentioned in Proposition 2.2 ii) are proper. Assume that Mj, ej , Uj, Vj, Pi satisfy (1) and (2), for every i E I. Then M,e,u,v,p, where M = e = (ej)iEl,U = (Ui)iEl,V =
IIMj,
(Vi )iEI, P =
II Pi, iEI
iEI
also satisfy these conditions.
Moreover the semigroups
102
(M,p) and II(Mi,Pi) are isomorphic . Thus, if there exist i and j in [such iEI
that p;(M;) = Miu; and pj(Mj) = Mjej then Mu =I p(M) =I Me. Some other examples of quintuples (M, , u, v, p) which satisfy conditions (1) and (2) such that the left ideal p(M) be different from Mu and Me have been given in [7]. 4
Characterizations of minimal right ideals associated with very good left magnifiers
Throughou t this section we assume that S = .5 (M, u, v, p), where M is a semigroup containing three elements e, u, v such that conditions (1) hold and p: M -+ M is a map which fulfils (2). The purpose of this section is to determine the minimal right ideals associated with left magnifiers of S. We also characterize the semigroups containing very good left magnifiers such that every left magnifier admits a unique associated minimal right ideal. According to Proposition 2.1, £9J1(S) = a{m E M I p(m) E£9J1(M)}. Let b = am, where m E M and p(m) E£9J1(M). Consider m' E Me such that p(m)m' = e and m' p(m) =I e. Then am'M is a minimal right ideal associated with b. Therefore, all the left magnifiers of S are very good ([7] and [20]). We have the following: 4.1. Theorem. Let mE M such that p(m) E£9J1(M) . The minimal right ideals of S associated with the left magnifier am are : {am'M I m' E 91(Me ;p(m),e)}. Proof: Let R = aM' be a minimal right ideal associated with am in S, where M' is properly included in M. Then, according to Proposition 2.3, M' fulfils the following four conditions: i) M'M eM'; iv) p(m)m~
ii) p(M') C M';
= p(m)m~
implies m~
iii) p(m)M'
= m~, for every
= M; m~, m~ in
M'
By iii) there exists m' E M' such that p(m)m' = e. More than that, according to i) and iv), m' = m'e, whence m' E 91(Me ;p(m),e). From i), m'M C M' On the other hand, as p(m)[m'M] = M, by iv), it follows that M' = m'M. Thus, R= am'M, with m' E 91(Me ;p(m),e) . • Notice that if m', m" belong to 91( Me; p( m), e) and am'M = am" M then m' = m". 4.2. Corollary. The minimal right ideals associated with a are : {av'M I v' E 91(Me ; u, e)} • 4.3. Corollary. For the semigroup S = .5(M, u, v , p) the following two statements are equivalent :
103
i) Every left magnifier of S is associated with only one minimal right ideal. ii) ryt(Me ;p(m),e) is a singleton, for every mE p-l(£;9)1(M)).
•
We close this section with some 4.4. Special cases 4.4.1. M = B(p, q), u = p, v = q. Then £;9)1 (M) = {pn I n E IN"} and M is a monoid. Hence there exists a unique right translation of M which fulfils (2), namely p = Pp (see [7]). Moreover, if m = p", with n E IN", then p(m) = pn-l and ryt(B;pn-l, 1) = {qn-l} . Thus, for the semigroup 2:: (see [7]) we have £;9)1(2::) = {a, a2 , ... } and every left magnifier of 2:: is associated with a unique minimal right ideal. 4.4.2. p = pu . In this situation, Mp(m) C Mu hence e tt Mp(M). It follows that p(m) E£;9)1(M) if and only if m ErytJ(M). Therefore, £;9)1(S) =rytJ(S) = arytJ(M) . If m ErytJ(M) then the minimal right ideals associated with am in the semigroup S = .5(M ;u,v,pu) are {am'M I m ' E ryt(Me ;mu,e)}. Notice that we also have ryt(Me ;mu,e):J vryt(Me ;m,e) . 4.5. Proposition. For the semigroup S = .5(M, u, v , Pu) the following two statements are equivalent.
i) Every left magnifier of S is associated with only one minimal right ideal. ii) ryt(Me ;m,e) is a singleton, for every mE rytJ(M) .
Proof: Suppose that i) holds and let m E rytJ(M). Remark that if Xc M then X is a singleton if and only if vX is a singleton. Hence, using 4.4.2, we obtain ii) . The fact that the converse is true results immediatly from 4.2. • 5
Conditions under which two semigroups containing very good magnifiers are isomorphic
Let (M,), (M/, ) be two semigroups and p : M -+ M, p' : M' -+ M' be two maps. The couples ((M, ), p) and ((M/ , ), p') are said to be equivalent if there exists r.p : (M,) -+ (M/,), a semigroup isomorphism, such that r.p 0 p = p' 0 r.p. We denote that by ((M, ), p) == ((M/, ), p') . Notice that if 0" is an automorphism of M then (M , p) == (M,O"opoO"-l). 5.1. Lemma. Suppose that (( M, ), p) == (( M' , ), p'), where M contains three elements e, u, v satisfying (1), and p : M -+ M is a map for which (2) hold. Then M/, e' , u ' , v', p' fulfil (1) and (2), with e' = p(e), u ' = r.p(u), v' = r.p(v) . •
104
Notice that under the hypothesis of this previous lemma the semigroups 5 = S(M, u , v, p) and 5' = S(M', u', v', p') are isomorphic. For instance, such an isomorphism is given by cp : 5 -+ 5', cp( am) a'
=
i)
= CPIM
: M -+ M' is
a semigroup isomorphism;
ii) M',e',u',v',p' obey conditions (1) and (2) ; iii) ((M,),p) == ((M',),p');
iv) 5':-= S(M',u' ,v',p').
•
Assume that M, e, u, v, p satisfy (1) and (2) and consider 5 =S (M, u, v, p). Let m E M and m' E Me such that p(m)m' = e. Then R = am'M is a right minimal ideal associated with the left magnifier b = am of 5 . We put e = am'm,'iI = am'p(m)m,v = am'm'm . Then be = b,b'iI = b2 and bv = e. Moreover, if p' : R -+ R, p'(r) = rb, for every r E R, then 5 =S(R, u' , v', p') . Notice that if TJ : M -+ R is defined by TJ( mI) = am' ml, for every mI EM , then TJ is an isomorphism between the semigroups (M, Pm' up) and (R, .). Furthermore ((M,Pm' oP),Pm op) == ((R,·),p') . Denote : J((M,), p, u, v) = {((M, Pm' 0 p), Pm 0 p) 1m E M , m' E Me, p(m)m' = e}. The following result holds : 5.3. Theorem. 5uppose that M, u, v, p and M', u', v', p' satisfy conditions (1) and (2) . Then the semigroups S(M, u, v, p) and S(M', u', v', p') are isomorphic if and only if the couple (( M' , ), p') is equivalent to a couple of J((M,),p,u,v) . • Some other applications of Theorem 4.1 like the construction of automorphism group of semigroups of type (M , p) will be presented in a next paper. References
1. T . S. Blyth and J. B. Hickey, RP-dominated regular semigroups, Proc. R. 50c. Edinb., 99 (1984),185-191. 2. F . Catino and F . Migliorini, Magnifying elements in semigroups, 5emigroup Forum, 44 (1992) , 314-319.
105
3. K. Chase, Maximal groups in sanwich semigroups of binary relations, Pacific J. Math., 100 (1982) , 43-59. 4. R. Desq, Sur les demi-groupes ayant des elements unites d'un cote, C.R . A cad. Sci. Paris, 256 (1963),567-569 . 5. M. Gutan, Semigroups with strong and nonstrong magnifying elements , Semigroup Forum , 53 (1996),384-386 . 6. M. Gutan, Semigroups which contain magnifying elements are factorizable , Comm. in Algebra, 25 (1997), 3953 - 3963 . 7. M. Gutan, Semigroups with magnifiers admitting minimal subsemigroups , Comm. in Algebra, 27 (1999) , 1975-1996 . 8. J. B. Hickey, Semigroups under a sandwich operation, Proc. Edinb. Math . Soc. , 26 (1983), 371-382 . 9. J . B. Hickey, On variants of a semigroup, Bull. Austr. Math . Soc . , 34 (1986) , 447-459. 10 . K . H. Hofmann, M. W. Mislove, All compact Hausdorff lambda models are degenerate, Fundamenta Informaticae, 22 (1995) , 23-52 . 11. J. M. Howie, Fundamentals of Semigroup Theory, (Oxford Science Publishers, 1995) . 12. W . C . Huang , On the sandwich semigroups of circulant Boolean matrices , Linear Algebra Appl. , 179 (1993) , 135-160 . 13 . N . Jacobson , Some remarks on one-sided inverses, Proc. Amer. Math. Soc., 1(1950) , 352-355 . 14. H . Ji.irgensen, F. Migliorini and J. Szep, Semigroups, (Akademiai Kiad6 , Budapest, 1991). 15 . E. S. Ljapin , Semigroups, (Amer . Math . Soc., Providence, R. S. 1963) . 16. K. D. Magill Jr ., P . R . Misra and U. B. Tewari, Finite automorphism groups of laminated near-rings, Proc. Edinb. Math. Soc., 26 (1983) , 297-306. 17 . F . Migliorini, Some researches on semigroups with magnifying elements, Periodica Math . Hung., 1 (1971),279-286. 18. F . Migliorini, Magnifying elements and minimal subsemigroups in semigroups , Periodica Math. Hung., 5 (1974) , 279-288 . 19 . F . Migliorini, Studio sui semigruppi con elementi accrescitivi, Rend. 1st. Mat. Univ. Trieste, 6 (1974) , 11-36. 20. A. Patelli, Una nuova famiglia P(A) di semigruppi con elementi accrescitivi , Rend. 1st. Lombardo Sc. Lett., A 122 (1988),335-367 .
LOCALLY UNIFORMLY 7r-REGULAR SEMIGROUPS MELANIJA MITROVIC University of NiS, Faculty of Mechanical Engineering, Beogradska 14, 18000 NiS, Yugoslavia E-mail: [email protected]
STOJAN BOGDANOVIC University of NiS, Faculty of Economics, Try VJ 11, P. O. Box 121, 18000 NiS, Yugoslavia E-mail: [email protected]
MIROSLAV CIRIC University of NiS, Faculty of Philosophy, Cirila i Metodija 2, P. O. Box 91, 18000 NiS, Yugoslavia E-mail: [email protected]@archimed.filfak.ni.ac.yu
Uniformly 7r-regular semigroups, defined as 7r-regular semigroups whose any regular element is completely regular, form an important kind of semigroups that have been first investigated in 1977 by Shevrin, who announced that they are characterized as completely 7r-regular semigroups whose any regular V-class is a subsemigroup, and also as semilattices of completely Archimedean semigroups. The proofs of these results were first published in [10], together with quite a number of other results concerning the same semigroups as well as semigroups of a more general type, whereas some similar results were obtained independently by Veronesi in [11]. Various other characterizations of these semigroups have been given in a series of papers by Bogdanovic and Ciric (see Theorem 1, the book [1], and the survey papers [2] and [5]). The main purpose of this paper is to characterize a more general kind of semigroups - 7r-regular semigroups whose any local submonoid is uniformly 7r-regular.
For any idempotent e of a seroigroup S, the subseroigroup eSe is a maximal submonoid of S, and it is known under the name local submonoid of S. If K is some class or some property of seroigroups, then S is said to be a locally K-semigroup if any local submonoid of S belongs to K or has the property K. Locally inverse seroigroups, called also pseudo-inverse seroigroups, are the most known seroigroups of this type. They were investigated in a series of papers by Schein, Nambooripad, Pastijn, Blyth and Gomes, McAlister, Petrich and others. These seroigroups include as special cases many important classes of seroigroups, such as completely simple and completely O-simple seroigroups, Supported by Grant 04M03B of RFNS through Math. Inst . SANU.
106
107
inverse seurigroups and others, which have been studied from the very beginning of the theory of seurigroups, as well as many kinds of seurigroups which have important applications in the theory of formal languages and the theory of automata, such as the seurigroups whose local submonoids are seurilattices. Uniformly rr-regular semigroups (this name was introduced in [5]), which are defined as rr-regular seurigroups whose any regular element is completely regular, form an important kind of seurigroups that have been first investigated in 1977 by Shevrin, who announced that they are characterized as completely rr-regular seurigroups whose any regular V-class is a subseurigroup, and also as seurilattices of completely Archimedean seurigroups. The proofs of these results were first published in [10], together with quite a number of other results concerning the same seurigroups as well as seurigroups of a more general type, whereas some similar results were obtained independently by Veronesi in [11]. Various other characterizations of these seurigroups have been given in a series of papers by Bogdanovic and Ciric (see Theorem 1, the book [1], and the survey papers [2] and [5]). The main purpose of this paper is to characterize a more general kind of seurigroups - rr-regular seurigroups whose any local submonoid is uniformly rr-regular, which are called locally uniformly rr-regular. Throughout the paper, N will denote the set of all positive integers. Let S be a seurigroup. By E(S), Reg(S), Gr(S), LReg(S) and RReg(S) we denote the sets of all idempotents, regular, completely regular , left regular and right regular elements of S, respectively. For a E Reg(S), by V(a) we denote the set of all inverses of a, that is V(a) = {x E S Ia = axa, x = xax}. For e E E(S), G e denotes the maximal subgroup of 5 with e as its identity, and for a E G e , a-I denotes the group inverse of a in G e . For X ~ 5, (X) denotes the subseurigroup of 5 generated by X. The natural partial order:::; on E(5) is defined as follows: for e, f E E(5), e :::; f {:} ef = fe = e. We also define the sets Q(5) and M(5) by
Q(5) =
U
e5f
and
M(5) =
",jEE(S)
U
e5e.
eEE(S)
Let us note that e5f = e5 n 5f, for all e, f E E(5). If T is a subseurigroup of 5 then
Reg(T) = {a E T I (3x E T) a = axa}, reg(T) = {a E T I (3x E 5) a = axa}. Evidently, Reg(T) ~ reg(T). The division relation I on 5 is defined by: alb if and only if b = xay, for some x,y E 51. 1fT is a subseurigroup of Sand
108
a, bET, then we say that a divides b in T, in notation alb in T or alTb, if b = xay, for some x,y E Tl. A semigroup S is called Archimedean if for any pair a, bE S there exists n EN such that albn , and it is completely Archimedean if it is Archimedean and has a primitive idempotent, or equivalently, if it is an ideal extension of a completely simple semigroup by a nil-semigroup. A semigroup S is said to be 1r-regular (or eventually regular, in some sources) if for any a E S there exists n EN such that a" is regular. We say that S is a completely 1r-regular semigroup (an epigroup, in some sources) if for any a E S there exists n EN such that an is completely regular. Finally, a 1r-regular semigroup whose any regular element is completely regular is called uniformly 1r-regular. For undefined notions we refer to [8] and [9].
First we quote several known results. Lemma 1 Let S be a 1r-regular or a completely 1r-regular semigroup. Then (E(S)) has the same property. Lemma 2 Let K be a subsemigroup of a completely 1r-regular semigroup S. If K itself is completely 1r-regular, then Gr(K) = K n Gr(S). The first of these two lemmas is taken from the Easdown's paper [7], whereas the second one is a result due to Shevrin [10]. Theorem 1 The following conditions on a semigroup S are equivalent: ~ Gr(S)); S is 1r-regular and Reg(S) ~ LReg(S); S is 1r-regular Reg(S) ~ RReg(S); S is completely 1r-regular and any regular 'V-class of S is a subsemigroup; S is completely 1r-regular and for all e, f E E(S), fie in S implies fie in (E(S» (vi) S is a semilattice of completely Archimedean semigroups.
(i) S is uniformly 1r-regular (i.e. S is 1r-regular and Reg(S)
(ii) (iii) (iv) (v)
As we have mentioned earlier, the conditions (iv) and (vi) were proved to be equivalent to (i) by Shevrin [10], and independently by Veronesi [11], for the conditions (ii) and (iii) this was done by Bogdanovic and Ciric [3], and for the condition (v) by Bogdanovic, Ciric and Mitrovic [4]. Next we give several results that describe some properties of the regular and group parts of quasi-ideals eSf, e, f E E(S), and bi-ideals eSe, e E E(S), of a semigroup S. Lemma 3 Let e,f be arbitrary idempotents of a semigroup S. Then the following conditions hold:
109
(1) Reg(eSf) = Reg(eS) n Reg(Sf); (2) Gr(eSf) = eSf n Gr(S). Proof (1) Let a E Reg(eS)nReg(Sf). Then a = ea = af and a = axa for some x E eS and yES f, and by this it follows that a E eS f and a = axaya E aeSaSfa
~
= aya,
a(eSf)a,
so a E Reg(eSf). Thus, Reg(eS) n Reg(Sf) ~ Reg(eSf). The opposite inclusion is obvious. (2) Let a E eSf n Gr(S). Then a = ea = af and a EGg, for some 9 E E(S), and we have that 9 = aa-1a-1a = eaa-1a-1af, which yields 9 = eg = gf. Now G g = gGgg = egGggf
~
eSf,
whence a E Gr(eSf), so we have that eSfnGr(S) inclusion is evident.
~
Gr(eSf). The opposite _
Lemma 4 Let e be an arbitrary idempotent of a semigroup S. folio'l.J.Jing conditions hold:
Then the
(1) Reg(eSe) = reg(eSe) = Reg(Se) n Reg(eS); (2) Gr(eSe) = eSe n Gr(S); (3) Gr(Se) = Se n Gr(S) and Gr(eS)
= eS n Gr(S) .
Proof (1) By Lemma 3 it follows that Reg(eSe) = Reg(Se) n Reg(eS). Let a E reg(eSe). Then a = ea = ae and a = axa for some XES , and we have that a = axa = aexea E a(eSe)a, so a E Reg(eSe). Thus reg(eSe) ~ Reg(eSe) . It is clear that the opposite inclusion also holds. (2) This is also an immediate consequence of Lemma 3. (3) Evidently, Gr(Se) ~ Se n Gr(S). Let a E Se n Gr(S). Then a = ae and a E Gj , for some f E E(S), so by f = a-1a = a-1ae E Se it follows that f = fe. Therefore Gj
= GJf = GJfe ~ Se,
which implies a E Gr(Se). Hence, Gr(Se) = Se n Gr(S). In a similar way we _ prove that Gr(eS) = eS n Gr(S). Lemma 5 Let S be a semigroup 'l.J.Jith E(S) =I- 0. Then Gr(S)
=
U Gr(Se) = U Gr(eS) = U Gr(eSe) = U eEE(S)
eEE(S)
eEE(S)
e,JEE(S)
Gr(eSf).
110
Proof By Lemma 3 it follows that
U
Cr(eSf)
=(
~,JEE(S)
U
eSf) nCr(S)
= Q(S)nCr(S) =Cr(S),
e,JEE(S)
since Cr(S) ~ M(S) ~ Q(S). Similarly we prove the remaining equalities. For a semigroup S, let the set RegM(S) be defined by
Re9M(S) =
U
Reg(eSe).
eEE(S)
Then the following equalities hold: Lemma 6 Let S be a semigroup with E(S)
f
0.
Then
RegM(S) = M(S) n Reg(S) = Reg(M(S)). Proof It is obvius that RegM(S) ~ M(S) n Reg(S) and RegM(S) ~ Reg(M(S)). Let a E M(S) n Reg(S). Then a E eSe, for some e E E(S), so by Lemma 4 we have that
a E eSe n Reg(S)
= reg(eSe) = Reg(eSe)
~
RegM(S).
Thus M(S) n Reg(S) ~ RegM(S), whence RegM(S) = M(S) the other side
Reg(M(S))
~
n Reg(S). On
M(S) n Reg(S) = RegM(S),
-
so we have proved Reg(M(S)) = RegM(S).
It is easy to verify that the following relationships between the sets Cr(S) , RegM(S) and Reg(S) hold on an arbitrary semigroup S:
Cr(S)
~
RegM(S)
~
Reg(S).
The conditions under which the first inclusion can be turned into an equality are determined by the following theorem.
Theorem 2 Let S be a semigroup with E(S)
ditions are equivalent:
(i) Cr(S) = RegM(S); (ii) ("Ie E E(S)) Reg(eSe) = Cr(eSe); (iii) ("Ie E E(S)) reg(eSe) = Cr(eSe).
f
0.
Then the following con-
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Proof (i)::::}(ii). Let Gr(S) = RegM(S) and let e E E(S). Then by Lemma 4 we have that Gr(eSe)
= eSe n Gr(S) = eSe n RegM(S) = Reg(eSe).
(ii)::::}(i) . Let Reg(eSe) = Gr(eSe), for each e E E(S). Then Lemma 5 yields RegM(S)
=
U Reg(eSe) = U Gr(eSe) = Gr(S). eEE(S)
eEE(S)
(ii)¢:?(iii) . This follows immediately by Lemma 4.
•
Throughout the rest of the paper we shall consider 7r-regular semigroups. According to the results given by Catino [6], a bi-ideal of a 7r-regular semigroup is not necessary 7r-regular. But, the principal bi-ideals generated by idempotents, that is to say local submonoids of a semigroup, have the following property: Lemma 7 Let S be a 7r-regular or a completely 7r-regular semigroup. Then for each e E E(S), the local submonoid eSe has the same property. Proof Let S be a 7r-regular semigroup, and let e E E(S) and a E eSe. Then there exists n E N such that an E Reg(S), and by Lemma 4 we have that an E eSe n Reg(S) = Reg(eSe). Thus eSe is 7r-regular, for every e E E(S). Let S be a completely 7r-regular semigroup and let a E eSe, for some e E E(S). Then there exists n EN such that an E Gr(S), so again by Lemma 4 it follows that an E eSe n Gr(S) = Gr(eSe). Hence, eSe is completely • 7r-regular, for each e E E(S) . A semigroup S is called locally completely 7r-regular if it is 7r-regular and eSe is completely 7r-regular, for every e E E(S) , and it is called locally uniformly 7r-regular if S is 7r-regular and eSe is uniformly 7r-regular, for every e E E(S) . The main result of the paper is the following theorem that characterizes locally uniformly 7r-regular semigroups. Theorem 3 The following conditions on a semigroup S are equivalent:
(i) S is locally uniformly 7r-regular; (ii) Sis 7r-regular and if a E S , n EN and a' E V(a n ), then a'Sa" (anSa' ) is uniformly 7r-regular; (iii) S is 7r-regular and RegM(S) = Gr(S); (iv) S is 7r-regular and Reg(eSe) = Gr(eSe), for each e E E(S); (v) S is 7r-regular and reg(eSe) = Gr(eSe), for each e E E(S);
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(vi) S is locally completely 7r-regular, (E(S)) is locally uniformly 7r-regular
and (Ve,f,g E E(S))
e 2: f, e 2: 9 & fig =} fl(E(eSe))g.
Proof (i){:}(iv). This equivalence is an immediate consequence of the definition of a uniformly 7r-regular semigroup. (i)=}(ii). Let a E S, n EN and a' E V(a n ). Set e = a'a n and f = ana'. Then eSe = a'anSa'a n ~ a'San = a'ana'Sana'a n ~ a'anSa'an = eSe, whence eSe = a' San, and by (i) it follows that eSe = a'Sa n is uniformly 7rregular. In a similar way we prove that anSa' = fSf is uniformly 7r-regular. (ii)=}(i). For each e E E(S), by e E V(e) and (ii) it follows that eSe is uniformly 7r-regular. (iii){:}(iv){:}(v). These equivalences are immediate consequences of Theorem 2. (i)=}(vi). It is clear that S is locally completely 7r- regular. Since S is 7rregular, then by Lemma 1 we have that (E(S)) 7r-regular, which implies that e(E(S))e, by Lemma 7, is also 7r-regular, for every e E E(S). By (i){:}(iv) we also have that Reg(eSe) = Gr(eSe) for every e E E(S). Further, by
a E Reg(e(E(S))e)
~
Reg(eSe) = Gr(eSe)
it follows that for a E Reg(e(E(S))e) there are x E eSe and y E e(E(S))e such that a = axa = aya and ax = xa E E(eSe). Now we have that
a = axa = xa 2 ~ E(eSe)e(E(S))ea 2 ~ e(E(S))ea 2 , i.e. a E LReg(e(E(S))e). Therefore Reg(e(E(S))e ~ LReg(e(E(S))e) and e(E(S))e is 7r-regular, which by Theorem 1 means that e(E(S))e is uniformly 7r-regular for every e E E(S). Thus (E(S)) is locally uniformly 7r-regular. Let e,j,g E E(S), such that e 2: f, e 2: 9 and fig in S. Then f,g E E(eSe) and fig in eSe and by Theorem 1 we have that fig in (E(eSe)). (vi)=}(i). Let e E E(S). By Lemma 1 we have that (E(eSe)) is completely 7r-regular. On the other hand, by the hypothesis it follows that e(E(S))e is uniformly 7r-regular. On the other hand (E(eSe)) ~ e(E(S))e, so by Theorem 1 and Lemma 2 we have that
Reg ( (E(eSe)))= (E(eSe)) n Reg(e(E(S))e) = (E(eSe)) nGr(e(E(S))e) = Gr(E(eSe))). Let f,g E E(eSe) such that fig in eSe. Then e 2: f, e 2: 9 and fig in eSe, and by the hypothesis we have that fig in (E(eSe)). Therefore, by Theorem 1 we obtain that eSe is uniformly 7r-regular for every e E E(S). Hence S is locally uniformly 7r-regular. •
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References 1. S. Bogdanovie and M. Cirie, Semigroups, Prosveta, Nis, 1993 (in Serbian). 2. S. Bogdanovie and M. Cirie, Semilattices of Archimedean semigroups and (completely) 7r-regular semigroups I (A survey), Filomat (Nis) 7 (1993), 1- 40. 3. S. Bogdanovie and M. Cirie, Semilattices of left completely Archimedean semigroups, Math. Moravica 1 (1997) , 11- 16. 4. S. Bogdanovie, M. Cirie and M. Mitrovie, Semilattices of nil-extensions of simple regular semigroups (to appear). 5. S. Bogdanovie, M. Cirie and T. Petkovie, Uniformly 7r-regular rings and semigroups: A survey , Topics from Contemporary Mathematics, Zbom. Rad. Mat. Inst. SANU 9 (17) (1999) , 1- 79. 6. F. Catino, On bi-ideals in eventually regular semigroups , illv. Mat. Pura. Appl. 4 (1989), 89-92. 7. D. Easdown, Biordered sets of eventually regular semigroups, Proc. Lond. Math. Soc. (3) 49 (1984), 483- 503. 8. P. M. Higgins, Techniques of semigroup theory, Oxford Univ. Press, 1992. 9. J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, Oxford: Clarendon Press, 1995. 10. L. N. Shevrin, To the theory of epigroups I, Mat. Sb. 185 (8) (1994) , 129- 160 (in Russian) . English translation: On theory of epigroups. I, Russian Acad. Sci. Sh. Math., 82 (1995), no. 2,485-512. 11. M. L. Veronesi, Sui semigruppi quasi forte mente regolari, illv. Mat. Univ. Parma (4) 10 (1984), 319- 329.
INTRODUCTION TO E-INVERSIVE SEMIGROUPS HEINZ MITSCH Department of Mathematics, Strudlhofgasse 1090 Vienna, Austria E-mail: [email protected] 1. 2. 3. 4. 5. 6.
1
4
Definitions and examples Characterizations Cancellativity conditions Restrictions on idempotents Congruences Covers
Definitions and exrunples
In 1952, G. Thierrin [33] defined a semigroup S to be E - inversive if for every a E S there exists xES such that ax E Es (the set of all idempotents of S) . Sometimes these semigroups are called E-dense - but the latter name is also used for E - inversive semigroups with commuting idempotents. It was noted by R. Croisot that this concept is not one-sided: if ax E Es then ay, ya E Es for y = xax. Examples (see [19]): Every regular semigroup; every periodic (in particular, finite) semigroup; more generally, every eventually regular semigroup (i.e., some power of any element is regular); even more generally, every ideal extension of an eventually regular semigroup; every Rees-matrix semigroup over an E - inversive semigroup; every Bruck-semigroup over a monoid. Since every semigroup with zero is trivially E - inversive, in 1964, G. Lallement [15] defined a semigroup S with zero to be O- inversive if for every a E S* there exists xES such that ax E E'S (where A* = A\{O} for any A ~ S. Examples: Every regular semigroup with zero; every periodic (in particular, finite) semigroup with zero as the unique nilpotent element; more generally, every eventually regular semigroup with zero as the unique nilpotent element ; every Rees-matrix semigroup with zero over an E - inversive semigroup with regular sandwich matrix (see [24]) ; every Brandt-semigroup over an E - inversive semi group. Up to now there are about 40 papers dealing with E - or O- inversive semigroups, entirely or in part. The bibliography at the end of the paper offers a list of the relevant references. We start with a short overview on the beginning of the theory. Following the originating paper by Thierrin [33], Tamura [32] studied 114
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E - inversive sernigroups with a unique idempotent showing that such a sernigroup is an ideal extension of a group (see Theorem 4.1, below) . Tillerrin [35, 36] investigated E - inversive sernigroups S, willch are rectangular (i.e., ax = by = az = m implies bz = m for a, b, x, y, z E S) providing a construction of these sernigroups. Clifford [3] pointed out that these sernigroups can be characterized as inflations of rectangular groups and that they are identical with the M - inversible sernigroups, whose structure was elucidated by Yamada [40] . Petrich [25], [26] studied tills class of E - inversive sernigroups further , in particular, investigating twosided rectangularity. In ills paper [27] he called a sernigroup S E-inversive with uniqueness if for every a E S there exists a unique x ES such that ax E Es , and showed that a sernigroup is a group if and only if it is E - inversive with uniqueness (see Corollary 3.3, below). The concept of O- inversive sernigroup was used by Lallement and Petrich [16] in the study of completely D-simple sernigroups, and more generally, of primitive regular sernigroups with zero. In particular, it is shown in [16] that a sernigroup with zero is primitive inverse if and only if it is O- inversive with uniqueness (see Corollary 3.6, below). It follows that a sernigroup S is a Brandt-sernigroup if and only if S is O-inversive with uniqueness and a i=- 0, b i=- 0 in Simply axb i=- 0 for some xES (see Petrich [27]) . Further results on E - or O- inversive sernigroups are collected in the following with the aim to give a short survey on the most important theorems in tills field. The notation and terminology generally follows the books [12] and [29] . 2
Characterizations
There are several characterizations of E - ( 0- )inversive sernigroups. The most useful is due to Catino and Miccoli [2] which shows that these sernigroups are characterized by " half' of the regularity condition. Lemma 2.1 [2] A semigroup S is E - (O- )inversive if and only a E S (a i=- 0) there exists y E S(y i=- 0) such that y = yay .
~f
for every
Let S be a sernigroup; an element xES is called a weak inverse of a E S if x = xax. Hence, by Lemma 2.1, a sernigroup S is E - (D-)inversive if and only if every element (i=- 0) of S has a weak inverse (i=- 0) . The set of all weak inverses of a E S is denoted by
W(a) = {x E Six = xax}. More generally for A ~ S, W(A) denotes the union of all W(a), a E A. Thus
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we can say that a semigroup S is E - (o-)inversive if and only if W(a) =f (W*(a) =f 0) for every a E S(a =f 0). Fitzgerald [4], Lemma 1, showed that every weak inverse of a product of n idempotents in an arbitrary semigroup S is a product of n + 1 idempotents of S . Thus we have
o
Proposition (1) (2) }inversive. (3)
2.2 [4] Let S be an E - (O- }inversive semigroup. Then: W(E:S) ~ E~+1 for n = 1,2, ... The subsemigroup of S generated by Es is again E - (O-
If Es forms a subsemigroup, then W(Es) = Es·
In particular, Proposition 2.2 (3) says that W(e) ~ Es for every e E Es , i.e. , if Es forms a semigroup every weak inverse of an idempotent in S is again idempotent. Seifert [31] characterized those E - ( 0- )inversive semigroups S for which equality holds for every e E Es. Lemma 2.3 [31] For an E-(O- }inversive semigroup S , W(e) = Es for every e E Es if and only if Es is a rectangular band (i.e., e = efe for all e, fEEs) . Concerning weak inverses of a product of two arbitrary elements in an E - inversive semigroup, in the following result Weipoltshammer [39] proved (1) , Fountain, Pin and Weil [8] proved (2) , (3); concerning (4) see Petrich [29], Lemma IV. 3.1.
Proposition 2.4 Let S be an E-inversive semigroup. Then: (1) W(ab) ~ W(b) . W(a) for all a, bE S. (2) M(e, f) = {g E Esige = 9 = gJ} =f 0 for all e, fEEs . (3) For any a, b E S, b' ga' E W(ab) whenever a' E W(a) , b' E W(b), 9 E M(aa' , b'b). (4) If Es forms a subsemigroup, then W(b). W(a) ~ W(ab) for all a, bE S i conversely, this implies that Es is a subsemigroup. If S has a zero the statements (2) , (3) in Proposition 2.4 are trivial (take 9 = 0). But under an additional assumption on idempotents we have
Corollary 2.5 Let S be a O- inversive semigroup . Then (1) W*(ab) ~ W*(b) . W*(a) for all a, bE S. If ef =f 0 for all e, fEEs then (2) M*(e,1) =f 0 for all e, fEEs (3) For any a, b E S , b' ga' E W*(ab) whenever a' E W*(a), b' E W *(b) , 9 E M*(aa' , b'b). (4) W*(b). W*(a) ~ W*(ab) for all a, bE Si conversely, this im-
117
plies that ef =1= 0 Ve, fEEs. Note that by (4) the additional condition on S implies that Es is a subsemi group of S. If the idempotents of an E-inversive semigroup form a subsemigroup we have the following Proposition 2.6 [8] Let S be an E-(O-)inversive semigroup such that Es forms a subsemigroup. Then S is weakly self conjugated, i.e., for every e E E s , xex',x'ex E Es whenever x E S,X' E W(x). Another characterization of E - ( 0- )inversivity follows almost immediately from the definition. Lemma 2.7 A semigroup S is E - (O- )inversive if and only if every principal right - or every principal left ideal (=1= {O}) of S contains an idempotent (=1= 0) .
Recall that a semigroup S is regular if and only if every principal right or every principal left ideal of S contains an idempotent generator. Higgins [11] provided a characterization of E - inversive semigroups by means of twosided principal ideals. Proposition 2.8[11] A semigroup S is E - (O-) inversive if and only if every principal ideal (=1= {O}) of S contains an idempotent (=1= 0) . The third characterization uses the natural partial order $.s which is defined on every semi group S (see Mitsch [18]):
a$.sb
ifandonlyif a=xb=by,xa=a=ay
some
x,yES 1 .
Note that the restriction of $.s to Es is the well-known partial order of idempotents of S: e $.s f if and only if e = ef = fe. Lemma 2.9 [22] A semigroup S is E - (O- )inversive if and only if for every a E S( a =1= 0) there exists a regular b E S(b =1= 0) such that b $.s a. This result can be used to identify the regular semigroups in the class of E - ( 0- )inversive semigroups. Recall that a subset T of a semigroup S is left- unitary if ta , t E T imply that a E T ; right- unitary if at, t E T imply a E T ; T is called unitary if T is both left- and right unitary. Proposition 2.10 [22] Let S be an E-inversive semigroup without 0 or a O- inversive semigroup . Then the following are equivalent: (i) S is regular; (ii2) a $.s b, a regular (a =1= 0) implies that b is regular;
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(iii) The subset of all regular elements (=I(right )unitary.
0) of S is left
REMARK. Similarly, S is completely (resp. eventually) regular if and only if a ':5:.s b, a regular (=I- 0) implies b completely (resp. eventually) regular.
The results given that a semigroup S is Q with zero if T is an is isomorphic with Q.
above can be generalized to ideal extensions. Recall an ideal extension of the semi group T by a semigroup ideal of S such that the Rees quotient semigroup SIT First, we have the following characterization.
Lemma 2.11[22] Let S be an ideal extension of a semigroup T. Then S is E - inversive if and only if T is E-inversive. In the G-inversive case we need an additional condition. If S is a semigroup with 0 and 0 =I- K <;;; S then the annihilator of K in S is defined by A(K) = a E Siak = ka = 0 forallk E K.
Lemma 2.12[22] Let S be an ideal extension of a semigroup T. If S is O- inversive so is T . Conversely, if T is O- inversive and A(T) = {O} then S is O- inversive. The condition A(T) = {O} in the converse part of Lemma 2.12 is not necessary for S to be O- inversive: the semilattice S = {O, e, f} with ef = 0 is O- inversive, T = {O, e} is an ideal of S , and A(T) = {O, J} =I- {O}. But A(T) = {O} is necessary for instance in case that S is an ideal extension of a semigroup T by a zero semigroup (see [22]). Lemma 2.9. has the following generalization. Proposition 2.13[22] Let S be an ideal extension of a semigroup T (with A(T) = {O}). Then S is E-(O-)inversive if and only if for every a E S(a =I- 0) there exists a regular bET (b =I- 0) such that b 5:.s a. This last result can be used to describe retract extensions S of regular semigroups T under some mild additional assumption on S . Recall that an ideal extension S of a semigroup T is called a retract extension if there exists a homomorphiom of S onto T whose restriction to T is the identity function. Note that by Lemma 2.11, Sis E - inversive in case that T is regular. Theorem 2.14[22] Let S be an ideal extension of a regular semigroup T such that for every a E S there exists e E Es with ea = a or ae = a. Then S is a retract extension of T and T is completely simple if and only if for every a E S there is a unique bET with b ':5:.s a. Three particular cases of this result should be mentioned separately. For
119
the first, note that for a finite semigroup S the kernel is a completely simple, hence regular semigroup. Corollary 2.15 [22] Let S be a finite semigroup such that for every a E S there exists e E Es with ea = a or ae = a. Then S is a retract extension of its kernel K if and only if for every a E S there is a unique b E K with b "5.s a. Concerning the second case observe that in a regular semigroup S for every a E S there exists xES with a = ax.a, where ax E Es(xa E Es) . Generalizing the kernel to a completely simple semigroup we obtain Corollary 2.16 [22] Let S be a regular semigroup which is an ideal extension of a completely simple semigroup T . Then S is a retract extension of T if and only if for every a E S there is a unique bET such that b "5.s a. The third particular case is that S is a monoid (hence ea = a is satisfied for the identity of S) ; here we obtain Corollary 2.17 Let S be a monoid which is an ideal extension of a completely simple semigroup T . Then the following are equivalent. (i) S is a retract extension of T (ii) For every a E S there is a unique bET such that b "5.s a. (iii) T is a group. (iv) S is a homogroup (i.e. , S has a kernel which is a group; see [34]) . REMARK . Since any finite semigroup is an ideal extension of its kernel, conditions (i) to (iv) are equivalent for every finite monoid S taking for T the kernel of S.
3
Cancellativity conditions
Supposing different kinds of cancellativity in E - ( 0- )inversive semigroups characterizations of the corresponding semigroups analogous to the regular case are obtained (Mitsch and Petrich [22]). A semigroup S is weakly cancellative if ax = bx and ya = yb for some x, yES together imply that a = b; S is called trivially ordered if the natural partial order on S is the identity relation. Proposition 3.1 [19] The following conditions on an E - inversive semigroup S are equivalent. (i) S is weakly cancellative.
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(ii) S is trivially ordered. (iii) S is completely simple. (iv) S is primitive regular. Considering one-sided concellativity recall that a semigroup S is right inverse (also: left unipotent) if every principal left ideal of S has a unique idempotent generator (Venkatesan (1974)). Proposition 3.2 [22) The following conditions on an E -inversive semigroup without zero are equivalent. (i) S is left cancellative. (ii) S is a right group. (iii) S is primitive right inverse. (iv) S is trivially ordered and Sa ~ as for every a E S. (v) S is right simple. Combining Proposition 3.2 with its right dual we obtain characterizations of cancellativity. The equivalence of (iii) and (vi) was shown in [27) . Corollary 3.3 [22) The following condition on an E - inversive semigroup without zero are equivalent. (i) S is cancellative. (ii) S is trivially ordered and Sa = as for every a E S. (iii) S is a group. (iv) S is primitive inverse. (v) S is a monoid with a unique idempotent. (vi) S is E - inversive with uniqueness (i.e ., for every a E S there is a unique xES such that ax E Es).
bx
In the presence of a zero we have to consider weak 0- cancellativity: ax = some x, yES together imply that a = b.
i= 0 and ya = yb i= 0 for
Proposition 3.4 [22) The following conditions on a O- inversive semigroup S are equivalent. (i) S is weakly O- cancellative. (ii) S* is trivially ordered. (iii) S is primitive regular with zero . (iv) Each non zero principal left (right) ideal of S is O- minimal. (v) S is an orthogonal sum of completely O-simple semigroups. In the case of left O- cancellativity, i.e., xa = xb i= 0 for some xES implies that a = b, we shall encounter the class of right inverse semigroups with zero: every non zero principal left ideal of S has a unique idempotent generator.
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Every such semi group is an orthogonal sum of right Brandt-semigroups. The latter are characterized as completely o-simple semi groups S in which ef = 0 or ef = f for all e, f E Es(Venkatesan (1974)). Theorem 3.5 [22] The following conditions on a O- inversive semigroup S are equivalent. (i) S is left O- cancellative. (ii) S is primitive right inverse with zero. (iii) S is an orthogonal sum of right Brandt-semigroups. (iv) ea = 0 or ea = a for every e E Es,a E S. (v) S is regular and ef = 0 or ef = f for all e, fEEs· (vi) S * is trivially ordered and Esa ~ aEs for every a E S. Combining Theorem 3.5 and its right dual we obtain several characterizations of O- cancellativity, i.e., ax = bx =I- 0 or xa = xb =I- 0 for some xES implies a = b. The equivalence of (iii) and (vii) was shown in [16]. Corollary 3.6 [22] The following conditions on a O-inversive semigroup S are equivalent. (i) S is O-cancellative. (ii) S* is trivially ordered and Esa = aEs for every a E S. (iii) S is primitive inverse with zero. (iv) S is an orthogonal sum of Brandt-semigroups. (v) ea, ae E {O,a} for all e E E s , a E S. (vi) S is regular and ef = 0 for all e, fEEs with e =I- f. (vii) S is O- inversive with uniqueness (i.e. , for every a E S* there is a unique xES such that ax E Es)· 4
Restrictions on idempotents
For any semigroup the structure of the set of its idempotents (if not empty) generally gives a good deal of information on the structure of all of S. In the following the order theoretical point of view will be adopted: imposing different restrictions on the ordering of the idempotents the impact of these restrictions on the E - ( 0- )inversive semi group is investigated (see Mitsch and Petrich [23]) . As a first restriction suppose that Es has only one element. Recall that a regular semigroup with this property is a group. A semigroup with zero is called poor if the zero is the unique idempotent. Necessity of the following characterization was proved by Tamura [32]. Theorem 4.1 [32] A semigroup Sis E - inversive and has a single idempotent
122
if and only if S is an ideal extension of a group by a poor semigroup. If S is O~inversive and has a single idempotent, namely 0, then S consists of one element only. Thus we will consider the condition that there is a unique non zero idempotent in S. Note that a regular semigroup with this property is a group with zero. Recall the definition of annihilator following Lemma 2.11.
Theorem 4.2[23) A non trivial semigroup S with zero is O~inversive and contains a single non zero idempotent if and only if S is an ideal extension of a group with zero K by a poor semigroup with A(K) = {O}. More generally, we assume now that Es admits a least element. Recall the definition of homogroup (see Corollary 2.17). Theorem 4.3 [23) A semigroup S is E ~inversive such that Es has a least element if and only if S is a homogroup. If S has a zero then this result is evident: in fact, in this case 0 is the least element of Es and the kernel of S is the one element group. Thus we consider the condition that there is a least non zero idempotent in S.
Theorem 4.4 [23] The following conditions on a non trivial semigroup S with zero are equivalent. (i) S is O~inversive and Es has a least element. (ii) S is an ideal extension of a group with zero K and A(K) = {O} . (iii) S is an ideal extension of a group with zero whose identity e satisfies eS* ~ S* The next restriction is that all the (non zero) idempotents of S are primitive, i.e., (0 #)e $.s f, e, fEEs, implies that e = f . The particular case when all elements (# 0) of S are incomparable in the natural partial order was dealt with in Proposition 3.1 for S E~inversive , and in Proposition 3.4 for S O~inversive. These semigroups we characterized (as in the regular case) as completely simple, respectively as orthogonal sums of completely O~simple semigroups (i.e. , primitive regular semigroups with zero). Theorem 4.5 [23] A semigroup S without zero is E ~inversive and all idempotents of S are primitive if and only if S is an ideal extension of a completely simple semigroup by a poor semigroup. Theorem 4.6 [23] A semigroup S with zero is O~inversive and all non zero idempotents of S are primitive if and only if S is an ideal extension of a primitive regular semigroup T by a poor semigroup such that A(T) = {O}.
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For the special case of Theorem 4.6 when T is completely o-simple (i.e., the primitive regular semigroup with zero T consists of one competely 0simple component, only) we have the following Theorem 4.7 [23] Let 8 be a O- inversive semigroup all of whose non zero idempotents are primitive; then the following are equivalent. (i) 8 is an ideal extension of a completely O- simple semigroup by a poor semigroup. (ii) Any two non zero idempotents are V - equivalent. (iii) {O} is a prime ideal of 8 {i.e., axb = 0, a, b, x E 8, implies that a = 0 or b = O. Our final restriction on idempotens is that Es forms an w--chain, that is, in the natural order Es has the form eo > e1 > e2 > . . .. If 8 is a regular semigroup, for which Es forms an w--chain (i.e., 8 is a regular w- semigroup), then 8 is of one of the following types: (1)
8 is an w--chain of groups (if 8 has no kernel)
(2)
8 is a Bruck semigroup over a finite chain of groups (if 8 is simple)
(3)
8 is an ideal extension of a semigroup of type (2) by a finite chain of groups with a zero adjoined.
On the pattern of this description a classification of E - inversive w- semigroups was given in Mitsch and Petrich [23]. Recall that a semigroup 8 is an w--chain of disjoint semigroups 8 0 ,81 , 8 2 , ... if there exists a homomorphism r.p of 8 onto the w--chain eo > e1 > e2 > . .. of idempotents such that 8 i = eir.p-l for i = 0, 1,2, .... For the definition of homogroup see Corollary 2.17. Theorem 4.8 [23] A semigroup 8 is E - inversive and the idempotents of 8 form an w- chain if and only if 8 is one of the following types: (i) 8 is an w- chain of homogroups each containing a unique idempotent, (ii) 8 has a kernel K which is a simple regular w- semigroup and ES\ K is a finite (possibly empty) chain of idempotents each of which acts as an identity on K. Concerning o-inversive semigroups 8 , whose non zero idempotents form as w--chain, note first that 8 can be characterized by the property that 8* is an E - inversive semigroup whose idempotents form an w--chain. As a consequence, Theorem 4.8 yields the following classification.
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Corollary 4.9 [23] A semigroup S is O-inversive and the non zero idempotents of S form an w-chain if and only if S is of type (i) or (ii) in Theorem 4.8 with a zero adjoined in each case.
5
Congruences
Several types of congruences on E-inversive semigroups S have been studied. In particular, primitive congruences were investigated by Reither [30]: a congruence p on S is called primitive if the natural partial order on S/ p (see section 2) is the identity relations (i.e., all elements of S/ p are incomparable). Since for an E - inversive semigroup S also S/ p is E-inversive, it follows by Proposition 3.1 that S/ p is completely simple. Therefore, the primitive congruences on an E-inversive semigroup are exactly the completely simple congruences. Particular primitive congruences are group congruences, that is, for which S/ p is a group. Note that a primitve congruence p on S such that S/ p is a primitive inverse semigroup, is already a group congruence (in fact, a trivially ordered, inverse semigroup contains only one idempotent and therefore is a group). A characterization of primitive congruences is given in the following Theorem 5.1 [30] Let S be an E-inversive semigroup and p a congruence on S. Then the following are equivalent: (i) p is primitive (equivalently, completely simple); (ii) p is weakly cancellative (i.e., xa p xb and ay p by imply a p b); (iii) a 5:.s b (a, b E S) implies that a p b. Corollary 5.2 [30] Let S be an E-inversive semigroup. Then a congruence p on S is a group congruence if and only if p is primitive and e p f for all e,f E Es·
Recall that a partially ordered set (X,5:.) is called directed downwards (upwards) if for all a, b E X there exists c E X such that c 5:. a, c 5:. b (c ;::: a, c ;::: b). As an immediate consequence of Theorem 5.1 and Corollary 5.2 we obtain Corollary 5.3 [30] Let S be an E-inversive semigroup S such that Es is directed downwards (respectively, upwards) with respect to the natural partial order. Then every primitive congruence on S is a group congruence. REMARKS. 1) Note that every E - inversive semigroup, which has an identity or in which the idempotents commute, satisfies the condition of Corollary 5.3. 2) Reither [30] also proved that a primitive congruence p on an E -
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inversive semigroup S is completely determined by those p-dasses, which contain idempotents (that is, by the kernel of p). Generalizing a method of G. Gomes (1988) for congruences or regular semigroups, she reconstructed any primitive congruence on an E-inversive semigroup from its kernel. Since the intersection of any set of weakly cancellative congruences on a semigroup S is again weakly cancellative, it follows by Theorem 5.1 that on every E - inversive semigroup S there is a least primitive congruence. A description of this congruence (similar to that on regular semigroups, due to Nambooripad (1980)) is given in the following Theorem 5.4 [30] Let S be an E - inversive semigroup. Then the congruence p" generated by the relation: a p b if and only if c '5:.8 a, c '5:.8 b for some c E S, is the least primitive (equivalently, completely simple) congruence on S. If S is a regular semigroup for which the natural partial order is compatible with multiplication, i.e. ,
a '5.8 b implies ac '5.8 bc and ca '5.8 cb for all c E S, K.S.Nambooripad (1980) proved that the relation p on S defined in Theorem 5.4 is already a congruence (hence p* = p). Reither [30] showed that this holds for every semigroup, whose natural partial order is compatible with multiplication. As a consequence we have Corollary 5.5 [30] Let S be an E-inversive semigroup for which the natural partial order is compatible with multiplication. Then the relation p defined on S by: a p b if and only if c '5:.8 a, c '5:.8 b for some c E S is the least primitive (eqivalently, completely simple) congruence. REMARK. For semigroups S having a compatible natural partial order see Mitsch [20] .
An alternative characterization of the least primitive congruence similar to that for regular semi groups given by T. Hall (1968) is the following. Theorem 5.6 [30] Let S be an E-inversive semigroup. Then the congruence 8* generated by the relation: a 8 b if and only if xa = xb, ay = by for some x, yES, is the least primitive (equivalently, completely simple) congruence
on S. Considering O-inversive semigroups S note first that in Theorem 5.4 the relation p. is the universal relation on S, since 0 '5.8 a for all a E S. Therefore, only O- restricted congruences on S are of interest, i.e., such congruences
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for which {O} is a congruence class. Reither [30] studied congruences p on S such that (S/p)\{O} is trivially ordered, equivalently: primitive regular congruences (see Proposition 3.4). Similarly to the regular case (considered by T. Hall (1968)) we first have Lemma 5.7 [30] A O- inversive semigroup S has a O-restricted primitive regular homomorphic image if and only if S is categorical (i.e., ab #- 0, bc #- 0 imply abc #- 0, a, b, c E S). Generalizing a result on primitive (regular) congruences on regular semigroups (due to K.S.Nambooripad (1980)) Reither [30] proved the following Theorem 5.8 [30] Let S be a O-inversive, categorical semigroup. Then the congruence (3* generated by the relation: a (3 b if and only if c ~s a, c ~s b for some c E S* and 0 (3 0, is the least O- restricted primitive regular congruence on S. If the natural partial order on S is compatible with multiplication, then (3* = (3.
A description of the least G-restricted primitive regular congruence similar to that given by T.Hall (1968) for regular semigroups with zero, was also provided by Reither [30] . In addition, right group congruence were dealt with there. Theorem 5.9 [30] Let S be a O-inversive, categorical semigroup . Then the relations 71". generated by the relation: a7l"b if and only if xa = xb #- 0, ay = by #- 0 for some x, yES and 0 71" 0, is the least O-restricted, primitive regular congruence on S. Under additional hypotheses on the O- inversive semigroup S, GomesHowie [9] gave a description of the least primitive inverse congruence on S. Theorem 5.10 [9] Let S be a O- inversive, categorical semigroup for which Es forms a subsemigroup such that ab E Es implies ba E Es. Then the relation (3 defined by: a (3 b if and only if ea = bf #- 0 for some e, fEEs and 0 (3 0, is the least O- restricted primitive inverse congruence on S. REMARKS 1) If a semigroup S satisfies the conditions of Theorem 5.10 and also: if a, b E S* then there exists c E S* such that J c ~ Ja , J b , then the relation (3 in Theorem 5.10 is the least Brandt congruence on S (see [9]). 2) If the semigroup S in Theorem 5.10 has the additional property that the idempotents commute, then (3 has the form: a (3 b if and only if ea = eb #- 0 for some e E E'S and 0 (3 0 (see [9]). Since on a group the natural partial order is the identity relation (because
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of cancellation), a particular case of primitive congruences on a semigroup are the group congruences. Now, an E-inversive semigroup S is a group if and only if S is cancellative (see Corollary 3.3); thus we immediately obtain Theorem 5.11 [30] Let S be an E-inversive semigroup. A congruence P on S is a group congruence if and only if P is cancellative (i.e., xa P xb or ay P by implies a p b, a, b, x, yES).
An explicite form of an arbitrary group congruence on an E - inversive semigroup S was first given by Mitsch [19] (the following formulation is due to Jiang - Trotter [14]) . Theorem 5.12 [19] Let S be an E-inversive semigroup and let T be a subsemigroup of S such that (1) Es ~ T and (2) ata' , a'ta E T for every t E T, a E S, a' E W(a); then the relation PT defined on S by: apT b if and only if xa = by for some x, YET, is a group congruence. Conversely, let P be a group congruence on S and let T = {a E SI ap is the identity of S / p}; then T has the properties (1) and (2) above and p = PT. Group congruences on E-inversive semigroups were studied in detail by Reither [30] and Zheng [41]. First we have Theorem 5.13 [30] The lattice of all group congruences on an E - inversive semigroup is modular. Generalizing the concept of normal subsemigroup in an inverse semigroup (see Howie [12]) , Reither [30] defined an E-inversive subsemigroup N of an E - inversive semigroup S to be normal if
1) Es
~
N, and 2) x'Nx ~ N, xNx' ~ N for all XES, x' E W(x).
Using the notations Nw = {x E Sia '5:.s x for some a E N} we have Theorem 5.14 [30], [41] Let S be an E-inversive semigroup and let N be a normal subsemigroup of S . Then the relation PN on S defined by: a PN b if and only if ab' E N w for some (all) b' E W (b), is a group congruence on S such that ker PN = Nw. Conversely, if P is a group congruence on S and if K = kerp, then K is a normal subsemigroup of S with Kw = K and P = PK. Corollary 5.15 [30] Let S be an E - inversive semigroup. Then the mapping
Reither [30] and Zheng [41] provided about 30 different descriptions of an arbitrary group congruence P on an E - inversive semigroup S, i.e. , P = PN for some normal subsemigroup N of S. It should be noted that in [41] N
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has the additional property that N w = N. All characterizations are strongly reminiscent of those given by Feigenbaum (1975) and La Torre (1982) for regular semigroups. For example: apNb if and only if NaN n NbN #- 0. If in particular, the idempotents of S commute, then: apNb if and only if (Na)w = (Nb)w ([30]) . The existence of the least group congruence 0" on an E - inversive semigroup S was noted by Hall and Munn [10]. An explicite description of 0" follows immediately from Theorem 5.12. Theorem 5.16 [19] Let S be an E-inversive semigroup. Then the relation 0" defined by: a 0" b if and only if xa = by for some x,y E D(S), where D(S) is the intersection of all subsemigroups of S satisfying (1) and (2) in Theorem 5.12 is the least group congruence on S. As a consequence of her results on general group congruences, Reither [30] obtained the following characterizations. Theorem 5.17 [30] Let S be an E - inversive semigroup. Then the relation a 0" b if and only if ab' E U for some b' E W(b) , where U denotes the least normal subsemigroup of S satisfying Uw = U, is the least group congruence on S. If furthermore, Es forms a subsemigroup of S then a 0" b if and only if ea = af for some e, fEEs (equivalently, ab' E Es for some b' E W(b)). For the special case that S is E - inversive with commuting idempotents, Margolis and Pin [17] described 0" in the following way: a
0"
b if and only if ea = eb for some e E Es .
For further characterizations in this case see Reither [30] , where also a description of 0" was given which corresponds to that of the least group congruence on an inverse semigroup due to Wagner (1953):
a 0" b if and only if c '5:s a, c '5:s b for some c
E S.
Further results on congruences p in connection with group congruences "Ion E - inversive semigroups S can be found in Reither [30] (see also Zheng [41]). For example: (i) P V "I = "lop 0 "I, (ii) a (p V "I) b if and only if xa p by for some x, y E ker "I. For the particular case that E s forms a subsemigroup, a result on the congruence lattice of an inverse semigroup (by Petrich (1978)) and more generally, of an orthodox semigroup (by La Torre (1982)) was generalized. Theorem 5.18 [30] Let S be an E - inversive semigroup such that Es forms a subsemigroup of S. Then the mapping 'P: p ---t P V 0" from the lattice of all
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congruences of S to the lattice of all group congruences on S is a surjective lattice homomorphism. REMARKS . Other types of congruences on E - inversive semigroups S were investigated by Seifert [31]. In particular, conditions for S are found under whlch the relation J.l resp.Y (whlch are the greatest idempotent separating resp. least inverse congruence on an orthodox semigroup) is the greatest resp. least such congruence on S.
Certain investigations on congruences of E - inversive semigroups in connection with Lallement 's Lemma were successful. For a semigroup S, a congruence p on S is called idempotent- consistent (or idempotent- surjective) if every idempotent p--class contains an idempotent e E Es. By Lallement's Lemma, for a regular semigroup every congruence is idempotent-consistent. More generally, it was shown by Edwards (1983) that every congruence on an eventually regular semigroup is idempotent-consistent. But if S is E inversive, a congruence p on S is not necessarily so. As a trivial example, consider the multiplicative semigroup S of natural numbers including 0 but without 1; then S is E - inversive and the relation p on S given by the partition {S* , {O}} is a congruence; the p--class S* is idempotent, since it is a subsemigroup of S , but it does not contain any idempotent of S . A less trivial example (without zero) is given by any ideal extension T of a group G by the semigroup S above and the congruence on T given by {G, S*}. Hence the class of E - inversive semigroups is too large for the Lemma of Lallement to hold. Nevertheless, Higgins [11] proved that a semigroup having the property stated in Lallement's result, necessarily belongs to this class. Theorem 5.19 [11] Let S be a semigroup all of whose congruences are idempotent-consistent; then S is E - inversive. REMARK . Examples of semigroups whose congruences are idempotentconsistent, but whlch are not eventually regular, were provided by Higgins [11] and by Kopamu (see the reference in [11]).
A weakened version of the Lemma of Lallement was proved by Mitsch [21] , whlch allows to specify several types of congruences on E - inversive semigroups whlch are idempotent-consistent. Theorem 5.20 [21] Let S be an E - inversive semigroup and p be any congruence on S . Then for every idempotent p- class ap E E(S/ p) there exists an idempotent e E Es such that ep :::; ap in the natural partial order of S/ p. Following T. Hall, a semigroup S is called E - primitive if every idempotent
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e E Es is primitive (see the definition following Theorem 4.4) . Furthermore, a congruence p on Sis E-primitive if S/ p is an E - primitive semigroup. For such congruences we obtain immediately from Theorem 5.20: Corollary 5.21 [21] On an E - inversive semigroup Severy E - primitive congruence is idempotent- consistent. In particular, every primitive (equivalently, completely simple) congruence on S is idempotent- consistent. Evidently, Corollary 5.21 also holds for congruences p on E - inversive semigroups S such that every idempotent p-<:lass is maximal in the natural partial order of S/ p. FUrther examples are given in [21]. Concerning idempotent-consistent congruences on general semigroups (with Es a subsemigroup, respectively) see [21]. It should be noted that "half" of the conditions given there is satisfied for E-inversive semigroups.
6
Covers
A semigroup T is called a cover of a semigroup S if there exists an idempotentseparating homomorphism cp from Tonto S (i. e. , cp restricted to ET is injective; in general, cp is not assumed to be surjective on Es). Supposing different conditions on T, several kinds of covers have been studied. A semigroup T is called an E - unitary cover of the semigroup S if T is E - unitary. Recall that a semigroup T is left (right) E - unitary if ET is a left (right) unitary subset of T (see the definition following Lemma 2.9) ; Tis called E - unitary if T is both left- and right E - unitary. The proof of a result of Howie - Lallement (1965) on regular semigroups immediately yields Lemma 6.1 Let S be a semigroup. Then S is left E - unitary if and only if S is right E - unitary. Furthermore, if S is E - unitary and E - inversive, then Es forms a subsemigroup. The problem of constructing an E - unitary cover T for a semigroup S was dealt with and solved first for the class of inverse semigroups by McAlister (1974). More generally, it was considered by Takizawa (1979) and Szendrei (1980) for orthodox semigroups. For finite semigroups with commuting idempotents the existence of an E - unitary cover was shown by Ash (1987). This result was generalized to E - inversive semi groups with commuting idempotents by Fountain [5] using methods of Margolis and Pin [17]. Finally, a general covering theorem was proved by Almeida, Pin and Weil [1] : Every semigroup S such that Es forms a subsemigroup has an E - unitary cover. In the particular case of an E - inversive semigroup the construction yields
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Theorem 6.2 [1] Every E-inversive semigroup S such that Es forms a subsemigroup has an E - unitary, E - inversive cover. All the constructions of E - unitary covers of a semigroup known up to now (with two exceptions- see the attempts in [19] and [30] below) start with a semigroup S for which Es forms a subsemigroup. For the general covering problem this assumption is not necessary. But if T is to be an eventually regular E - unitary cover of S then, by Edwards (1983) (see section 5.) , the congruence corresponding to the idempotent- seperating homomorphism from Tonto S is idempotent-consistent. Since by Lemma 6.1 , ET forms a subsemigroup of T , it follows that Es forms a subsemigroup of S, too. Hence, in this case S has to be supposed to have Es as a subsemigroup. Concerning the covering problem for E - inversive semigroups S, for which Es not necessarily is a subsemigroup, a first sufficient condition for the existence of an E - unitary cover of S was given by Mitsch [19] using a generalized concept of unitary surjective subhomomorphism (introduced by McAlister and Reilly (1977) for inverse semi groups as a special relational morphism). Theorem 6.3 [19] Let S be an E - inversive semigroup . If there is a unitary surjective sub homomorphism of S onto a group then S has an E-unitary, E -inversive cover. REMARK.
It was shown by P. Trotter (see [19]) that under a certain hypothesis on the cover the condition given in Theorem 6.3 is also necessary.
Using the concept of prehomomorphism (introduced by McAlister (1974) for inverse semigroups) , Reither [30] provided another sufficient condition for an E - inversive semigroup to admit an E - unitary cover. Theorem 6.4 [30] Let S be an E - inversive semigroup . If there is a unitary prehomomorphism of a group onto S then S has an E - unitary, E - inversive cover. With respect to the O- inversive case the corresponding covering problem was solved by Gomes and Howie [9] under the necessary hypothesis that the given O- inversive semigroup S is categorical (see Lemma 5.7): for all a, b, c E S, abc = 0 implies that ab = 0 or bc = O. For a semigroup S with 0, the concept of E - unitary has to be adapted in the following way: S is called left- (right - ) E* - unitary if e, ea E Es (e, ae E Es) imply that a E Es ; S is called E*- unitary if S is both left- and right E*- unitary. Similarly to Lemma 6.1. we have
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Lemma 6.5 [9] Let S be a semigroup with zero. Then S is left E*-unitary if and only if S is right E* -unitary. FUrthermore, if S is E* -unitary and O-inversive, then Es forms a subsemigroup of S. Using a technique essentially derived from Fountain [5] , Gomes and Howie [9] proved the following covering result. Theorem 6.6 [9] Every O- inversive, categorical semigroup S, for which Es forms a subsemigroup, has an E* -unitary, O- inversive, categorical cover. The structure of E - unitary, E-inversive semigroups was first decribed by Almeida, Pin and Weil [1]. Their characterization generalizes a result an E unitary, E - inversive semigroups with commuting idempotents due to Margolis and Pin [17] . Theorem 6.7 [1] The following conditions on a semigroup S are equivalent. (i) S is E-unitary, E - inversive. (ii) There exists a group G and a surjective homomorphism 'P from S onto G such that 1'P- 1 = Es (1 the identity of G). Several other characterizations of E - unitary, E - inversive semigroups were given by Reither [30] . Theorem 6.8 [30] For an E - inversive semigroup S the following conditions are equivalent. (i) S is E-unitary. (ii) Es is a subsemigroup and Esw = E s , where Esw {a E Sle ~s a for some e E Es} . (iii) E s is a (j - class ((j the last group congruence on S) . (iv) (j = T (T the greatest idempotent- pure congruence on S). (v) I (e ) = Es, where I(e) = {a E Slae, ea E Es}. REMARK. For inverse semigroups S the two conditions: "E-unitary" and " Esw = Es" are equivalent; note that in this case Es is a (commutative) subsemigroup of S. For E - inversive semigroups, results on the relation of these two conditions can be found in Reither [30].
Concerning the O- inversive case we have the following characterizations (note that for a semigroup with zero, (j is the universal relation) . Theorem 6.9 For a O- inversive semigroup S the following are equivalent. (i) S is E* -unitary. (ii)Es is a subsemigroup, Esw = E s , and ab E Es implies ba E Es . REMARK.
Characterizations of O-inversive, E *- unitary, categorical semi-
133
groups as G-direct unions of particular such semigroups were given by Gomes and Howie [9]. The covering problem for E - inversive left type-A monoids S with commuting idempotents was solved by Fountain and Gomes [7] showing that every such monoid S has a cover T of the same type which is left proper (a particular case of E - unitary). In addition, a representation of such monoids T as McAlister monoids over a right cancellative monoid (instead of a group) was given, thus generalizing McAlister's P - theorem on inverse semigroups (1974). Weakening the condition that the cover be E - unitary, Fountain [6] considered (E) - unitary covers of particular E - inversive semigroups where (E) denotes the subsemigroup generated by the idempotents. For regular semigroups, 'Trotter [37] resp. Jiang and 'Trotter [14]) studied Coo - unitary (finite) covers. Finally, a general covering theorem was proved by Fountain, Pin and Weil [8], from which all results found up to now follow as particular cases. They showed that every E - inversive semigroup S has an E - inversive, D unitary cover T , that is to say, that D(T) (for the notation see Theorem 5.16) is a unitary subset of T (see the definition following Lemma 2.9) and there exists a surjective homomorphism from Tonto S whose restriction to D(T) is an isomorphism onto D(S). Theorem 6.10 [8] Every E -inversive semigroup has an E - inversive, D unitary cover. Similarly to the E - unitary case there is the following characterization of D- unitary, E - inversive semigroups. Theorem 6.11 [8] An E - inversive semigroup S is D-unitary if and only if there exists a group G and a surjective homomorphism
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RINGS GRADED BY INVERSE SEMIGROUPS
w. D . MUNN Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K. e-mail: [email protected]. uk. 1
Introduction
The notion of a semi group-graded ring, defined below, embraces many familiar mathematical structures. This short survey is mainly devoted to an account of work by A. V. Kelarev and the author concerning certain properties of rings graded by inverse semigroups, where the grading is assumed to be 'faithful', in the sense of Cohen and Montgomery. Let R be a ring faithfully graded by an inverse semigroup S. Associated with each maximal subgroup G of S, there is a naturally defined subring He of R, graded by G. In 1998, Kelarev showed that a sufficient condition for R to be semiprime [respectively, semiprimitiveJ is that each RG be semiprime [respectively, semiprimitiveJ. Subsequently, for the case in which S is bisimple, the author obtained sufficient conditions, of a similar nature , for R to be prime and for R to be right primitive. These results generalise earlier theorems of Domanov (1976) and the author (1990) on inverse semi group rings. For the reader's convenience, all the necessary ring-theoretic concepts are introduced in the text. 2
Definitions and Examples
Definition.Let R be a ring and S be a semigroup. Then R is said to be S-graded (equivalently, graded by S) if and only if (i) its additive group is expressible as a direct sum of (abelian) subgroups Rx (x E S) , and (ii) the multiplication in R is such that
(VX, YES)
RxRy ~ Ra,.y .
We call the R x the homogeneous components of R with respect to the grading. This terminology had its origin in Ex.1 below. Example 1. Let R := F [Xl, X2, . . . , Xn], the ring of polynomials in the n commuting indeterminates Xl, X2, . .. , Xn over a field F. Let S denote the additive semi group of nonnegative integers and, for each n E S, let Rn be the 136
137
set of all elements of R that are homogeneous of degree n. Then clearly each Rn is a subgroup of (R, +) , R = ffinESRn and, for all m and n , RmRn ~ Rm+n. Observe that R{) (~F) is the only homogeneous component that is a subring of R. Example 2. Let R denote A [S] , the semigroup ring of a semigroup S over a ring A. The elements of R are the mappings from S to A of finite support, addition in R is defined pointwise and multiplication is convolution. It is convenient to regard R as the set of all formal sums of the type (ax E A), with the convention that at most finitely many coefficients ax are nonzero. The operations in R are then according to the rules
x
x
x
and
For each XES, let Rx denote Ax (= {ax: a E A}) . Then R = ffixEs Rx and, for all x and yin S, RxRy ~ Axy = Rxy. For the case in which A has a unity 1 we also assume that Ix = x for all xES, thereby embedding S in (R,.) . Remark. Ex.I , for n ~ 2, and Ex.2 together show that the same ring can be graded by different semigroups: for we can regard F [Xl, X2, ... , x n ] as the semigroup ring over F of the free commutative monoid on Xl , x2, . . . , X n . This situation is again illustrated in the next example. Example 3. Let S be a semigroup that is a band (in particular, a semilattice) E of subsemigroups Se (e E E) ; that is, the Se are pairwise disjoint with union S and, for all e and f in E, SeSf ~ Sef. Now let A be a ring and let R := A [S]. For all e E E, let Re denote the semigroup ring A [Se] . Then R = ffieEERe and, for all e and finE, ReRf ~ Ref· Hence R can also be regarded as an E-graded ring. In this case every homogeneous component is a subring of R . Example 4. Let S be an arbitrary semigroup and let (RX)xES be a family of arbitrary abelian groups indexed by S. Define R to be the external direct
138
sum of the Rx and define a multiplication in R by taking all products to be O. Then, for all x and y in 8, RxRy (= 0) ~ Rxy . The following definition provides a useful restriction which excludes, in particular, trivial cases such as Ex.4 above. Definition. Let R be a ring graded by semi group 8, with homogeneous components Rx (x E 8). Then R is said to be faithful (equivalently, faithfully graded by 8) if and only if
(\fx, y E 8) (\fa E Rx \0)
aRy
i- 0 and Rya i- O.
This concept was introduced by Cohen and Montegomery [2] for a groupgraded rings. It is easily checked that , in Ex.l, R is faithful. The same is true for R in Ex.2, provided that the coefficient ring A has no nonzero left or right annihilator (in particular, if A is a non trivial ring with a unity) . By contrast, the E -graded ring in Ex.3 need not be faithful. To see this, consider the case where C is a cyclic group of order 2, with generator x , and 8 is the semilattice of groups obtained by adjoining a zero z to C . We may thus take E = {e, z} , where e is the identity of C. Let A be a field and let R := A [8]. Then R = Re ffi R", where Re := A [G] and R" = Az. Now let a E Re \0 be defined by a := e - x . Then since (e - x) z = 0 = i (e - x ), we have that aR" = 0 = R"a.
3
Problem to be studied: results on inverse semigroup rings
Before outlining the problem to be considered, we introduced some further notation . • The set of all maximal subgroups of a smeigroup 8 is denoted by Ms . • Let R be an 8-graded ring, with homogeneous components R x (x E 8). For a nonempty subset T of 8 , we write RT := ffixETR'X .
Note that if T and U are nonempty subsets of 8 then RTRU ~ R TU . Hence, if T is a subsemigroup of 8 (in particular, if T EMs) then RT is a subring of R. This enables us to formulate the general problem: relate properties of R to the corresponding properties of its group-graded subrings RG (C EMs) .
139
Observe that, if R = A [S] for some ring A and if GEMs then Rc = A [G] , the group ring of G over A. At this stage, we specialise our discussion in three directions: all gradings will be faithful ; only inverse semigroups will be considered; and only four ringtheoretic properties will be discussed, namely semiprimeness, semiprimitivity, primeness and right primitivity. Recall that a ring R is
• semiprime if and only if it contains no nonzero (right) ideal A such that A2 =0; • semiprimitive if and only if its Jacobson radical is 0 only if there is no nonzero a E R such that (\;Ix E
R) (3y E R)
that is, if and
ax +y = axy;
• prime if and only if the product of any two nonzero (right) ideals is nonzero; • right primitive if and only if there exists a nonzero right R-module M such that (i) M has no submodules except M and 0, and (ii) for all a E R, M a = 0 implies a = 0 - that is, M is irreducible and faithful . It can be shown that every right primitive ring is both prime and semiprimitive, that every prime ring is semiprime, and that every semiprimitive ring is semiprime; but none of the converse statements is true. Since inverse semigroup rings (over suitable coefficient rings) form a muchstudied subclass of the class of all rings faithfully graded by inverse semigroups , we conclude this section with a summary of some previous results concerning these. An important landmark here is the following theorem, obtained by Domanov in 1976 (4).
• Let F be a field and S an inverse semigroup. If F [G] is semiprimitive for all GEMs then F [S] is semiprimitive. For finite S, this had essentially been established in the 1950s by Oganesyan, Ponizovskir and the author (independently) ; and it led to an exercise study of matrix representations of finite inverse semigroups, based on the classical theOI}' of representations of finite groups. Removal of the finiteness restriction , however, required a more sophisticated approach. In passing, we remark that, also in 1976, Barnes [1) showed directly that, for an arbitrary inverse semigroup S, C [S] is semiprimitive - a result which can alternatively
140
be obtained by combining Domanov's theorem with the well-known fact that every complex group ring is semiprimitive. AS was pointed out by Ponizovskii inspection of Domanov's proof reveals the following theorem concerning right primitivity.
• Let F be a field and S a bisimple inverse semigroup. If F [G] is right primitive for some GEMs then F [S] is right primitive. Furthermore, it is easily seen that, in the proof, the field F can readily be replaced by a general ring A witha unity. It is not clear that Domanov's argument can be adapted to deal with semiprimeness and primeness. But a different method, developed in 1990 by the author [7], yields both theorems quoted above, together with thw results below.
• Let A be a ring with unity and S an inverse semigroup . If A [G] is semiprime for all GEMs then A [8] is semiprime. • Let A be a ring with unity and S a bisimple inverse semigroup. If A [G] is prime for some GEMs then A [S] is prime. 4
Serniprirneness and serniprirnitivity
This section is devoted to a discussion of theorems of Kelarev [5] that extend the results on semiprimeness and semiprimitivity of inverse semigroup rings to the case of a ring faithfully graded by an inverse semigroup. The following standard notation is required for the statement of the key result (Lemma 1) on which the proofs of these theorems and those of the next section are based. • For two sets X and Y, X\Y denotes {x EX: x ¢:. Y}. • For an idempotent e in a semigroup S, Pe denotes the right unit subsemigroup of eSe and He (E Ms) denotes the group of units of eSe. • For an element a in an S-graded ring R with homogeneous components Rx (x E S), we denote the Rx-component of a by ax and define the support, supp(a), of a by supp (a)
:=
{x
E
S : ax # O}.
Note that Isupp (a)1 < 00; and supp(a) = ; if and only if a = O.
141
The lemma below, due to Kelarev [5], is closely modelled on the corresponding result for semigroup rings given by the author in [7]. Its proof contains almost all the semigroup theory required for Theorems 1 and 2 below. Lerruna 1 (Kelarev, Munn). Let S be an inverse semigroup, let R be a faithful S-graded ring and let A be a nonzero ideal of R. Then there exist e = e 2 E S and a E A such that e E supp(a) ~ He U (eSe',Pe).
It is perhaps of interest to indicate, without proof, how the elements e and a are obtained. First, let b E A\ O and let the elements of supp(b) be Xl> X2, ... , X n . Choose e to be maximal in { X iX;1 : i = 1,2, .. . , n} , under the usual partial ordering of idempotents, and assume that the Xi are numbered so that, for some k, e=
Assume,
further ,
that
Xl
XiX;1
is
¢:}
1 ~ i ~ k.
such
that
xllXI
is
minimal
in
i = 1,2, ... , k} . Since bX1 #- 0, the faithfulness condition shows that there exist cERe and d E RX-l such that cb X1 d #- O. Now take a := cbd. {x;IXi :
1
Using this result, Kelarev established the following theorems (and others of a similar type, concerning ring-theoretic properties not discussed here). Theorem 1 (Kelarev). Let S be an inverse semigroup and let R be a faithful S -graded ring. If Ra is semiprime for all GEMs then R is semiprime. Theorem 2(Kelarev). Let S be an inverse semigroup and let R be a faithful S -graded ring. If Ra is semiprimitive for all GEMs then R is semiprimitive. To end this section, we give a proof of Theorem 1 which differs somewhat from Kelarev 's and can readily be modify to established Theorem 3 below. First, we extend our earlier notation. Let X ~ S and let a E R. Then we write
._ {2:xEY ax
ax·-
0
if Y #- ;, ifY = .
"
where Y denotes Xnsupp(a). Let Ra be semiprime for all GEMs and let A be a nonzero ideal of R. To show that R is semiprime, we prove that A2 #- O. By Lemma 1, there exist
142
e = e 2 E S and a E A such that e E supp(a)
~
GUT,
where G := He and T := eSe",Pe. Thus, since G n T = j,
(1)
a = aG +aTj
and aG i- 0, since a e i- O. Now the right ideal aGRc of RG is non zerOj for otherwise, if 1 denotes the principal ideal of RG generated by aG then 12 = 0, contrary to the hypothesis that RG is semi prime. Hence, again since RG is semiprime, (aG RG) 2 i- 0 and so there exist u, v E RG such that aGuaGv
i- O.
(2)
We note also that, from (1), auav = aGuaGv
+ aGuaTV + aTuav.
The following elementary facts are easily established, where G and Tare as above and Q := e~He.
TS ~ Q {
(4) (5)
GQ~Q
Now, from (4),
(6)
aTuav E RTR ~ RTs ~ RQ.
Similarly, aTv E RQand so, from (5) , aGuaTV E RGRQ ~ RGQ ~ RQ .
But aGuaGv E RGand RG
n RQ
= O.Hence, from (3), (6) and (7),
(auav)G
and so, by (2), auav 5
(7)
i- O.Since au, av
= aGuaGv E Athis shows that A2
i- O.
Primeness and right primitivity
We now outline some recent work by the author [8] that complements Kelarev's work by extending the previous results on primeness and right primitivity of inverse semigroup rings to the case of a ring faithfully graded by a bisimple inverse semigroup. Some preliminary remarks may help to set the background. First, for a free inverse semigroup Sof infinite rank and an arbitrary field F, the semigroup
143
ring F [S]is right primitive [3]. This shows that the the bosimplicity of S is not a necessary condition for a ring faithfully graded by an inverse semigroup to be right primitive (nor, of course, to be prime) . Next, let R be a ring faithfully graded by a bisimple inverse semigroup S and let G, HEMs. While necessarily G ~ H, it may happen that RG ~ RH. As an illustration, take 8 to be bicyclic semigroup, take A to be a polynomial ring in one indeterminate over a field and take B to be the ideal of A consisting of all polynomials with zero constant term. Let e denote the the identity of 8 and let R := Ae EB T, where T := EBxE~eBx. Clearly R is a subring of A [8] . It is, in fact, a faithful 8-graded ring, with Re = Ae and Rx = Bx for all x E ~e. Now all the subgroups of 8 are trivial; and Re ~ A. But if f is any idempotent in ~e then Rf = Bf ~ B and, since B ~ A, it follows that Rf ~ Re . The results in this section make use of the following lemma derived from Lemma 1. Note that the restriction to the bisimple case leads to achange of quantifier in the conclusion. As in the previous section, the lemma contains almost all the semigroup theory that is needed for the theorems. Lenuna 2. Let 8 be a bisimple inverse semigroup, let R be a faithful 8graded ring and let A be a nonzero ideal of R. Then, for all e = e 2 E 8, there exists a E A such that
e E supp (a)
~
He U (e8e',Pe).
From this, we first obtain the theorem below concerning primeness. Theorem 3. Let 8 be a bisimple inverse semigroup and let R be a faithful 8 -graded ring. If RG is prime for some GEMs then R is prime. We sketch the start of the proof. Suppose that RG is prime for some GEMs. Let A and B be nonzero ideals of R. It suffices to show that AB # O. Let e be the identity of G. By Lemma 2, there exist a E A and b E B such that
e E supp(a)
~
e E supp (b)
GUT,
~
GUT,
where T := eSe,,"Pe . Thus
a = aG
+ aT,
b = bG
+ bT ,
aG
#0,
bG
#0.
The next step is to show that aGRGbGRG # 0 and the remainder of the argument is similar to that given earlier for Theorem 1.
144
To complete the picture , we turn to right primitivity. It is convenient here to make a further definition. Definition. A ring R is right inclusive if and only if, for all a E R , a EaR. Clearly, every ring with a (right) unit is right inclusive. Below, we shall be concerned with right primitive right inclusive rings. First, we note that the semigroup ring F [S] of a free semigroup S (not inverse!) of rank 2 over a field F is right primitive, but not right inclusive. We also note that the ring of all linear transformations of finite rank (written as right operators) of an infinite-dimensional vector space over a field is a right primitive right inclusive ring that has no right unity. The final theorem seems technically trickier to prove than its predecessors, at least by methods used in [5] and [8]. Theorem 4. Let S be a bisimple inverse semigrvup and let R be a faithful S -graded ring. If RG is right primitive and right inclusive for some GEMs then R is right primitive. At present, I do not know if this remain true without the hypothesis that RG is right inclusive.
6
Converses?
The converses of all four theorems are false. In fact , this can be demonstrated by a single example! As shown by the author in [6]' for a given prime p there exists a bisimple inverse monoid S , with group of units G, such that the following hold: (i) for any field F , F [S] is right primitive, (ii) for any field F of characteristic p, F [G] is not primitive.
Acknowledgment This article is based on a lecture given at the International Conference on Semigroups held in the Universidade do Minho in June 1999. I wish to record my thanks to the members of the Mathematics Department for their kindness and generous hospitality.
145
References
1. B.A. Barnes, Representations of the It-algebra of an inverse semigroup, Trans. Amer. Math. Soc. 218 (1976),361-396 2. M. Cohen and M. S. Montgomery, Group-graded rings, samsh products and group actions, Trans. Amer. Math. Soc. 282 (1984),237-258. 3. M. J. Crabb and W. D. Munn, On the algebra of a free inverse monoid, 1. Algebra 184 (1996), 297-303. 4. O. I. Domanov, On semisimplicity and identities of inverse semigroup algebras , Rings and Modules, Mat. Issled. Vyp. 38 (1976), 123-137. 5. A. V. Kelarev, Semisimple rings graded by inverse semigroups, J . Algebra 205 (1998),451-459. 6. W. D. Munn, Two examples of inverse semigroup algebras, Semigroup Forum 35 (1987), 127-134. 7. W. D. Munn , On the contracted semigroups rings, Proc. Roy. Soc. Edinburgh, Sect. A 115 (1990), 109-117. 8. W . D. Munn, Rings graded by bisimple inverse semigroups, Proc. Roy. Soc. Edinburgh, Sect. A (to appear).
VARIETIES OF BANDS MARlO PETRlCH The description of the lattice of varieties of bands is probably the best single achievement in the theory of varieties (with possible modification as pseudo, existence, ... ) of semi groups (and their variants). It was determined by Biryukov, Gerhard and Fennemore (within a short period of time) . Only the last of these authors provided a system of identities for bases of these varieties. Their proofs are long and involved. This subject was "revisited" by Gerhard and myself with a more transparent proof, a new system of identities and an explicit solution of the word problem for free objects in proper varieties of bands. Polak derived the lattice of band varieties from his theorem on varieties of completely regular semigroups. Various aspects of the varieties of bands and their lattice were studied by many researches. They included word problems for free objects in all band varieties, relationship with Malcev products, construction of bands belonging to specified varieties, quasivarieties of (special) bands, groupoids of varieties and some particular quasivarieties, monoid and *-band varieties, relatively free bands, and relationship of bands with formal languages. Many, but not all, of these results are complete with considerable esthetic apeal.
1
Introduction
The description of the lattice of varieties of bands was a significant achievement attained in a short time span by Biryukov [2] , Gerhard [6] and Fennemore [4]. These solutions comprise a system of invariants for these varieties: in addition, Fennemore [4] constructed a set of identities which determine all the varieties of bands. This feat was somewhat marred by long and complicated proofs it entailed. In sections 2-7, we outline a new solution of the problem of constructing the lattice of varieties of bands. It is based on new simple invariants and identities for each band variety. The solution, due to Gerhard and Petrich [10], is considerably shorter and more transparent than the previous ones. This reference also contains a formula which converts the Fennemore system of identities into the new one. Word problems, additional information, quasivarieties, varieties of band monoids and of *-bands are discussed in the remaining sections. 2
Notation
We will use the following symbolism. X - a fixed countably infinite set. Elements of X are called variables. F - the free semigroup on X . Elements of F are called words. They are finite 146
147
strings of elements of X written as XIX2 ...Xn where Xl, X2, ... , Xn E X. The product is concatenation. c (w) - the content of w E F is the set of variables occurring in w. # (w) - the number of elements in c (w), that is the number of distinct variables occurring in w. w - the dual of w is the word obtained from w by reversing the order of variables. That is , if w = XIX2 ... Xn with Xl,X2, ... ,Xn E X, then w = XnXn-l ... XI. Let w = uxv where c (w) = c (ux) and c (w) i:- c (u). Then s (w) = u - the longest left cut of w which contains all but one of the variables of w. The definition is to include s (xm) = 0, the empty word. Hence s is a mapping from F onto X*, the free monoid on X ; s (w) is the start of w. (J' (w) = X the last variable to occur in w in order from the left. Note that
c(w) = c(s (w)) U {a(w)}. e (w) = s (w) , c: (w) = (J' (w) or are defined dually to s and e (w) is the end of w. [Ua = va] - the Va. The defining
V=
[Ua = va]
3
Invariants
(J',
respectively;
variety of bands determined by the family of identities Ua = identity for bands x 2 = x is consistently omitted.
if V= [Ua = Va]. the lattice of varieties of bands. U = V ~ P = q - implication of identities. Most proofs and definitions are by induction. We also make extensive use of the left-right duality.
e (8) -
For t E {h, i, Ti, ~} and n ~ 2, let tn (0) = 0. For w E F, set h2 (w) = h (w) , the first variable in w (called the head of w), i2 (w) = i (w), the variables of w written in the order of first occurrence (called the initial part of w) tn (w) = tn (w) for n ~ 2, t E {h, i} , and define inductively (J' (w) tn-l (w) for n ~ 3, t E {h, i} . The definition of tn (w) harbors two inductive steps: the first induction on the index n and the second induction on # (w) . Each step of the first induction requires an inductive proof performed by means of the second induction. Six technical lemmas concerning various properties of these invariants must first be established for later use. Note that tn and tn are transformations of X*.
tn (w) = tns (w)
148
4
Identities
Except for a few varieties at the bottom of the lattice, each variety of bands is determined by an identity which can be derived from those of the form G n = Hn and G n = In for n 2: 2, where the words G n , Hn and In are defined as follows. Inductively, we define
G2 = X2 Xt, H2 = X2, h = X2 XI X2, Gn = XnGn-I, Tn = GnXnTn-l for T E {H, I} and n 2: 3. A couple of simple lemmas concerning these words are useful in the sequel.
5
Implications
Further three key lemmas describe which identities imply and are implied by those of the form G n = Hn and G n = In . The first : Lemma. Let n 2: 2, u, v E F, t E {h, i} . Then
Gn
= Tn =} u = v if and only if tn (u) = tn (v).
The second lemma describes implications among identities of the form G n = Hn and G n = In· For example, G n = Hn =} G n = In, G n = Hn =} G n +1 = Hn+l' etc. The third lemma is a kind of dual of the first: Lemma. Let n 2: 2, u, v E F.
(i) u = v=} G n = In if and only if h n (u) =l-hn (v). (ii) u = v =} G n+1 = Hn+1 if and only if In (u) =I- In (v). 6
The lattice
The next three lemmas describe the varieties in the first three columns of Diagram 1. Their proofs rely heavily on the implications of the identities just discussed. Lemma. Let n 2: 2, u, v E F.
(i) [u
= v] = [G n = In]-¢:=:> in (u) = in (v), h n (u) =l-hn (v).
(ii) [u = v] = [Gn+1 = Hn+1 ]-¢:=:> hn+l (u) = hn+1 (v), In (u) Lemma. Let u, v E F. (i) If c (u) =I- c (v), hdu) = h2 (v) and h2 (u) =l-h2 (v), then
[u (ii) If c (u)
=I- In (v).
= v] = [axy = ayx] /\ [a = axa] = [G 2 = H2]'
= c (v),
h2 (u)
= h2 (v),
i2 (u)
=I- i2 (v) and h2 (u) =I- h2 (v),
149
then [u
= v] = [G 2 = H 2] V [xy = yx] = [axy = ayx] = [G3 = H3] 1\ [G2 = 12] = [G3G2 = H3h] .
(iii) If n ~ 2, hn+1 (u) Zn (u) =1= Zn (v), then [u
hn+l (v), in (u)
=1=
=
in (v), lin (u)
=
lin (v),
= v] = [Gn = In] V [Gn = Hn] = [GnXn+1Gn = InXn+1Hn] = [Gn+l = In+l] 1\ [Gn+ 1 = Hn+1] = [Gn+1Gn+l = In+1Hn+l] .
(iv) If n ~ 3, h n (u) = h n (v), in (u) lin (u) =1= lin (v), then [u
= v]
=
[Gn = Hn] V [Gn-l
=
=1=
in (v), Zn-l (u) = Zn-l (v),
In-l]
= [GnXn+lxnGn-l = HnXn+1Gnxnln-l]
=
[Gn+l
= H n+1] 1\ [Gn = In] = [Gn+1Gn = Hn+1lnJ
.
The third lemma has a similar form. 7
Conclusion
We now put together the information already gained. Lemma. Let u, v E F. Then u = v is equivalent to exactly one identity in Diagram 1. We thus reach the desired goal in Theorem. Every variety of bands is one based. The lattice of varieties of bands is depicted in Diagram 1. Following Petrich [18], we now give an alternative system of words which can be used to provide bases for band varieties, see Diagram 2. Let and for n > 2, define inductively
Tn = Rn-lxnTn-l
(TE {P,Q,R}).
In comparison with the system of words G n , H n , In, which is given by two formulas, the words Pn , Qn, Rn are defined by means of a single formula. The length of the words in the new system is somewhat larger than in the old one. We also let
Tn = [Tn = Rn] Tn = [G n = Tn]
(T = P, n ~ 3 and T = Q, n ~ 2),
(T
= H,
n ~ 3 and T
= I,
n ~ 2)
150
Theorem. For n ~ 3, 7-l n = 'Pn and for n ~ 2, In = Q... The proof is based on the solution of the word problem for free objects in the respective varieties. For meets, we have Theorem. (i) For U, V E {P,Q}, n ~ 3 and U
Un
= V = Q,
n ~ 2, we have
n Vn = [Vn = Vn] = [VnVn = Rn] .
(ii) For n ~ 3, we have 15n n Q..-l = [Pn-1XnQn-l = Rn). We also have In = Qn for n ~ 2 and 7-l n = 15n for n ~ 3. From the diagram, we see that joins can also be obtained as suitable meets. These formulas are simpler than those for the old system. We can also use this system for *-bands. 8
Word problems and free objects
As a by-product, we have the following useful information. Proposition. Let n ?: 2 and t E {h, i}. The following gives a solution of the word problem for free objects in the indicated variety:
(u,v E F) . Thus tn (F) with multiplication tn (u) 0 tn (v) = tn (uv) is a [G n = Tn)-free object on X. The word problem for the variety of semilattices is given by u '" v {:} c(u) = c(v)
(u,v E F)
whence one obtains the familiar copy of the free semilattice on X as the set of all non-empty finite subsets under union. Various solutions of the word problem for free bands were devised by Green and Rees [14), Siekmann and SzabO [22) and Gerhard and Petrich [9). The last one may be described as follows. On F define a function b inductively by b (x) = xx for x E X, b (w) = bs (w) a (w) e (w) be (w) for wE F\X. On b (F) define a multiplication by u . v = b (uv) . Theorem. The function b is a homomorphism of F onto b (F). The congruence induced by b on F gives the word problem for the variety of bands:
u '" v {:} b(u) = b(v). Hence b (F) is a .free band on X.
151
9
Additional information concerning C (8)
The following is taken form Gerhard and Petrich [11]; for an alternative treatment, consult Polak [20]. Let G (X2) = X2 and for n 2: 3, inductively define G (X2' X3, ... , xn) = xnG (X2' X3, ... , Xn-l) . Note that G n = G (X2Xl, X3, ... , x n ) . Let Co be the greatest congruence on a band B contained in the Green relation C. Theorem. For n 2: 2, the following conditions on a band B are equivalent. (i) B satisfies G n = Hn. (ii) BICo satisfies xya = yxa if n = 2 and Gn - 1 = Hn-l otherwise. (iii) B satisfies w = h n (w) for any w E F. (iv) If DX2 2: DX3 2: ... 2: D xn , XICX3 if n is even and xl'R.x3 if n is odd, then
There is an analogous result for G n = In, n 2: 2. Part (iv) of the above theorem can be used to describe which bands satisfy the above conditions in terms of structure mappings figuring in the standard construction of an arbitrary band. In [19], Polak deduces the form of the lattice of varieties of bands from a general structure theorem for varieties of completely regular semigroups. From his description and an expression for a basis of the system of identities of the maximum variety of bands with the same left trace as the given variety, one can easily derive that the varieties in the extreme left columm of Diagram 1 are determined by the identities G n = Hn and G n = In, as explained earlier. In fact , the left trace classes can be computed directly by using the Malcev product; alternatively, they may be obtained from part (ii) of the above theorem. Gerhard [8) constructed a set K of subdirectly irreducible bands such that for any n 2: 2 and T E {H, I}, there is a band in K which generates [G n = Tn]· FUrther information concerning subdirectly irreducible bands, in particular relative to join of varieties, can be found in Gerhard [7] . The set C (8) can be given the Malcev multiplication: U * V is the variety of bands generated by the Malcev products U 0 V where U E U and V E V (recall that B E U 0 V if and only if there exists a congruence p on B whose classes are in U and B I p E V). Sukhanov [23] described the resulting groupoid as well as some of its properties.
152
10
Quasivarieties
These are precisely the clases determined by a set of implications. For the variety N8 of normal bands, they were described by Gerhard and Shafaat [13]; for a different proof, see Petrich [16]. This lattice is depicted in Diagram 3. The description of the corresponding Malcev groupoid, as well as many of its properties, can be found in Gerhard and Petrich [11]. Whereas this groupoid is a 13-element monoid, Sapir [21] proved that, with very few exceptions, each variety of bands is generated by continuum many quasivarieties. 11
Varieties of band monoids
Wismath [24] proved that the mapping V~VnM
(V E
.c (8)),
where M is the class of all band monoids, is a homomorphism of .c (8) onto the lattice of varieties of band monoids. She also described the lattice of pseudovarieties (in the sense of Eilenberg) of band monoids. The classes of the congruence on .c (8) induced by the mapping above is indicated in Diagram 4 by broken lines. The lattice of varieties of band monoids is depicted in Diagram 5. Even though we have enriched the bands with the nullary operation of identity, no "new" varieties have been created this way. Wismath's proof follows the description of .c (8) given by Fennemore [4]; in particular she uses his system of identities. 12
Varieties of *-bands
We now enrich the bands with an involution, that is a unary operation * which is an antiautomorphism of order at most 2. If this involution is regular, that is satisfies the identity x = xx·x, we speak of a *-band. Adair [1] described the lattice of varieties of *-bands. Petrich [17] contains yet another system of identities for *-bands and a proof that the system indicated in Diagram 6 also represents such a system. Hence no star is necessary to describe the varieties of *-bands. In fact, here too there is a mapping (which is a meet homomorphism) of .c (8) onto the lattice of varieties of *-bands. The classes of the equivalence relation induced on .c (8) by this mapping are indicated in Diagram 4 by heavy lines. The lattice of varieties of *-bands is decipted in Diagram 6. A solution for the word problem of free *-bands can be found
153
in Gerhard and Petrich [9J . Diagram 7 represents the lattice of varieties of *-band monoids. As in the preceding section, but here somewhat more surprizingly, the addition of a unary operation of a regular involution does not create "new" varieties of *-bands. Adair also follows the description of the lattice and the determining identities of Fennemore [4J.
G 3 G2 =H3 H 2 G2X3G2 = H2 x 3 G 2
Diagram 1: The lattice of varieties of bands
154
P3 = P3 ( P3 P 3'lf.= R3 )
Diagram 2: The lattice of varieties of bands
155
x=x
ax = ay
=?-
xy = xyx
xa xax
= ya =?- yx = xyx
= yay =?- xy = yx xya =yxa
axy = ayx
xa
= ya =?- xy = yx
ax
ax =a
= ay =?- xy = yx
xa=a xy =yx
x=y Diagram 3: Quasivarieties of normal bands
NB = [axya = ayxa]
156
X=X
•
v
Diagram 4: The equivalence classes for ..... . band monoids _______ *-bands
157
X=X
•
G 3 =13 G3G3 = 13 13 G2 x 3G 2 = 12 X 312
X=y
Diagram 5: The lattice of varieties of band monoids
158
X=X
•
Gs =Hs
x =xyx
G 2 =H2 Diagram 6: The lattice of varieties of *-bands
159
x=y
Diagram 7: The lattice of varieties of *-band monoids
References
1. Adair, C. L. , Bands with an involution, J. Algebra 75 (1982) , 297-314. 2. Biryukov, P. A. , Varieties of idempotent semigroups, Algebra i Logika 9 (1970) , 255-273 (Russian) . 3. Cirie, M. and Bogdanovie, S. , The lattice of varieties of bands, Semigroups and Applications, Proc. Conf. St. Andrews 1997, World Scientific, Singapore (1998) , 47-61. 4. Fennemore, C. F., All varieties of bands, Math. Nachr. 48 (1971) ; I , 237-252; II, 253-262. 5. Fennemore, C. F., Characterization of bands satisfying no non-trivial identity, Semigroup Forum 2 (1971) , 371-375. 6. Gerhard, J . A., The lattice of equational classes of idempotent semigroups, J . Algebra 15 (1970) , 195-224. 7. Gerhard, J. A., Subdirectly irreducible idempotent semigroups, Pacific J. Math. 32 (1971) , 669-676. 8. Gerhard, J. A., Some subdirectly irreducible idempotent semigroups, Semigroup Forum 5 (1973) , 362-369. 9. Gerhard, J. A. and Petrich, M., Free bands and free *-bands, Glasgow Math. J. 28 (1986) 161-179. 10. Gerhard, J . A. and Petrich, M. , Varieties of bands revisited, Proc. London Math. Soc. (3) 58 (1989) 323-350.
160
11. Gerhard, J. A. and Petrich, M., Characterizations of varieties of bands, Proc. Edinburgh Math. Soc. 31 (1988) 301-319. 12. Gerhard, J. A. and Petrich, M., The Malcev product of quasivarieties of normal bands, Archiv fur Math. 52 (1988) 140-149. 13. Gerhard, J. A. and Shafaat, A., Semivarieties of idempotent semigroups, Proc. London Math. Soc. 22 (1971) 667-680. 14. Green, J. A. and Rees, D., On semi-groups in which xr = x, Proc. Cambridge Phil. Soc. 48 (1952) 35-40. 15. Neto, O. and Sezinando, H., Trees, band monoids and formal languages, Semigroup Forum 52 (1996) 141-155. 16. Petrich, M. , Lectures in semigroups, Akademie-Verlag, Berlin (1977). 17. Petrich, M. , Identities without the star for *-bands, Algebra Universalis 36 (1996) 46-65. 18. Petrich, M. , New bases for band varieties, Semigroup Forum 59 (1999), 141-151. 19. Polak, L. , On varieties of completely regular semigroups III, Semigroup Forum 37 (1988), 1-30. 20. Polak, L. , Structural theorems for varieties of bands, Lattices, Semigroups and Universal Algebra, Proc. Int. Conf. Lisbon 1988, Plenum, New York (1990) 211-223. 21. Sapir, M.V., On the lattice of quasivarieties of idempotent semigroups, Mat. Zap. Ural. Univ. 11 (1979) 158-169 (Russian). 22. Siekmann, J. and SzabO, P., A noetherian and confluent rewrite system for idempotent semigroups, Semigroup Forum 25 (1982) 83-110. 23. Sukhanov, E. V. , The groupoid of varieties of idempotent semigroups, Semigroup Forum 14 (1977) 143-159. 24. Wismath, S. L., The lattice of varieties and pseudovarieties of band monoids, Semigroup Forum 33 (1986) 187-198. 25. Wismath, S. L., The lattice of varieties of *-regular band mono ids, Semigroup Forum 46 (1993) 130-133.
CHARACTERIZATION OF A SEMIDIRECT PRODUCT OF GROUPS BY ITS ENDOMORPHISM SEMIGROUP PEETER PUUSEMP Department of Mathematics of Tallinn Technical University, Ehitajate tee 5, 19086 Tallinn, Estonia E-mail: [email protected] The sernidirect product G = H >-- ((GI X .. . x G n ) >-- K) of groups, where < G i , K >= G i >-- K (i = 1,2, . . . , n), is characterized by the properties of the sernigroup End G (Theorem 2.1) . This characterization makes it possible to give conditions for the summability of orthogonal idempotents of the sernigroup End G for an arbitrary group G (Theorem 3.3). This leads to the following result. Let G and G be groups such that the sernigroups End G and End G are isomorphic: cP : End G --+ End G, cp - isomorphism. If Xl, . . . , Xn are orthogonal and summable idempotents of EndG, then the idempotents XICP, . .. , Xncp of EndG are also summable and (Xl + . . . + xn)CP = Xl cP + ... + Xncp (Theorem 3.4) .
1
Introduction
Let G be a group. If G = K H , where K and H are subgroups of G such that H is normal in G and K n H =< 1 >, then we write G = H A K and say that G is a semidirect product of Hand K. In this paper we shall characterize a seruidirect product G = H A ((GI
X ...
x Gn
)
A K),
(1)
where < G i , K >= G i A K (i = 1, 2, .. . , n), by the properties of suitable idempotents of the seruigroup End G of all endomorphisms of G (theorem 2.1). As a corollary of this characterization, we establish that if the group G decomposes into a direct product G = G I X ... x G n and its endomorphism seruigroup EndG is isomorphic to the endomorphism seruigroup EndH of another group H , then the group H decomposes into a direct product H = HI X ... x Hn such that the seruigroups EndGi and EndHi are isomorphic for each i E { 1, 2, ... , n} (theorem 3.2). In order to characterize further results of our paper, let us give a definition. If x, y E EndG, then the map x +y : G ~ G , defined by g(x +y) = (gx)(gy), 9 E G , is an endomorphism of G if and only if gh = hg for each 9 E 1m x and hElm y. In this case we say that the endomorph isms x and yare summable [1]. The characterization of the decomposition (1) , given by theorem 2.1 , makes it possible to describe the summability of orthogonal idempotents of EndG by properties of the seruigroup EndG. This is done 161
162
in theorem 3.3. We say that the endomorphisms x and y are orthogonal if xy = yx = O. Similar problems of summability were studied for arbitrary semigroups and algebraic categories in [2,3]. We shall use the following notations: G - a group; J( G) - the set of all idempotents of EndG; K(x) = {y EEnd G I yx = xy = y}; K(x)* the group of all invertible elements of the semigroup K(x) with identity x, where x E J(G); C(x) = {y EEnd G I yx = xy}; VK(x). (y) = {z E K(x)* I zy = y} ; Cn - a cyclic group of the order n; g an inner automorphism of G, generated by 9 E G. We shall write a mapping on the right of the element on which it acts. The following four lemmas are useful in the proofs of our results. We do not prove them, because everybody can do it as an easy exercise. Lemm.a 1.1. If x, y 1m x, (Kerx)y C Kerx.
E EndG and xy
=
yx, then (lmx)y
c
Lemma 1.2. If x, y E EndG and yx = x, then g-l(gy) E Kerx for each gEG.
Lemm.a 1.3. If x E J(G), then G = Kerx A Imx and Imx = {g E G I gx = g} . Lemm.a 1.4. If x E J(G), then the subset K(x) = {y E EndG I yx = xy = y} is a subsemigroup with the unity x of End G which is canonically isomorphic to End (1m x). In this isomorphism the element y of K (x) corresponds to its restriction to the subgroup 1m x of G (later we shall identify y with its image under this isomorphism). 2
Main Theorem
Theorem 2.1. Let G be a group and n be an integer, n ~ 2. Suppose that G is decomposed into a semidirect product as in (1), where
< G i , K >= G i A K (i = 1,2, ... , n). Denote by x and
Xi
(2)
the projections of G onto K and G i A K, i.e., n
Imxi = G i A K, Kerxi = H A n
Gj
,
(3)
j=l,#i
Imx=K, Kerx=HA(G1X ... xGn ),
(4) (5)
163
Then idempotents x, Xl, ... ,
Xn
of End G satisfy the equations
XiXj = XjXi = X
(i =f: j)
(6)
and the following property (P): for each i, j E {l, 2, ... , n}, i =f: j, there exists Zij = Zji E J(G) such that 1° Xi, Xj E K(Zij), 2° there exists an unique pair Vi, ltj of subgroups of K(Zij)* with properties (i) Vi C C(Xi), ltj C C(Xj), (ii) Vixi = VK(Xi)'(X), ltjXj = VK(Xj)'(x), (iii) XiVXi = Xi for each v E ltj, (iv) XjUXj = Xj for each U E Vi. Conversely, suppose that there exist idempotents x, X!, .. • , x" of EndG, satisfying the equations (6) and the property (P). Then the group G decomposes into a semidirect product (1), where the equations (2)-(5) are true. Proof. Assume that the group G decomposes as shown in (1) and (2). Denote by X and Xi the projections of G onto K and G i A K, respectively (i = 1, ... ,n). Evidently, the equations (3)-(6) are true. Let us prove the property (P). Clearly, it is sufficient to do it only for i = 1 and j = 2. It follows from the assumptions that the group G decomposes as follows
Denote by Z12 the projection of G onto its subgroup (GI x G2) A K, i.e., Imzl2
= (GI
x G2) A K,
Kerzl2
=H
A (G 3 X
...
x Gn ) .
(7)
Using equations (3), (5) and (7), it is easy to check that XIZl2 = Zl2XI = Xl and X2Z12 = Zl2X2 = X2. Therefore, Xl, X2 E K(ZI2) and the proposition 1° is true. Denote by \Ii the set of all maps rjJ : G ---> G, where
I '1j; E VK (X2)'(X);
(ghc)if; = (g'1j;)h, 9 E G 2 A K,
hE GI, c E KerzI2}. Then V2 is a subgroup of K(ZI2)*, too. Our aim is to show that \Ii and V2 satisfy the properties 2° (i)-(iv). If
= hXI = 1,
= g
g(XI rjJ)
= 1,
= grjJ = g
h(XI
= 1
164
Therefore, CPXl = cP = X1CP, Vi c C(Xl) and VK (Xl)o(X) = Vixl. Similarly, V2 C C(X2) and VK (X2)O (x) = V2X2. Properties 2° N and 2° (ii) are proved. Assume that v E V2 . By definition of V2, v = 't/J for some 't/J E VK (X2)o(X). Choose 9 E Imxl. Then 9 = glk for some gl E Gl and k E K. Since 't/J E K(X2)*' 't/Jx = x and Imx2 = G 2 A K, then (Imx2)'t/J C Imx2, k- l . (k't/J) E Kerx n Imx2 = G 2 and k't/J = kg 2 for some g2 E G 2 C Kerxl· Therefore,
g(X1VX!)
= g(VX1) = (glk)(vxd = (glk)(ijJxl)
= gl((kxd(92Xd)
=
= gl(k ·1) = glk = 9 = gXl·
Consequently, Xl VXl = xl, and 2° (iii) holds. Similarly, property 2° (iv) also holds. Next we shall show the uniqueness of the subgroups Vi, V2 of K(z12)* with properties 2° (i)-(iv). Assume that Wl, W2 is another pair of subgroups of K(Z12)* such that conditions 2° (i)-(iv) hold, i.e.,
Wl
C
C(Xl), W2
c C(X2),
(8)
(9) (10)
Xl VXl = Xl for each v E W2,
(11) Let us show that Vi = Wl and V2 = W 2. Choose wE Wl. Then WZ12 = Z12W and, by (8), WXl lemma 1.1,
G2W = (Kerxl
n Imz 12 )W C
Kerxl
= X1W'
In view of
n Imz12 = G2.
Therefore, if h E G 2, then hw E G 2 and, by (11),
h
= hx2 = h(X2WX2) = h(WX2) = (hW)X2 = hw.
Write cP = WX1· In view of equations (9), cP E VK (Xl)O (x). Consequently,
= (gh)w = (gw)h = (g(xlw))h = (gcp)h = (ghc)cp A K = Imxl, hE G 2 and c E Kerz12. Hence, W = cP E Vl.
(ghc)w
for each 9 E G l As W is an arbitrary element of W l , then W l C Vi, Choose
U
E Vi, Then
165 U = rp for some cp E VK(x.).(x). By construction of rp, rpXl = cpo It follows from equations 2° (ii) and (9) that UXl = rpXl = cp = WXl for some wE W l . It follows that W = rp. Hence, U = W E W l • Therefore, Vi C W l . Consequently, Vi = W l · Similarly, V2 = W2. The uniqueness of Vi and V2 is proved. So the first part of the theorem is proved.
Let us start the proof of the second part of the theorem. Suppose that there exist idempotents x, Xl, ... , Xn of EndG, satisfying equations (6) and the property (P). We shall show that for the group G the semidirect decomposition (1), where the subgroups H, K, Gt. ... , G n of G satisfy equations (2)-(5), holds. It follows from equation (6) that XiX = XXi = X, Kerxi C Kerx and Imx C Imxi (i = 1, 2, ... ,n). By lemma 1.1, (Kerx)xi C Kerx and (Imxi)X C Imxi. Therefore, Kerx = Kerxi A (Kerx n Imxi), Imxi = (Imxi n Kerx) A Imx. Let K = Imx, G i = Kerx n Imxi. Then ImXi = G i A K,
Kerx = Kerxi A G i .
Choose another subscript j E { 1, 2, ... , n}, j
=f i.
Since XiXj = XjXi =
X and Kerxi C Kerx, then
< 1 >=
Gj
(Kerx
n ImXj)x
= (Kerx n ImXj)XjXi =
= Kerx n Kerxi n ImXj = Kerxi n ImXj.
By lemma 1.1, (Kerxi)Xj C Kerxi. Hence, Kerxi = (Kerxi
n Kerxj) A (Kerxi n ImXj)
= (Kerxi
n Kerxj) A
Gj .
Therefore,
(12) If we continue this process, then we obtain a semidirect decomposition
where
7r
is an arbitrary substitution of integers 1, 2, ... , Imx.".(n) = G.".(n)
A K,
n
and
166
Kerx.".(n) = ( ... ((ni=l ,i¢.".(n) Kerxi) A G.".(l)) A ... ) A G.".(n-l)'
The proof of the theorem will be finished, if we show that
< G i , Gj
>=G i
X
Gj ;
i,j = 1,2, .. . , n; i i j .
(14)
Let us prove equations (14). By the property (P), there exists Zij = Zji E J(G) such that 1° and 2° hold. As Xi, Xj E K(Zij) , then XiZij = ZijXi = Xi, XjZij = ZijXj = Xj and G j , G i , K C Imzij , KerZij C KerxinKerxj.
It follows from here and (12) that
Imz ij =
=
(((Imzij n Kerxi n Kerxj) A Gj) A Gi) A K =
(((Imz ij n Kerxi n Kerxj) A G i ) A G j ) A K =
where H o = Im zij n Kerxi n Kerxj. Further we shall identify the elements of K(Zij) with the corresponding elements of End(Imzij) in the canonical isomorphism K (Zij) ~ End(Im Zij). Define for each cp E VK(x,)O (x) an endomorphism tjJ of G by the equations gtjJ = gcp, 9 E Imxi = G i A K, htjJ = h,
hE Ho A G j .
Clearly, tjJ is well defined and the set Vi = { tjJ I CP_ E VK(X;) ° (x) } is a subgroup of K(Zij)* . Similarly, we construct a subgroup Yj of K(Zij)*: Vj
= {1[; I 'Ij; E VK(Xj)O (x)},
9 E ImXj = Gj A K, h1[; = h,
g1[;
= g'lj;,
hE Ho A G i .
By construction,
Vi
C C(Xi)' Vj C C(Xj), Vixi
= VK(x,)o(x),
VjXj
= VK(Xj)o(x).
Hence, the subgroups Vi and Vj of K(Zij)* satisfy the properties, similar to 2° (i) and 2° (ii). Suppose 1[; E Vj, 'Ij; E VK(Xj)o(x). By lemma 2.2, g-l(g'lj;) E Kerx
(16)
for each 9 E G. Therefore, k-1(k'lj;) E KerxnImxj = G j for each k E K and, in view of the inclusion G j C Kerxi, (k-l(k'lj;))Xi = 1, kXi = k('Ij;xi).
167
It follows now that for each 9 have
g(XdJXi) =
= kgi E ImXi = G i
A K, k E K, gi E Gi, we
= g(~Xi) = (kgi)(~Xi) = ((k'I/J)gi)Xi
(k('l/Jxi))(giXi)
=
= (kXi)(giXi) = (kgi)Xi = gXi,
i.e., Xi~Xi = Xi for each ~ E~. Similarly, XjtpXj = Xj for each tp E K Therefore, the subgroups Vi and ~ of K(Zij)* satisfy properties 20 (iii) and 20 (iv), too. Consequently, Vi = Vi, ~ = Yj. Consider the subgroup ~o =< Gj, ~ > of K(Zij)", generated by Gj = { [Ii I gj E G j } and ~. We shall show that the pair Vi, Vjl satisfies the properties corresponding to 20 (i)-(iv). Choose gj E G j . As
k(gjxj)
= kgj = k(xjgj),
9j(gjXj)
= 9jgj = 9j(Xjgj), (17)
for each h E Ho A G i , k E K, 9j E G j , then gj E C(Xj) and gjXj E K(xj)*. Applying to equations (17) an endomorphism x, we get gjXjX = x, gjXj E VK{Xj)o(x) . Therefore, in view of ~ c C(Xj) and ~Xj = VK{Xj)o(x), we obtain ~o C C(Xj),
~OXj = VK{Xj)o(x) .
It is already shown that the pair Vi, ~o satisfies the properties corresponding to 20 (i), (ii), (iv). Let us show now 20 (iii) for Vi, Vjl, i.e., we shall show that XiVXi = Xi for each v E ~o. Choose 9 E G and v E ~o. As gXi E Imxi = G i A K, then gXi = kgi for some k E K and gi E G i . By definition of ~o , the element v is a product of a finite number of elements ~ E ~ and gj, where'I/J E VK{Xj)o(x) and gj E Gj . It follows from (16) that Gj~ c G j ; kijJ = kCj, Cj E Gj . On the other hand,
Gjgj
= Gj;
kgj
= kdj ,
dj E Gj .
Hence, kv = kaj for some aj E Gj . As ~ acts identically on Ho A G i , Hoh j = Ho for each h j E Gj and, by (15), gigj = gihO for some ho E H o, then giV = gibO for some bo E Ho. Therefore,
g(XiVXi) = (kgi)(VXi) = ((kV)(giV))Xi = = ((kaj)(gibo))xi
vt
= (kXi)(giXi) = (kgi)Xi = gXi
for each 9 E G and v E Consequently, XiVXi = Xi and the pair satisfies the properties corresponding to 20 (i)-(iv).
Vi, Vjo
168
It follows from the uniqueness of pair Vi, Vj with properties 2° (i)-(iv) that Hence, [lj E ~ for each gj E Gj. By construction of~, giflj = gi, i.e., gigj = gjgi for each gi E G i and gj E G j . Consequently, < G i , G j >= Gi x Gj . The theorem is proved.
".to =~.
3
Corollaries
Suppose that the idempotents x, XI, ... , Xn of EndG satisfy equations (6) and the property (P), formulated in theorem 2.1. Then equations (1)-(5) hold. Denote by z the projection of G onto its subgroup (G l x ... x G n ) A K, i.e., G=HAImZ, H=Kerz, Imz=(Glx ... XGn)AK.
The following theorem describes the connection between the idempotents z and X, XI, ... , x n . Theorem 3.1. The set B = {y E J(G) I XI, ... , Xn E K(y)} is nonempty and there exists a unique u E B such that uy = yu = u for each y E B. This u is equal to z, defined above. Proof. By definition, the idempotents XI, .. . , Xn belong to K(z) and, therefore, z E Band B is non-empty. Assume that y E B and conclude yz = zy = z. Since y E B , then XiY = yXi = Xi for each i E {I, ... , n}. Therefore, (Imxi)Y C ImXi C Imy,
(Kerxi)y C Kerxi,
Hence, g(zy) = g(yz) = gz = 1 for each 9 E Ker z, g(zy) = g(yz) = gz = 9 for each 9 E Imxi,
(i = 1, ... , n). As the groups Ker z, ImXI, ... , Imxn generate the group G, then zy = yz = z . Consequently, yz=zy=z for each yEB.
(18)
If u is another element of B such that uy = yu = u for each y E B, then uz = zu = u. Taking y = u, it follows from (18) that uz = zu = z. Hence, z = u. The theorem is proved.
169
Theorem 3.2. Let G and G be groups such that their endomorphism semigroups are isomorphic. If G decomposes into a direct product G = G l X ... x G n of its subgroups G l , ... , G n , then there exists a direct decomposition G = G l X ... x G n of G such that the semigroups EndGi and EndGi are isomorphic for each i E {1, ... , n}. Proof. Suppose that EndG ~ EndG.
(19)
Let XI, ... , Xn be the projections of G onto its subgroups GI, ... , G n , respectively. Denote by Xl, ... , xn the images of XI, ... , Xn in the isomorphism (19). We can use for G theorem 2.1, taking there X = 0 (i.e. K =< 1 » and H =< 1 >. Therefore, idempotents X = 0, X!, ... , Xn satisfy properties (6) and (P) of the theorem 2.1. By isomorphism (19), the idempotents X = 0, XI, ... , xn of EndG satisfy similar properties. Using now theorem 2.1 for G and its endomorphisms X = 0, XI, ... , xn, we conclude that
where
R = Imx =< 1 >,
Gi ~ Imxi (i = 1, ... , n).
By lemma 1.4 and the isomorphism (19), EndG i
= End(lmxi)
~ K(Xi) ~ K(Xi) ~ End (1m Xi)
= EndGi
(i=1, ... ,n). Let us use now theorem 3.1 for G and G. As in our case Z = 1, then, by the isomorphism (19), z = I and, therefore, fI =< 1 >. Consequently,
G = Gl
X ...
x
Gn ,
where EndGi ~ EndG i for each i E { 1, .. . , n}. The theorem is proved. Note that theorem 3.2 was first proved in [4]. Taking in theorem 2.1 K =< 1 > (x = 0) and using also theorem 3.1, we immediately obtain the following two theorems. Theorem 3.3. The orthogonal idempotents Xl, ... , Xn of EndG are summable if and only if for each i, j E {1, 2, ... , n}, i i- j, there exists Zij = Zji E I( G) such that
1° Xi, Xj E K(Zij), 2° there exists a unique pair Vi, V; of subgroups of K(Zij)* with properties (i) Vi c C(Xi ), V; c C(Xj), (ii) Vixi = VK(x,)·(x), V;Xj = VK(Xj).(x),
170
(iii) IT these
XiVXi
=
Xi
X!, ... , Xn
for each v E V;, (iv) XjUXj = Xj for each are summable, then in the subset
B = {y E J(G)
I X!, ...
, Xn
U
E
Vi.
E K(y)}
of End G there exists an unique element z such that zy = yz = z for each y E B and this z is equal to the sum Xl + ... + X n . Theorem 3.4. Suppose that Xl, ... , Xn are orthogonal and summable idempotents of End G. Let G be another group such that the semigroups EndG and EndG are isomorphic: rp : EndG - - EndG, rp -isomorphism. Then the idempotents Xlrp, • .. , Xnrp of EndG are also summable and (Xl
+ ... + xn)rp =
Xl rp
+ ... + Xnrp·
References 1. Kurosh, A.G. Group theory. Moscow, Nauka, 1969. (In Russian). 2. Livshits, A.H. Direct decompositions of idempotents in semigroups. Proc. Moscow Math. Soc. 11 (1962), 37-98. (In Russian). 3. Livshits, A.H. Direct decompositions in algebraic categories. Proc. Moscow Math. Soc. 9 (1960), 129-141. (In Russian). 4. Puusemp, P. Jdempotents of the endomorphism semigroups of groups. Acta et Comment. Univ. Tartuensis, 1975, 366, 76-104. (In Russian).
GENERALIZED N-SEMIGROUPS J. C. ROSALES AND J . I. GARCiA-GARCiA Departamento de Algebra, Universidad de Granada, E-18071 Granada, Spain E-mail: [email protected]@ugr.es This paper introduces the concept of generalized N-semigroup, shows a method to obtain this class of semigroups and gives theorems of structure for them.
1
Introduction
All the semigroups, monoids and groups appearing in this work are commutative, for this reason in the sequel we will omit this adjective. An N-semigroup is an Archimedean idempotent free cancellative semigroup. Tamura [5] proved that every N-semigroup is isomorphic to a semigroup obtained as follows: Let (G, +) be a group, (N, +) be the monoid of nonnegative integers and l:GxG----+N be a mapping satisfying:
(Tl) For all 91,92 E G, 1(gl,g2) = 1(g2,gl)' (T2) For all gl , g2, g3 E G, l(gl, g2) + l(gl
+ g2, 93) =
1(g2' g3) + l(gl, g2 + g3).
(T3) For all 9 E G 1(0, g) = 1. (T4) For every 9 E G there exists k E
N\ {O} such that l(g, kg) 2: 1.
On the set N x G we define the operation
(a1 ' gI) +1 (a2' g2) = (a1
+ a2 + l(g1, g2), gl + g2);
then (N x G , + I) is an N-semigroup. The aim of this paper is to characterize the semigroups which are isomorphic to a semigroup of the form (N x G, +I) with G a group and [ : G x G ----+ N satisfying only properties (Tl) and (T2). Note that the class of N-semigroups is contained in this class of semigroups. This is the reason for referring to this new kind of semigroups as generalized Nsemigroups. We prove that the class of generalized N-semigroups is the same as the class of cancellative semigroups which are not groups and with at least one Archimedean element. Furthermore, we prove that in order to obtain 171
172
all the generalized N-semigroups, up to isomorphisms, we can impose a new condition on 1, namely, that 1(0,0) E {O, I}. Using these conditions we will be able to distinguish between semigroups with identity element (1(0,0) = 0) and semigroups without identity element (1(0,0) = 1). If 8 is a generalized N-semigroup, then so is 8 1 ; therefore we need only study generalized N-semigroups with an identity element, which we call N-monoids. 2
Generalized N-semigroups
Let (8, +) be a semigroup, we say that x E 8 is an Archimedean element of 8 iffor every y E 8 there exists k E N\{O} and z E 8 such that kx = y+z. A generalized N-semigroup is a cancellative semigroup which is not a group and which contains at least one Archimedean element. Let (G, +) be a group, (N, +) be the monoid of the nonnegative integers and [:GxG~N
be a mapping such that:
(1) For all 91 ,92 E G, 1(91,92) = 1(92,91). (2) For all 91,92,93 E G, 1(91,92) + 1(91 + 92, 93) = 1(92,93) + 1(91, 92 + g3). On N x G define the operation (al' 91)
+1 (a2' 92)
+1
by
= (a1 + a2 + 1(91,92)' 91 + 92);
then (N x G, + I) is a semigroup. Two easy facts that can be proved are that the above semigroups are cancellative (see Tamura [5]) and that (N x G, +I) is a monoid if and only if 1(0,0) = 0. Theorem 1 Every semi9roup (N x G, +I) is a generalized N -semi9roup. Proof: As the above remark says, (N x G, + I) is a cancellative semigroup. Hence, we must show that it is not a group and that it contains at least one Archimedean element .
°
• Assume that (N x G, + I) is a group. Then 1(0,0) = and (0,0) is the identity element. Since (N x G, +I) is a group, (1,0) must have an inverse. However, if (1,0) +da, 9) = (0,0), then a+ 1 +1(0, 9) = which is a contradiction .
°
• Let (a,9) EN x G. By induction on b:
b(l,O) = (b + (b - 1)1(0,0),0)
173
for all bEN. Let k,k E N\{O} be such that
a +k
+ J(g, -g)
= k
+ (k -1)J(0,0).
Then we obtain that
(a, g) +dk, -g)
= (a + k + J(g, -g), 0) = (k + (k -1)J(0, 0), 0) = k(1, 0).
Thus (1,0) is an Archimedean element.
o Now, we prove that this construction characterizes (up to isomorphisms) all the generalized N-semigroups. From now on, we assume that (S, +) is a generalized N -semigroup and that m is an Archimedean element of S. Clifford and Preston [1] proved that every cancellative semigroup can be embedded in a group. Since the condition of being a generalized N-semigroup is preserved by isomorphisms, we may assume that S is a subsemigroup of a group, this allows us to use expressions such that x - y, O· x = 0, x + 0, x - 0, etc. We define in S the binary relation
xRmy if and only if x
+ km = y + k'm for some
k, k' EN,
note that xRmy if and only if x-y E Zm. The reader will not have difficulty in proving that R1TI is a congruence on S. Hence, we can construct the quotient semigroup (S/ Rrn , +) which is a monoid because [m] is its identity element ([m] denotes the Rrn-class of m). Using this fact and that m is an Archimedean element, we obtain that this monoid is a group. Lemma 2 For every xES there exists kx = max{k E N I x - km E S}. Furthermore, if xRmy, then x + kym = y + kxm. Proof: • Since m is an Archimedean element of S, there exist k E N\ {O} and yES such that km = x + y. If there exists k > k such that x - km E S then (km - x) + (x - km) E S. Hence (k - k)m E S. But k > k and thus (k - k)m E S. Therefore 0 = (k - k)m + (k - k)m E S. Furthermore (k-k-1)m E Sand m+(k-k-1)m+(k-k)m = 0 and consequently m has ~n inverse. Let a E S. Since m is Archimedean, there exist t E N\ {O} and b E S such that tm = a + b. If c is the inverse of m then b + tc is the inverse of a. So, we have shown that S is a group, contradicting that S is a generalized N-semigroup .
• If xRmy , then there exist k, k' EN such that x + km = y + k'm, which implies x + kymRmy + kxm. Hence there exist k, k' E N such that x + kym + km = y + kxm + k'm. Suppose that k' > k. Then x - (k x +
174
k' - k)m = y - kym E S which contradicts the maximality of kx. So, k' :S k and in the same way, it can be shown that k :S k', therefore k = k'. Since S is cancellative, we obtain x + kym = y + kxm.
o Lemma 2 allows us to define the following mapping AB': SIR", --+ S, AB'([x]) = x - kxm,
where xES and [x) denotes its R",-class. Observe that !m(AB') = {s E S I s-m r:t S}. Lemma 3 FoT' every s E S there exists a unique (k,x) EN x !m(AB') such that s = km + x . Proof: Clearly, s = ksm + (s - ksm). Suppose that km + x = k'm + y with x,y E !m(AB') . If k' > k, then x - (k' - k)m = yES which contradicts kx = O. Hence, k' :S k. Similarly k :S k'. Therefore k = k' and by the cancellativity of S, x = y. 0 If x,y E S, then (x +y) - (k x + ky)m = (x - kxm) + (y - kym) E S. Thus kx+y 2: kx + kyo This fact, Lemma 2 and Lemma 3 allow us to define the following mapping:
I: SIR", x SIRm --+ N l([x), [y]) = kx+y - kx - kyo The reader will not have difficulty in proving that the mapping I satisfies:
(1) l([x), [y)) = l([y), [x]), (2) l([x), [y))
+ l([x + y), [z])
= l([y), [z))
+ l([x), [y + z)).
By Lemma 3, we know that for a given s E S there exists an unique (k,x) EN x Im(AB') such that s = km + x. So, we can define the mapping (J: S --+ N x SIRm, (J(s) = (k, [x)),
which is bijective. Assume now that (J(Sl) = (kSl' [Xl]) and (J(S2) = (kS2' [X2]) with Sl - kSlm = Xl and S2 - kS2m = X2. We know that (Sl + S2) - (kSl + kS2)m E S. Thus Xl + X2 - kXl +X2 m E S and therefore (k l
(J(SI + S2) = (k l + k2 + kX1 + X2 , [Xl + X2)) = + k2 + l(XI' X2), [Xl) + [X2)) = (kl, [Xl)) +1 (k2, [X2)) = (J(st) + (J(S2);
whence (J is a semigroup isomorphism and we obtain the following result.
175
Theorem 4 The semigroup (8, +) is isomorphic to (N x 81 Rm, + I ). Theorem 1 and Theorem 4 are structure theorems for generalized Nsemigroups. Now, we are going to improve these results. As indicated above, we may assume that 8 is a subsemigroup of a group (H, +). If 0 is the identity element of H, then either 0 E 8, or 0 rt 8. If 0 E 8, then k2m = 2, k m = 1 and 1([m), [m]) = k2m - k m - k m = 0 (observe that if -m E 8 then as in the proof of Lemma 2, we obtain that 8 is a group, a contradiction). If 0 rt 8, then k2m = 1, k m = 0 and 1([m), [m]) = k2m - k m - k m = 1. Therefore the mapping I verifies 1([m), [m]) E {O, I}. To complete this section we introduce the concept of an N-monoid, which we study. If (S, +) is a semigroup without identity element then we can make a extension of it and turn it in a monoid in the following way. We add to 8 a new element, denoted by 0, and we define 0 + 0 = 0 and 0 + s = s + 0 = s for all s E 8 . Clearly, (8 U {O}, +) is a monoid and 0 is its identity element (see Clifford and Preston [1]). As a consequence of this remark we have the following result. Corollary 5 If (8, +) is a generalized N -semigroup without identity element then (8 U {O}, +) is a generalized N -semigroup with identity element. We know restrict ourselves to generalized N-semigroups with an identity element, which we call N-monoids. Thus, an N-monoid is a cancellative monoid which is not a group and which contains at least one Archimedean element. 3
Some properties of N-monoids
Now, we focus our attention on proving some properties of N-monoids. In this section, G will denote a group and I : G x G --+ N will denote a mapping verifying: (1) For all gl,02 E G, l(gl,g2) = l(g2,gl). (2) For all gl,g2,g3 E G, l(gl,g2) + l(gl
+ g2,93) =
l(g2,g3) +1(gl,g2 + g3).
(3) 1(0,0) = O. It is easy to prove that Condition (3) is equivalent to 1(0, g) = 0 for all 9 E G. Now, we are going to embed (N x G, +I) in a group. Recall that every cancellative monoid is a submonoid of a group.
176
Proposition 6 Let Z be the set of integers. Then Z x G has structure of group with the following operation (Zl' gl) +1 (Z2' g2) = (Zl + Z2 + 1(g1, g2), g1
+ g2).
Furthermore, (N x G, +1) is a submonoid of (Z x G, +1) and every element of (Z x G, +1) is a difference of two elements of (N x G, + 1). Proof: We only have to take into account that (z, g) +J( -z - 1(g, -g), -g) = (z -z -1(g, -g) +1(g, -g), g- g) = (0,0) and that if (z, g) E Z x G, taking a, bEN such that a - b = z and the elements (a,g), (b,O) EN x G, then (a,g) - (b,O) = (a,g)
+ (-b, 0) =
(a - b,g) = (z,g).
o A consequence of Proposition 6 is that the quotient group of an N-monoid (N x G, +1) is the group (Z x G, +1). The following result is readily proved. Proposition 7 The group (Z x G, +1) is finitely generated if and only if the group G is finitely generated. Hence, we know when (N x G, +1) has a finitely generated quotient group. We say that a monoid is reduced if its only unit is its identity element. We study now in terms of G and 1 when an N-monoid N x G is reduced. The following Lemma describes how are the units of an N-monoid. Lemma 8 An element (a,g) is a unit of (N x G, +1) if and only if a = and 1(g, -g) = 0. Thus, we have the following result. Proposition 9 The monoid (NxG, +1) is reduced if and only if 1(g, -g) :/= for all 9 E G\{O}. Finally, we study in terms of G and 1 when N x G is torsion free. But first, we recall some concepts. A monoid (S, +) is torsion free if kx = ky with k E N\{O} implies that x = y (they arc also known as power cancellative semigroups). It is known that a cancellative monoid is torsion free if and only if its quotient group is a torsion free group. Let 9 E G , we define O(g) = min{k E N\{O} I kg = a}. If this minimum does not exist, then O(g) = 00.
° °
177
Proposition 10 The group (Z x G,+I) is torsion free if and only ifO(g) ~ {l,oo} implies that O(g) does not divide I:~i)-l 1(g,ig) .
Proof: Suppose that there exist a E N andg E G such that I:~i)-l 1(g, ig) = aO(g). Then, O(g)(-a,g) = (-O(g)a+I:~i)-l 1(g,ig),0(g)g) = (0,0) and therefore (Z x G, + I) is not torsion free. Conversely, if k(z,g) = (0,0) then kg = a and I::~11 1(g,ig) = k(-z). Since kg = 0, we have k = lO(g) and then I::~11 1(g, ig) = I:~~~g)-l 1(g, ig) = l I:~i)-l 1(g, ig) . Therefore I:~i)-l 1(g, ig) divides O(g). 0 Using Proposition 6, from the fact that a cancellative monoid is torsion free if and only if its quotient group is torsion free, and that every subgroup of a torsion free group is torsion free, we deduce the following result. Proposition 11 The monoid (N x G, + I) is torsion free if and only if O(g) (j. {l,oo} implies that O(g) does not divide I:~i)-l 1(g,ig). 4
Some classes of N-monoids
Using the above properties we obtain some interesting classes of N-monoids. Every nongroup submonoid of a finitely generated group is an N-monoid. Using Proposition 7 and the fact that (Z x G, + I) is the quotient group of (N x G, + I) we obtain the following result. Proposition 12 A monoid is a nongroup submonoid of a finitely generated group if and only if it is isomorphic to an N -monoid (N x G, + I) in which G a finitely generated group. Now, using the condition for obtaining a reduced N-monoid or a torsion free N-monoid we only have to add the condition which fulfills every reduced N-monoids or the condition which fulfills every torsion free N-monoid. Thus, we obtain two new classes of N-monoids. Proposition 13 A monoid is a reduced submonoid of a finitely generated group if and only if it is isomorphic to an N -monoid (N x G, +I) in which G a finitely generated group and 1(g, -g) =I a for all 9 E G. Proposition 14 A monoid is isomorphic to a sub monoid of (zn, +) with n E N if and only if it is isomorphic to an N -monoid (N x G, +I) in which G is a finitely generated group and 1 is such that if O(g) (j. {I, oo}, then O(g) does not divide I:~i)-l 1(g, ig). Finally, using that every finitely generated cancellative reduced torsion free monoid is isomorphic to a submonoid of Nn (see Grillet [3] or Rosales [4]), we obtain the following result.
178
Proposition 15 A monoid is isomorphic to a finitely generated submonoid of Nn with n E N if and only if it is isomorphic to an N -monoid (N x G, +1) in which N x G is finitely generated, I(g, -g) =f. 0 for all 9 E G and if O(g) ~ {I, oo}, then O(g) divides L:~f)-l I(g, ig).
5
Maximal reduced submonoids
Let H be a group and S be a submonoid of H. We say that a submonoid S is a valuation monoid of H if for all h E H we have that {h, -h} n S =f. 0. Observe that if S is reduced then the cardinal of the set {h, -h} n S is equal to 1 for all h E H. Theorem 16 A monoid (N x G, +1) is a reduced valuation monoid of (Z x G, +1) if and only if I(g, -g) = 1 for allg E G\{O}. Proof: Assume that I(g, -g) = 1 for all 9 E G\{O}. Since I(g, -g) =f. 0 for allg E G\{O} then (N x G,+I) is a reduced monoid. Let (a,g) E Z x G. If (a, g) ~ N x G, then a < O. Therefore, -(a, g) = (-a - I(g, -g), -g) =
(-a-1,-g)ENxG. Conversely, if I(g,-g) = r 2: 2, then (-l,g) (l-r,-g)~N x G.
~
N x G and -(-l,g) = 0
Observe that I( -g, g) + 1(0, g') = I(g, g') + I( -g, 9 + g'). If I(g, -g) then I(g, g') E {O, I}. SO, we obtain the following result.
= 1,
Corollary 17 I.f(N x G,+I) is a reduced valuation monoid of(Z x G,+I), then Im(I) ~ {O, I}. Theorem 18 Let (N x G, +1) be an N-monoid with I(g, -g) = 1 for all 9 E G. Then (N x G, +1) is a reduced valuation monoid of (Z x G, +1) with at least one Archimedean element. Furthermore, this construction classifies (up to isomorphisms) all the reduced valuation monoids with at least one Archimedean element which can be embbeded in a commutative group. Gilmer [2] shows that if S is a valuation monoid of a group G, then defining:::; by a :::; b if and only if b - a E S, we obtain a linear order on G. Using this fact it is easy to prove that a group admits a reduced valuation monoid if and only if it is torsion free. Corollary 19 A monoid is a reduced valuation monoid of (zn , +) if and only if it is isomorphic to an N-monoid (N x G, +1) in which I(g, -g) = 1 for all 9 E G and G is a finitely generated group.
179
5.1
Reduced valuation mono ids of
zn zn.
Let S be a reduced valuation monoid of Then S has an Archimedean element m. In this section our aim is to give the group G and the mapping I for this kind of N-monoids. Using that S defines an order on we define
zn
[0, m[= {x E
zn I 0 ~ x < m} =
{x E S I x - m {t S}.
These sets have group structure as is proved in the following result. Proposition 20 The set [0, m[ is a group with the following operation: if a + b < m, a + b = {a + b a + b - m if a + b 2: m . FUrthermore, every element of SI Elm has an unique representant in [0, m[. Now, we define the following mapping:
c: [0, m[ x [0, m[--+ {O, I}, c(x, y) =
o ifx+y<m, { 1 if x + y 2: m .
It can be proved that this mapping satisfies conditions (1), (2), and (3) in Section 3. Furthermore we know that (N x [0, m[, +c) is a reduced valuation monoid of (Z x [0, m[, +c) isomorphic to S.
5.2
Reduced submonoids of
zn zn.
Let S be a reduced submonoid of Since the group generated by S is a free group with rank less than or equal to n, we may assume without loss of generality that the group generated by Sis Gilmer [2] shows that if S is reduced submonoid of a torsion free group G, then S is contained in a reduced valuation monoid of G. Let S be a reduced valuation monoid of containing S, m be an Archimedean element of S, a E S and 9 (S) be the group of differences of S. There exists x, yES such that a = x - y. Since m is Archimedean, there exists kEN and c E S such that x + c = km. Thus we obtain km - a = km - x + y = c + yES and therefore m is also Archimedean in S. Proposition 21 Let G = {x E [0, m[ I x + km E S for some kEN}. Then G is a subgroup of [0, m[. We define the mapping
zn.
zn
t.p: G --+ N, t.p(x) = min{k E N I x + km E S}. The following is straightforward to prove.
180
Proposition 22 The mapping r.p has the following properties:
1. r.p(0) = 0,
2. r.p(x) +r.p(Y) +c(x,y)-r.p(x+Y) 2:0. Proposition 22 allows us to define the following mapping
I:GxG-N, J(x, y) = r.p(x) + r.p(y) + c(x, y) - r.p(x + y). Using the definitions of r.p and c, the reader can prove that the mapping I verifies the properties for obtaining an N-monoid which is also reduced. Moreover, (N x G, +I) is isomorphic to S. Acknowledgments This project has been partially supported by the project PB96-1424. We would like to thank P. A. Garda-Sanchez and the referee for their comments and suggestions. References 1. A. H. Clifford and G.B. Preston, "The Algebraic Theory of Semigroups," Amer. Math. Soc., Mathematical Surveys, 7, 1961. 2. R. Gilmer, "Commutative semigroup rings," Chicago Lectures in Mathematics, 1984. 3. P. A. Grillet, The free envelope of a finitely generated commutative semigroup, Transactions of the Amer. Math. Soc., 149(1970), 665-682. 4. J. C. Rosales, On finitely generated submonoids of N k , Semigroup Forum, 50(1995) 251- 262. 5. T. Tamura, Nonpotent Archimedean Semigroups with cancellative law, I.J. Gakugei Tokushima Univ., 8(1957), 5-11.
PG
= BG:
REDUX
B. STEINBERG Faculdade de Ciencias da Universidade do Porto -4099-002 Porto, Portugal E-mail: [email protected] This paper gives a proof, using geometric techniques, of the theorem of Henclrell and Rhodes that J • G = J G, a key ingredient of the proof that PG = BG. The purpose of this paper is twofold: firstly, it gives a proof of the result using less machinery and providing, we hope, more insight than the original proof; secondly, it serves as an introduction to the techniques used by the author to show that, in H for extension-closed and, even, arborescent pseudovarieties of fact, J • H = J groups H . At the end of the paper, we sketch what changes are needed to prove this stronger result.
e
e
1
Introduction
This communication is intended to introduce the reader to some of the geometric techniques used by the author [0] to prove that, for an arborescent pseudovariety of groups H (that is, (H n Com) * H = H), the equation J * H = J H is valid (where J is the pseudovariety of finite a-trivial monoids). The author has already shown that equality can fail in general [0]. We restrict ourselves here to the case that H = G, the pseudovariety of all finite groups, as most of the ideas can be found in this case. The resulting proof is simpler and relies on less machinery than the original proof of Henckell and Rhodes [3]. The result is particularly important as it was the final step in proving that PG = BG: the pseudovariety generated by power groups is the pseudovariety of block groups. At the end of the paper, we indicate some of the changes needed to prove the more general result. The author showed [8,11] that, in the arborescent case, PH = J * H = J G H (it can be deduced from the work of Higgins and Margolis [4] that BH is strictly larger). This paper assumes familiarity with the basic concepts of monoid theory as well as a familiarity with the notion of an automaton. All monoids and groups considered within will be either finite or free.
e
2
Pseudovarieties and Relational Morphisms
A pseudovariety is a class of finite monoids closed under the formation of finite direct products, submonoids, and quotients. A relational morphism tp : M -e-t N of monoids is a relation tp ~ M x N which is a submonoid projecting onto M. A quotient relational morphism is one in which the projection to N 181
182
is also surjective.
Throughout this paper, H will denote a pseudovariety of groups. Then if V is a pseudovariety of monoids, V H is the pseudovariety of all finite monoids M with a relational morphism <.p : M ~ H E H with l<.p-i E V. This is a generalization of the notion of extensions from group theory. On the other hand, V *H denotes the pseudovariety generated by semidirect products of monoids of monoids in V with groups in H (generalizing the notion of split extensions from group theory). The following is then clear by considering the projection from a semidirect product:
e
Proposition 2.1 V
* H ~ V e H.
Equality can fail: the oldest example, perhaps, being V = A * G and H = G due to Rhodes [5]. The author shows [9]: Theorem 2.2 Let H be a non-trivial pseudovariety of groups satisfying an identity of the form [xn = 1] for some n > 0 or of the form [xy = yx]. Then J *H S;; J GH. 3
Schiitzenberger Graphs and Inverse Automata
In this communication, all automata are assumed to be finite state. Let M be a block group, that is a member of J G G, generated by a finite set A. Then each regular element of M has a unique (semigroup) inverse. The right Schiitzenberger graph corresponding to m is the automaton
S(m)
= (Rm,mm-i,m)
with transitions of the form m ~ n if ma = n and where Rm is the :R-class of m. Equivalently, this is the subautomaton of the right Cayley graph of M obtained by just considering the strongly connected component containing m . The left Schutzenberger graph corresponding to m S'(m)
= (Lm,m-im,m)
is defined dually. If A is a finite alphabet, we let
A = A U A-i. An A-automaton is called inverse if the following holds when it is viewed as an automaton over A (by allowing the reverse transitions): the automaton is deterministic, connected, and has a unique terminal state. Proposition 3.1 Suppose M is an A-generated block group and m a regular element of M. Then S(m) and S'(m) are inverse automata.
183
Proof. We just handle the case of SCm), the other being dual. Clearly SCm) is connected and has a unique terminal state. Suppose na = ra with a E A and n,r,na E Rm. Choose tl,t2 such that nah = n and rat2 = r. For x E M, we use XW to represent the unique idempotent of the subsemigroup of M generated by x . Let el = (atl)W and e2 = (at2)W - Since el and e2 are idempotents and M E J G G, it is easy to see that these two elements generate a a-trivial subsemigroup of M. So
(ele2)W = (ele2)W el . But n(ele2)W
= r and n(ele2)W el = n so r = n.
0
We use red: X* -+ FG(A) for the canonical map where FG(A) denotes the free group on A. In general, we will not distinguish between an element of FG(A) and a reduced word. If A = (Q, i, t) is an inverse automaton, L(A) = {w E X*liw = t}.
The fundamental group of A, denoted 7TI(A) is then red(L(Q,i,i)) and is the subgroup of FG(A) corresponding to the usual fundamental group of the underlying graph of A at i. Proposition 3.2 Let A be an inverse automaton and w a reduced word with = t. Then red(L(A)) = 7TI (A)w.
iw
Proof. If v is a reduced word reading a loop at i, then vw E L(A) and so red(vw) E red(L(A)). If v E red(L(A)), then vw- l reads a loop at i, so v E 7TdA)w. 0 4
The Profinite Topology
Let A be a finite alphabet and N the collection of finite index normal sub= {I}. The groups of FG(A). Since the free group is residually finite, profinite topology is defined by taking N as a basis of neighborhoods of 1. This topology is metric and can be defined by the following norm: let w E FG(A) and define
nN
r(w) = min{[FG(A) : NJlw
then
rt N,N EN};
184
where 2- 00 = O. This topology is somewhat analogous to the p-adic topology on the integers. We now present some simple facts to give the reader a feeling for this topology. Lemma 4.1 Let W E FG(A). Then Wn!-1 -+ w- 1. Proof. Let N be a normal subgroup of FG(A) of index m. Then, for n ~ m, Wnl - 1w E N so Iw nl - 1(w- 1)-11 -+ 0, whence Wn!-1 -+ w- 1. 0 Proposition 4.2 Let L ~ FG(A) and wE FG(A) . Then wE L if and only if for every homomorphism I{) : FG(A) -+ G E G, WI{) ELI{). Proof. First we prove necessity. Let N = kerl{). Then Nw n N L =f:. 0 by the definition of the profinite topology. Hence WI{) ELI{). For sufficiency, we need to show that if N ~ FG(A) is a finite index normal subgroup, then Nw n N L =f:. 0. But this follows by considering the morphism I{) : FG(A) -+ FG(A)/N E G. 0 The following theorem is due to Hall [2] (see Stallings [7] for a simple proof): Theorem 4.3 (Hall's Theorem) Let H Then H is closed.
~
FG(A) be finitely generated.
A generalization was proved by rubes and Zalesskil [6]: Theorem 4.4 (Ribes and Zalesskil) Let HI, ... , Hn be finitely generated subgroups of FG(A). Then HI ... Hn is closed. We will only need the case n = 2 for which there are short elementary proofs [1,10]. Corollary 4.5 Let H, K be finitely generated subgroups of FG(A) and v, w E FG(A) . Then HvwK is closed. Proof. Since conjugates of finitely generated subgroups are finitely generated and left translation is a homeomorphism, the equality HvwK = vw«VW)-1 Hvw)K
implies the result by the rubes and ZalesskiY Theorem. 0 Corollary 4.6 Let A be an inverse automaton. Then red(L(A» is a closed subset of FG(A). Proof. Since A is finite state, 1fl (A) is finitely generated and so, by Hall's Theorem, closed. Since right translation is a homeomorphism, it follows, by
185
Proposition 3.2, that red(L(A) is closed. D If M is an A-generated monoid and m E M, then
Lm
= {w E A*I[W]M = m}
where we use [W]M for the evaluation of a word W in M. In this text, we will view A* as embedded in FG(A) and the closure L of a subset L of A* will denote the closure in FG(A). Proposition 4.7 Let M be an A-generated block group and m E M be a regular element. Then Lm = red(L(S(m))) . Proof. Since, by the above corollary, red(L(S(m))) is closed and Lm ~ red(L(S(m))), we see that Lm ~ red(L(S(m))). For the reverse inclusion, first let v E A* such that [V]M = mm- l . Then, by Lemma 4.1 and the continuity of multiplication, v nl -+ 1. But [Vn!]M = mm- l for all n ~ o. So if
L = {w E A*lmm-l[w]M = m}, then, using that Lm ~ L and vn!L ~ L m, we can deduce that L = Lm. We now show that if W E red(L(S(m))), then W E L. Suppose that an edge r ~ n is used in the reverse direction on reading W from mm- l . Then, since S(m) is strongly connected, there exists t E A* such that nt = r. Then n(ta)n!-lt = r and (ta)n!-l -+ a-I in FG(A) by Lemma 4.1. So, by replacing backwards transitions, in the run of w, with terms of the above sort, we can obtain a sequence of words Wn E L such that Wn -+ w. Thus W E L = Lm . D Lemma 4.8 Let M be an A-generated block group and m E M be regular.
Then
Proof. By symmetry, we need only show that (Lm)-l ~ Lm-l. Since W t---+ w- l is a homeomorphism, (Lm)-l = L;;"? Thus it suffices to show that L;;.! ~ Lm-l. Let v E A* such that [V]M = m- l and let W ELm . Then, for all n ~ 0, (m-1m)nl-Im-1 = m- l . So (vw)n!-lv E Lm-, for all n > 0 whence w- l E Lm-, by an application of Lemma 4.1. D Of course, an analogous result holds for the left Schiitzenberger graphs.
186
5
Liftable 2-tuples and the Semidirect Product
Let
• Obj(Der(
= G;
• Arr(Der(
X
-+ gLg;
• (gL, (m,g))(gLg, (m',g')) = (gL, (mm',gg')).
One says that Der(
w Figure 1.
From the results of Tilson [12], one can deduce from the results of Tilson [12] that M E J * G if and only if there exists a quotient relational morphism
187
note that vry E mcp, wry E ncp. But, by Lemma 4.8, V-I E Lm-l, w- l E Ln-l whence wry E m-1cp,vry E n-1cp. Thus in Der(cp) we have the configuration of Figure 2 below.
(m, vry)
Figure 2.
We can then conclude that
mn = (mm-I)Wmn(n-1n)W = (mm-I)W(n-In)W = mm-1n-1n. Now we must show the converse. For each pair of regular elements {m, n} which is not a liftable 2-tuple, we can find a finite index normal subgroup Nm,n ~ FG(A) such that N n LmLn = 0. Since there are only finitely many such pairs, the intersection N of the respective normal subgroups is a finite index normal subgroup. Let G = FG(A)jN and ry : FG(A) -t G the canonical projection. Then, by choice of N, for m,n E M regular, (m,n) is a liftable 2-tuple if and only if 1 E (LmLn)ry. Let cp = r-1ry. This is easily seen to be a quotient relational morphism. Observe that 1 E (LmLn)ry if and only if 1 E mcpncp. We show that Der(cp) satisfies Knast's equation. Indeed, suppose that we have in De r( cp) a configuration as in Figure 3. Let m = (rs)W r and n = u(tu)w. Then m and n are regular (being a-equivalent to idempotents) . But 1 E mcpncp so (m, n) is a liftable 2-tuple. (r, h)
(t, h) Figure 3.
188
Now observe that mm- 1 = (rs)W since m is !R-equivalent to (rs)W and, dually, n-1n = (tu)w. So the equality mn = mm-1n-1n implies that
whence Der(4?) satisfies Knast's equation. 0 6
The Proof
We can now give the desired proof of: Theorem 6.1 J
*G
= J
eG
Proof. Let M be an A-generated block group and suppose that m, n E M are regular elements such that (m, n) is a liftable 2-tuple. We show that mn = mm-1n-1n. First we remark that A- acts on the left of S'(n), it being the dual of an automaton, and so we let X- act this way as well. Under this convention, 11"1 (S' (n» is then the reversal of the fundamental group of the underlying graph. Let H = 11"1 (S(m», K = 1I"1(S'(n» and u,v E A- be such that [UlM = m, [VlM = n. Then, by Proposition 4.7 and its dual, Lm = Hu, Ln = vK whence LmLn = HuvK by Corollary 4.5. Since (m,n) is a liftable 2-tuple, 1 E LmLn = HuvK and so there exists wE X- such that w E Hu, and w- 1 E vK. So mm-1w = m in S(m) and w-1n-1n = n in S'(n). The theorem can then be deduced from the following lemma upon letting b = m andd=n.D Lemma 6.2 Let w E
X- such that mm-1w = b in S(m) and w-1n-1n = d
in S'(n) . Then
Proof. The proof is by induction on Iwl. IT Iwl = 0, b = mm- 1 , d = n-1n, and mm-1bdn-1n = mm-1n-1n. Suppose w = vx- 1 with x E A, the other case being dual. Let b ~ b' be the edge whose reverse is the last edge used in the run of w from mm- 1 and let d = xd' (d' is well-defined since w- 1 has a run from n-1n to d and the automaton is inverse). Then
mm-1(bd)n-1n
= mm-1bxd'n-1n = mm-1(b'd')n-1n.
But mm-1v = b' and v-In-In = d' so, by induction, the right hand side of the above equation is equal to mm-1n-1n. 0
189
7
Arborescent Pseudovarieties
e
Here we indicate the changes necessary to prove that J * H = J H for H arborescent; recall that a non-trivial pseudovariety of groups H is called arborescent if (H n Com) * H = H where Com is the pseudovariety of commutative monoids. One first defines the pro-H topology on FG(A). Here one takes as a basis of neighborhoods of 1 the set of all normal subgroups whose quotients are in H . If H is a pseudovariety of groups, we say that H satisfies the Hall property if, for an inverse automaton A, the transition inverse monoid of A is in SI H (SI is the pseudovariety of semilattices) if and only if 11'1 (A) is H-closed (Hall's theorem states precisely that G satisfies the Hall property). A pseudovariety H is said to satisfy the property RZ2 if the product of two H-closed finitely generated subgroups is again H-closed. The author shows [10] that arborescent pseudovarieties satisfy both the Hall property and the property RZ2 • One can show [9] that a block group M is in J H if and only if its left and right Schiitzenbeger graphs have transition inverse monoids in SI H. Hence, in the case H satisfies the Hall property, this occurs if and only if their fundamental groups are closed. One can then prove analogs of the results of Section 4, but where the various inverse automata are taken to have transition H, and inverse monoid in SI H, the block groups are taken to be in J the various finitely generated subgroups considered are taken to be H-closed. If M is a monoid, one can define an H-liftable 2-tuple as in the case for G, but using the closure in the pro-H topology. Then the analog of Theorem 5.1 holds: just replace J * G with J * H, replace liftable 2-tuple by H-liftable 2-tuple, and replace finite index normal subgroups with H-open normal subgroups. The remainder of the proof that J * H = J H, assuming that H satisfies the Hall property and the property RZ2 , goes unchanged.
e
e
e
e
e
e
Acknowledgments The author was supported in part by Praxis XXI scholarship BPD 16306 98 and by FCT through Centro de Matematica da Universidade do Porto. References 1. R. Gitik and E. Rips, On separability properties of groups, Int. J. Algebra and Comput. 5, 703-717 (1995) .
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2. M. Hall, A topology for free groups and related groups, Annals of Mathematics 52, 127-139 (1950). 3. K. Henckell, S. Margolis, J. -E. Pin, and J . Rhodes, Ash's type II theorem, profinite topology and Malcev products. Pari I, Int. J. Algebra and Comput. 1, 411-436 (1991). 4. P. M. Higgins and S. W. Margolis, Finite aperiodic semigroups with commuting idempotents and generalizations, preprint 1998. 5. J. Rhodes, Kernel Systems a global study of homomorphisms on finite semigroups, J. Algebra 49, 1-45 (1977). 6. L. Ribes and P. A. Zalesskil, On the profinite topology on a free group, Bull. London Math. Soc. 25, 37-43 (1993). 7. J. Stallings, Topology of finite graphs, Inv. Math. 71, 551-565 (1983). 8. B. Steinberg, A note on the equation PH = J * H, Semigroup Forum, to appear (1999). 9. B. Steinberg, Finite state automata: A geometric approach, Tech. Rep. Univ. of Porto, 1999. 10. B. Steinberg, Inverse automata and profinite topologies on a free group, Tech. Rep. Univ. of Porto, 1999. 11. B. Steinberg, Polynomial Closure and Topology, Tech. Rep. Univ. of Porto, 1999. 12. B. Tilson, Categories as algebra: an essential ingredient in the theory of monoids, J. Pure and Appl. Algebra 48, 83-198 (1987).
TRANSFORMATION SEMIGROUPS: PAST, PRESENT AND FUTURE R P SULLIVAN* Department of Mathematics and Statistics, Sultan Qaboos University, Al-Khodh, 123, Sultanate of Oman e-mail: [email protected] and Department of Mathematics and Statistics, University of Western Australia, N edlands, 6907, Australia e-mail: [email protected] From the 1940s to the early 196Os, the Russian school of semigroups produced some major results on transformation semigroups which concerned generators, morphisms, ideals and congruences. Then in 1966 Howie published his seminal paper on idempotent transformations and in 1975 Sullivan initiated the study of G(X)-normal semigroups of transformation. Throughout the last two decades, many papers have been written on these and related ideas, as well as on links with linear algebra, ring theory and general topology. As a consequence, the theory of transformation semigroups now constitues GL substantial body of knowledge in its own right. In this paper, I briefly discuss some achievements of the past and activities of the present, and suggest avenues for further work in the future.
1
Introduction
This survey concerns the theory of transformation semigroups: that is, subsemigroups of the semigroup P(X) (under composition) of all partial transformation of a set X (that is, all mappings a: : A -+ B, where A, B ~ X). The survey is not meant to be complete: given my current location and limited access to journals, that is almost impossible. In fact , for some of what follows , I have been obliged to rely on reviews and bibliographies for the years 1980-1994, rather than on the actual publications.
' The author gratefully acknowledges the generous support of Centro de Matematica, Universidade do Minho, during his visit in June-July 1999
191
192
In other words, this survey should be regarded merely as an attempt to systematise some of the ideas and results that have appeared in the last three decades (see [109] [119] for a survey in 1970) . My sole aim is to discuss some of the past achievements and recent work in the area, and to suggest avenues for future research. Naturally, however, I welcome any information regarding major errors and/or omissions. We shall focus on unstructured sets X. Many people have studied transformations that preserve some structure on X: for example, an order relation [1], an ultrafilter [123], a block design [12]; and when X is a topological space [23]' a ring [100]' or a vector space [107]. However, due to time and space considerations, these will be barely mentioned, if at all. The same applies for the much bigger class of relation semigroups: that is, subsemigroups of the semigroup B(X) (under composition) of all binary relations on the set X. Although B(X) contains P(X), much less is known about B(X) and it is best treated in a separate survey [143]. Likewise, for '~ he various types of "partial isomorphisms" of a set X: as a regular polygon [92]' a topological space [2], an inverse semi group [24].
2. Notation & Terminology Clifford and Preston's text [10] contains a lot of information on transformation semigroups but its second volume appeared in 1967 and much has ha ppened since then. Consequently, although we will follow the notation and terminology in [10] as much as possible, we also aim to provide a uniform exposition across several disparate areas.
If 0: E P(X), we write domo: for the domain of 0: and ran 0: for its range, and we let T(X) denote the semigroup of all total transformations of X (that is , a E P(X) such that domo: = X). Each 0: E P (X) determines an equivalence 0: 0 0: -Ion dom 0: which we shall sometimes denote by eq u a : Clifford and Preston use 7r0< ([10] vol 1, section 2.2) but we feel equa is more memorable; and Howie uses ker Cl ([33] p 19) which can be confused with the standard notation for the kernel of a linear transformation (and ser~1igroups of linear transformations are often studied now). We let Equ(X) denote the collection of all equivalences on X. We also use part 0: to denote the partition dom 0:/ equ a of dom a determined by the equivalen ce equ 0:, and write P art(X) for the collection of all partitions of X. Corresponding to each FE P art(X) there is a natural equivalence on X which we denote by equ F; and likewise part p will denote the natural partition corresponding to p E Equ(X). Finally, Sub (X) will denote the collection of all subsets of X
193
If a E P(X), we write r(a) or rank a for the rank of a (that is, IXal) and we will often need the following notation: D(a) = X \ Xa,
d(a) = ID(a)l,
G(a) = X \ dom a,
g(a) = IG(a)l,
S(a) = {x E doma: xa -=I- x},
s(a) = IS(a)l,
C(a) = U{ya- 1 : lya-11 2': 2} ,
c(a)
= IC(a)l.
The cardinal numbers d(a) , g(a) , s(a) and c(a) are called, respectively, the defect, gap, shift and collapse of a. The notion of 'defect' seems to have been used first by Vorobev (compare [10] vol 1 , p 6, Exercise 10); 'gap' was formally defined in [139]; and 'collapse' was introduced by Howie in [32] . We let I(X) denote the symmetric inverse semigroup on X ([10] vol 1, p 29) and for each non-empty A ~ X, we write idA for the transformation a with domain A which fixes A (that is, xa = x for all x E A). In particular, id x denotes the identity of P(X) and the empty set 0 acts as a zero for P(X) and I(X) . We let G(X) denote the symmetric group on X . As in [10] vol 2, p 227, if k is any infinite cardinal then k' will denote the successor of k (that is, the least cardinal greater than k) . Also, recall that a cardinal k is regular if I U {Ai: i E I} I = k implies either III = k or some Ai has cardinal k; and k is singular if it is not regular. Also , if m is any infinite cardinal then cf(m), the cofinality of m, is the least cardinal n such that m can be expressed as a sum of n cardinals each less than m. Hence, cf(m) :::; m, and equality occurs if and only if m is regular. Finally, we say a E P(X) is spread over its rank if for each cardinal p < r(a), there exists y E X with lya-11 > p (this concept was introduced in [139]) . In what follows , we extend the convention introduced in [10] vol 2, p 241 : namely, if a E P(X) is non-zero then we write
a
=
(~:)
and take as understood that the su bscri pt i belongs to some (unmentioned) index set I, that the abbreviation { x d denotes {Xi: i E I}, and that Xa = {xd, Xia-1 = A i and dom a = U{Ai : i E I}. When III = 1, a is a constant map and we follow Magill's notation in [73] and write a = Ax. We note in passing that, using an idea credited to Vagner (see [10] vol 2, p 254), we can regard P(X) as a subsemigroup of T(XO) where XO = Xu {O}
194
and 0 ~ X. This is achieved by defining, for each E T(XO) via: if x E domO', xO' = {XO' o if x ~ domO'.
0'0
0'
E
P(X), an element
°
Then the map P(X) -+ T(XO), 0' -+ 0'0 is an embedding under which P(X) is isomorphic to the subsemigroup ofT(XO) consisting of all total transformations fixing O. However, in practice, this is rarely used and the following exposition will treat P(X) as a semigroup which is worth studying in its own right.
3. The Past & Present According to Clifford and Preston ([10] vol 1, pix), the theory of semigroups "really began in 1928 with the publication of a paper of fundamental importance by A K Suschkewitsch". Moreover, they note ([10] vol 1, p 208) that Suschkewitsch used transformations of a set to describe the structure of a general finite simple semigroup, albeit in a rather complicated manner. In 1937, Suschkewitsch published a short book in which T(X) played an important role in illustrating various definitions. For example, Suschkewitsch characterised (1) the right [left] units of T(X) as the injective [surjective] transformations of X ([10] vol 1, p 23, Exercise 2), (2) Green's.c and R relations for T(X) ([10] vol 1, p 51) , and he showed (3) if X is finite, a subsemigroup of T(X) is a left [right] group if all its elements have the same range [partition] ([10] vol 1, p 58, Exercise 8) . According to [10] vol 1, p 58, Exercise 7, Suschkewitsch also showed in 1940 that if X is countably infinite and 0' , (3 E T(X) satisfy 0'{3 = id x , and if H is a subgroup of G(X), then (3H 0' is a subgroup of T(X) consisting of elements of T(X) with infinite rank; and conversely: every subgroup of T(X) containing only elements with infinite rank can be obtained in this way. This can be viewed as a special case of Suschkewitsch's use of transformations in his description of completely simple semigroups: see [10] vol 1, p 99, Exercise 13. In other words , at the very beginning, transformation semigroups were recognised as being useful and interesting: to provide examples of certain concepts and to produce results of general relevance. We now consider various aspects of the theory as it has grown from those early days up to the present (see Hofmann [30] for some more history of the subject).
195
3.1 Embeddings
According to Schein [114] p 165, Suschkewitsch showed that every semigroup can be embedded in T( X) for some X; and that result means transformation semigroups play the same role for semigroup theory as permutation groups do for group theory. Although every semigroup can be regarded as a transformation semigroup, it is important to know which transformation semigroups correspond to semigroups with specific algebraic properties; and, more generally, which transformation semigroups act as a 'model' for a special class of semigroups. One of the earliest results of the first type is Malcev 's characterisation of T(X) itself: namely, as a semigroup S having a non-empty set K of right zeros such that (1) ka = kb for all k E K implies a = b, and (2) for each a E T(K) there exists a E S with ka = ka for all k E K ([10] vol 1, p 6, Exercise 8). In a similar vein for infinite X, Shutov characterised the subsemigroup of P(X) consisting of all partial transformations with finite shift and rank at most 1 ([125] Theorem 3), and then produced a characterisation of P(X, ~o), the semigroup of all partial transformations with finite shift, using earlier work of Gluskin. On the other hand, Margolis [83] assumes X is finite and says S ~ P(X) is a k-tmnsformation semigroup if it is a semigroup and for all a E S and all x EX, we have Ixa-11 ::::: k. He then determines when a semigroup is isomorphic to a k-transformation semigroup. Every transformation of X can be regarded as a subset of X x X and, for some transformations a and (3, their union aU (3 and/or intersection a n (3 are transformations of X . Following [113]' we say a subsemigroup S of P(X) is compatible if a U (3 E P(X) for all a, (3 E S (note that we do not assume aU (3 E S). If S ~ I(X) and aU (3 E I(X) for all a, (3 E S, we say S is strictly compatible. In [113] Schein characterises such semigroups as follows .
Theorem 1. A semigroup S is isomorphic to a compatible subsemigroup of P(X) if and only if xy2 = xy and xyz = xzy for all x, y, z E S. Theorem 2. A semigroup S is isomorphic to a strictly compatible subsemigroup of I(X) if and only if (xy)2 = xy and xy = yx for all x, yES . In [115] Schein goes further and regards P(X) as a set of relations ordered by inclusion: that is, a ~ (3 if and only if dom a ~ dom (3 and a = (31 dom a. He
196
then characterises the ordered semigroups S (that is, semigroups endowed with a partial order) which are order-isomorphic to an inclusion-ordered compatible subsemigroup of P(X) (and he obtains a corresponding result for J(X): see [115] Theorem 2) . In addition, he considers U- semigroups: that is, subsets of P(X) which are closed under 0 and U. Clearly, such S are semilattice-ordered by ~, and we have the following result [115] Theorem 3.
Theorem 3. A semilattice-ordered semigroup (A,·, v) is isomorphic to a subsemigroup (B, 0, U) of P(X) if and only if is left and right distributive over V and, if x ~ y denotes x V y = y, then for all x, y , u , v E S , (1) xy ~ x, and (2) x
~
y V uv implies x
~
y V xv.
The analogous result for J(X) simply adds a further condition that S be commutative: see [115] Theorem 4. As quoted by Schein [115] Theorems 7 and 8, Garvackii showed in 1971 that n-semigroups can be characterised as follows .
Theorem 4. (1) A semilattice-ordered semigroup (S, ., /\) is isomorphic to a n-subsemigroup of P(X) if and only if . is left distributive over /\ and, for all x, y E S1 and s, t , u E S, (s /\ t /\ u)x /\ (t /\ u)y = (s /\ u)x /\ (t /\ u)y. (2) A semilattice-ordered semigroup (S'" /\) is isomorphic to a n-subsemigroup of J(X) if and only if . is left and right distributive over /\ and, for all x,y,u,v E S1, xv /\ uv /\ uy /\ xy = xv /\ uv /\ uy. A similar problem arises with the relative complement a \ (3 of a, (3 E P( X). In [117], Schein characterises transformation semigroups which are closed under \ as being weak subtraction semigroups: that is, as ordered semigroups S with two operations '.' and '-' satisfying the following identities for all x, y, z E S: x-(y-x)=x x - (x - y) = y - (y - x)
(x - y) - z = (x - z) - y x(y - z) = xy - xz
197
x:S;y
(y-x)z=yz-xz.
===}
He also characterises semigroups of invertible transformations which are closed under \ as being subtraction semigroups: that is, as above, except that the last property is replaced by full right distributivity (namely, (x - y)z = xz - yz for all X,y,z E S). A lot of work has also been done on embedding semigroups from a given class into transformation semigroups in the same class (see [130] for a brief survey of this topic, with notes by B M Schein). One of the most important results in this area is the following [10] vol 1, Theorem 1.20.
Vagner-Preston Theorem. Any inverse semi group can be embedded in a symmetric inverse semigroup J(X) for some set X. For an embedding which involves one-to-one total transformations, we suppose ~o :s; q :s; p = IX I and consider the set: BL(p, q)
= {a E T(X)
: a is one-to-one and d(a)
= q} .
(1)
Then BL(p, q) is a semigroup called the Baer-Levi semigroup of type (p, q). It is a right cancellative, right simple semigroup without idempotents; and, as shown by Teissier in 1953, every semigroup with these properties can be embedded in some Baer-Levi semigroup of type (p,p) ([10] vol 2, section 8.1) . A related result , apparently credited to Tully in [10] vol 1, P 10, is the fact that a semigroup S can be embedded in the semigroup of all one-to-one total transformations of a set if and only if it is right cancellative and has no idempotent other than an identity. Moving away from an injective setting: Ueiskaja [153] has shown that a semigroup S can be embedded in the semigroup consisting of all total transformations a of X for which there exists x E X such that xa = x, xa -1 #- {x} if and only if S does not contain an identity and for every non-idempotent a E S, there exist distinct u, v E S with au = u and av = u. For any regular semigroup S, we put
v = {(x,y) 7r
= {(x , y)
E S
x S: xyx = x, yxy = y}
E S x S : V(x)
= V(y)}
(2)
where V( x) denotes the set of all inverses for xES. In [116], Schein calls S catholic if 7r = ids and shows that P(X) is catholic for any set X, and hence
198
any semigroup can be embedded in a catholic semigroup. The relations V and have been useful in other contexts: see [99] for representations by partial transformations of a semigroup and [3] for congruences on the semigroup of continuous self-maps of a topological space.
7r
Clearly, it is desirable for any embedding to be as efficient as possible and that was considered by Easdown in [15] . If S is any finite semigroup, he says the minimal faithful degree p,(S) of S is the least non-negative integer n such that S can be embedded in P(X) for some set X containing n elements. And he proves that if S is any finite fundamental inverse semigroup with semilattice of idempotents E, then p,(S) = n where n is the number of join irreducible elements of E. In [16]' he also provides a method of calculating p,(S) when S is a semilattice of groups. 3.2 Ideals For each
~
satisfying 2 ::;
~
::;
T~ =
lXI', we write {a
E T(X) : r(a)
< O.
(3)
As shown by Malcev [82], these sets constitute all the ideals of T(X) (compare [10] vol 2, Theorem 10.59). Some authors have used the notation Tr = {a E T(X) : r(a) ::; r} , especially when X has finite cardinal n, in which case T(X) can be written conveniently as Tn and this compares well with Sn for the symmetric group and An for the alternating group . However, for work on the congruences of T(X) when X is infinite, it is more convenient to use T~ as defined above. Even for finite X, with the notation in (3), Tn becomes the 'essential' part of T(X) = Tn U Sn and it equals Sing n , which Howie and others have used to denote the semigroup generated by all non-identity idempotents of T(X). Likewise, we note that P~ =
{a
E
P (X) : r (a) <
0
are the ideals of P(X) for 1 ::; ~ ::; lXI', and we use an analogous notation for the ideals of I(X) (this differs greatly from the notation used in [10] vol 2, p 227, but is one way of introducing some uniformity into the notation currently used) . To unify results like the above, in which the ideals of a transformation semigroup are described by bounding the ranks of their elements, Sullivan [134]
199
introduced the notion of a cotmnsitive semigroup S: that is, for each a E S with
and each {xd ~ X and {Yi} ~ X there exist A, /-L E S with bi = XiA and Ai = Yi/-L-l. He then proved the following simple result. Theorem 5. If S is any cotransitive subsemigroup of P(X) then the ideals of S are the sets: SE; = {a E S : r(a) < 0
where 2 ::::: ~ ::::: where ~ ~ 2.
lXI' In particular, the principal ideals of S are of the form
SE;'
This extends Clifford and Preston's observation in [10] vol 1, p 57, Exercise 3: namely, if X is finite , every ideal of T(X) is principal; and if IXI = No then T~o is the only non-principal ideal of T(X) . Another way of describing ideals in a transformation semigroup is to bound the defects of their elements. For example, as shown by Vorobev in 1953, the ideals of the semigroup of all total transformations with finite shift: T(X , No) = {a E T(X) : 8(a)
< No}
(4)
are the sets {a E T(X, No) : d(a) ~ n}
where n is a non-negative integer. Likewise, in [127] Shutov considered the semigroup M(X) = T(X) nI(X) consisting of all one-to-one total transformations of an infinite set X and says its ideals are precisely the sets:
Mo where
=
{a E M(X) : d(a) ~ 8}
(5)
°: : : 8 ::::: IXI . He also states that every left ideal equals M(X, Y) = {a E M(X) : Xa ~ Y}
for some Y
~
X, and that every right ideal is a two-sided ideal.
Likewise, Reynolds and Sullivan [106] proved that for Howie's semigroup E(X) = V u H (see section 3.4 below), the ideals of V have the form : Vn
= {a
E T(X, No) : d(a) ~
n}
(6)
200 for some positive integer n, and each Vn is a principal ideal generated by an element with defect n. On the other hand , H is the first example of a transformation semigroup for which both rank and defect are needed to describe its ideals: namely, its ideals are the sets H(8,~) =
{a
H: d(O')
E
~
8,r(O') <
0
(7)
where No S; 8 S; IXI and 2 S; ~ S; lXI' Moreover, the principal ideals are H(IXI,7)') and H(c, IXI') for some 7),c satisfying 1 S; 7) S; IXI and No S; c S; IXI. In addition, despite appearances, the collection of all ideals in H forms a chain: namely, if IXI = k then
H(k , 2)
~
...
~ H(k ,~) ~
...
~
H(k, k')
~
...
~
H(Nl' k')
~
H(No, k'). (8)
Indeed, for many transformation semigroups, the ideals form a descending or ascending chain, and it would be interesting to find some reason for this . Of course, there are other ways of describing ideals besides bounding the ranks and/or the defects of their elements . For example, in [127] and [129] Shutov shows that the left ideals of the Baer-Levi semigroup BL(p, p) are the sets L ~ B L(p, p) with the property: if f3 ELand X a ~ X f3 for some a E B L(p, p) where IXf3 \ X 0'1 = IXI then a E L. Sullivan [134] extended this in an obvious way to any Baer-Levi semigroup BL(p, q).
In passing we note that Shutov [125] described the ideals of the semigroup P(X, No) = {a
E
P(X) : s(o') < No}
in the following terms. We say a E P(X) strictly fixes x E X if xO'- 1 = {x}. Let Y ~ X and for a E P(X) let Y(o') denote the elements of Y strictly fixed byO'. Put
C(Y,O')=Y \ YO',
C(O',Y)=XO'\Y(O').
For each integer n satisfying 0 S; n S; IX \ YI, let P(X, Y,n) denote the set of all a E P(X , No) such that (1) all elements of domO' , except for finitely many of them, are contained in Y , and (2) if C(Y, a) and C(O', Y) have cardinal p, q respectively then q S; p + n. Then each P(X, Y, n) is an ideal of P(X, No) and in particular, if IY I = m < No then
P(X, Y, n)
= {a
E
P(X, No) : r(O') S; m + n} .
Shutov [125] states that every ideal of P(X, No) is a union of ideals of the form P(X, Y, n). In [134], Sullivan used an alternative approach to find all one and two-sided ideals of P(X , ~) =
{a
E
P(X) : s(O') < 0
(9)
201
where No ::; ( ::;
IXI'.
3.3 Green's relations As already noted, Suschkewitsch described Green's Land R relations on T(X) for finite X. All of Green 's relations are described fully by Clifford and Preston in [10] vol 1, section 2.2 for T(X) and arbitrary X. The essence of the matter is contained in the following Theorem.
Theorem 6. If X is an arbitrary set and a, (3 E T(X) then (a) (3 = Aa for some A E T(X) if and only if X{3 <;;; Xa, (b) (3 = aj.I for some j.I E T(X) if and only if a
0
a-I <;;; {3 0 {3-1,
(c) (3 = Aaj.I for some A,j.I E P(X) if and only ifr({3)::; r(a),
(d) D =
J
Since these relations are important for determining the ideals of a transformation semigroup, several people have characterised them in different contexts but with essentially the same result : for example, see [97] Lemma 1.2 for J(X) , as well as [20] for an exposition in a categorical setting. The similarity is due largely to T E Hall 's Theorem [33] Proposition 1I.4.5.
Hall's Theorem. If U is a regular subsemigroup of a semigroup 5 then Green 's 1: and R relations on U are the restrictions to U of the corresponding relations on S. According to [10] vol 1, p 33, Exercise 1, Doss showed in 1955 that T(X) is regular. We shall see below that many other interesting transformation semigroups 5 are regular, and this means their Land R relations can be readily obtained from those on P(X). For a version of the latter that looks like Theorem 6 above, we restate the following result from [89] .
Theorem 7. If X is an arbitrary set and a, (3 E P(X) then (a) (3 = Aa for some A E P(X) if and only if X{3 <;;; Xa, (b) (3 = aj.I for some j.I E P(X) if and only if dom {3 <;;; dom a and
(c) (3 = Aaj.I for some A,j.I E P(X) if and only if r({3) ::; r(a),
202
(d) V=:J . To show that part (c) in the above two Theorems can be different for other transformation semigroups, we quote the following result ([106] Theorem 6) for the semigroup H of all 'balanced' transformations of an infinite set X (see section 3.4 below).
Theorem 8. If a, (3 E H then (a) (3 = Aa for some A E H if and only if X(3 <;;; Xa, (b) (3 = aJ.L for some J.L E H if and only if a 0 a-I <;;; (30 (3-1, (c) (3 = AaJ.L for some A, J.L E H if and only if r((3) ::; r(a) and d((3) ~ d(a),
(d) V=:J. The 4-spiral semigroup Sp( 4) [7] is an example of a regular idempotentgenerated semigroup which is bisimple but not completely simple. In [49] Lallement showed that it is isomorphic to the subsemigroup of T(Z) generated by the following transformations of Z, the set of integers:
xa = x'Y
=~-
Ix -
lxi,
x(3 =
-lxi,
H x8 = ~ + Ix - H
And in [7] the authors proved that a V-class of E(X) contains a copy of the bicyclic semigroup only when it contains a copy of Sp( 4). Green's relations on E(X) are determined in [106] Theorem 7, so it should be possible to decide which V-classes contain Sp( 4) and to use this information in the general theory. We know the maximal subgroups of T(X) are the group 'H-classes of T(X) and the latter are described in [10] vol I, Theorem 2.10 (the corresponding result for P(X) was discussed in [122]). In [27] Harris and Schoenfeld showed that the number of idempotents (hence, also the number of group 'H-classes) in T(X) when X is finite and contains n elements is given by
and they derived some combinatorial identities and an asymptotic formula for the numbers Un, n ~ 1. More precisely, according to [10] vol I, p 56, Exercise
2,
203 (1) each £-class of T(X) corresponding to a subset of X with r elements contains rn-r idempotents, and (2) each R-class of T(X) corresponding to a partition {nl' ... ,nr } of n contains ni . .. nr idempotents. In [48]' Kim obtained even more precise information by counting the number of idempotents in certain "sections" of given £ and R classes of T(X) . To explain this, we let DTr denote the 'V- class of T(X) consisting of all transformations with rank r. For each P E Equ(X) whose corresponding partition of X contains r elements, and for each non-empty Y E Sub(X) with JYI = s, we let Dr(p, Y) = {a E DTr : p ~ equa and rana ~ Y} and call this the (p, Y) - section of DTr . In [48], Kim counts the number of idempotents in these subsets of T(X) and deduces the following result. Theorem 9. If IXI = n then the number of idempotents in each maximal principal right ideal equals
In [149] the authors use a computer to generate 'V-classes of T(X) . More recently, Lallement and McFadden [50] have produced an algorithm for computing Green's relations on any semigroup S ~ T(X) which is given by its generators. A much earlier result [28] provides a general method of counting the size of certain subsets of T(X) . In passing we mention an alternative approach to Green's relations that was developed by Clark and Carruth in [9] : namely, they introduce the concept of a Green's pair on a semigroup S that involves two sets L, R of partial left (right) translations of S containing ids where the elements of L commute with those of R. It remains to be seen whether this approach is useful in the theory of transformation semigroups.
3.4 Jdempotents Howie's paper [32] has had a profound influence on the theory of transformation semigroups (and many related areas) ever since it appeared in 1966. And yet it seems to have happened 'almost by accident'. I recall sitting with John
204
Howie at a table on the ground floor of the Hargreaves Library at Monash University not long after he had written the paper, and asking him how he came upon the problem and its solution. He replied that at the time he had nothing better to do and thought products of idempotents in T(X) would be interesting to study, and that after two weeks he had a satisfactory description of them, except for some minor details. Thus began one of the most seminal results in the history of our subject!! Actually, over a decade before this, Vorobev came close to proving Howie's Theorem for finite sets X. That is because, as noted in [10] vol 1, p 6, Exercise 10, Vorobev showed that if IXI = n is finite and if a E T(X) has rank r < n then a = "(0 for some "( E T(X) with rank r+1 and some idempotent 0 E T(X) with rank n -l. Consequently, every a E T(X) with 1 :::; d(a) :::; n -1 can be expressed as a product of a permutation and k = d(a) idempotents with defect l. In fact, if a, {3 E T(X) have defect 1, then {3 = )...ap, for some )... , p, E G(X) , and hence T(X) can be generated by G(X) and a single idempotent with defect l. Shutov reported a similar result for P(X) and I(X) in [124]. In fact , Howie used Vorobev's work to prove the 'easy' part of [32]: namely, if E(X) denotes the semigroup generated by the non-identity idempotents in T(X) then E(X) = T(X) \ G(X) when X is finite. For the much harder case when X is infinite, Howie proved the following result. Howie's Theorem. The semigroup E(X) is the disjoint union of two semigroups: v = {a E T(X) : 1 :::; d(a) :::; s(a) < i{o}, (10) H = {a E T(X) : d(a) = s(a) = c(a) :::=: i{o}. That V is a semigroup follows from [32] Lemmas 2 and 5, and a related semigroup seems to have been studied by Vorobev [154]. That H is a semigroup follows from [32] Lemma 6, and in [38] Howie referred to its elements as balanced transformations of X (the notation 'H ' stands for 'Howie') . An important step in proving Howie 's Theorem was to show that also V U H is a semigroup and that required a surprising result which is worth restating here (see [32] Lemma 7 and [38] Lemma 2.10 for a correction to the original proof). Theorem 10. If a E H , {3 E T(X) and s({3) < s(a) then both a{3 and {3a have shift, defect and collapse equal to that of a. As already stated, Howie 's description of E(X) has been extremely fruitful. However , when Gordon Preston first heard about it, he told me that "it must
205
surely be more complicated than necessary" and that "a simpler description using fewer parameters ought to be possible: at least one that avoids using the notion of 'collapse' ". So far, nobody has provided an alternative approach. Subsequent work on E(X) has proceeded in two directions depending on whether X is finite or infinite: the former leading to several combinatorial questions and the latter producing examples of congruence-free semigroups (see sections 3.5 and 3.8 below). In passing we note that the basic description of E(X) was extended to partial transformations by Evseev and Podran [17] [18] and independently by Sullivan [132]; and the ideals and Green's relations for the semigroup E*(X) generated by all the non-identity idempotents in P(X) were discussed in [106] section 4.
3.5 Combinatorics Some years after his work on E(X), Howie and his co-workers began to investigate its combinatorial aspects for finite X (see [40] for a brief survey up to 1990). In 1966, Howie showed that if
IXI =
nand n is finite then
Tn = {a E T(X) : r(a) < n}
is generated by its idempotents with rank n - 1. Since idempotents fix each element of their range, those with rank n - 1 are completely determined by their action on a 2-element subset {a, b} of X. Since there are n( n - 1)/2 choices for the subset and two ways of mapping it to get an idempotent, there are a total of n(n - 1) idempotents with rank n - 1 in Tn. Quite surprisingly, Howie [34] showed in 1978 that a minimal set of idempotents with rank n - 1 that generates Tn has size n(n - 1)/2. Moreover, he did this using a simple graph-theoretic argument, something that has been exploited more fully by Higgins (see [29] for many ideas and results in this area, especially section 6.1 for probabilistic arguments, page 202 for references to the literature, pages 212-221 for a proof of Schein's Covering Theorem for finite T(X), and section 6.3 for an exposition of some of Howie's work being discussed here). To state a more 'local' result, we let grav(a) denote the gravity of a: that is, the least number of idempotents with rank n -1 that generate a given a E Tn . Iwahori [46] and Howie [35] independently showed that grav(a) = n - fix(a)
+ cyc(a)
206
where fix(a) = n - s(a) is the number of elements of X fixed by a and cyc(a) denotes the number of cyclic orbits of a. And in [35] Corollary 3.13, Howie proves that the maximum value of grav(a) for a E Tn is [3(n - 1)/2] (where [... ] denotes the greatest integer function) . Hence , this number is also the depth of Tn (compare section 3.S before Theorem 2S). In [41] the authors show (among other things) that if n = 2m and m ::::: 3 then the number of a E Tn with maximum gravity equals
(4m 2 - 1)(2m)! 3· 2m - 1 (m - I)!' Since the set of all idempotents in Tn generates Tn, for each a E Tn there must exist a least integer k(a) for which a E Ek(<<) but a tJ. Ek (<<) - l Howie [39] and Saito [IDS] obtained upper bounds for k(a) via arguments involving some elementary graph theory. And in [110] Saito proved: grav(a) '() grav(a) d(a) ::; K a ::; d(a)
+1
and if grav( a) == 1 mod d( a) then
where m# means the least integer greater than or equal to m. Garba [22] has extended some of these ideas to P(X) by regarding P(X) as a subsemigroup of T(XD) as discussed at the end of section 2 above. Returning to a global setting, Gomes and Howie [26] let irank (S) denote the idempotent rank of S: that is, the minimal number of idempotents required to generate a transformation semigroup S, with a corresponding definition for nilpotent rank nrank (S). They showed (among other things) that irank (Tn)
=
~n(n - 1)
= rank (Tn)
where rank(S) denotes the minimal number of generators of S. On the other hand, rank(In) = 3. Howie and McFadden [44] extended this to
T r+ 1 = {a E T(X) . r(a) < r
+ I}
207 for 1 :S r
< n, and showed that for
r ;::: 2
irank(Tr + l ) = S(n,r) = rank (Tr+d where S(n, r) denotes the Stirling number of the second kind (that is, the number of partitions of {I, ··· ,n} with exactly r elements).
In a more specific vein, Howie, Robertson and Schein [45] characterised the a E Tn which are a product of just 2 or 3 idempotents. For example, in the case of 3 idempotents, it is necessary and sufficient that rank a :S ~(n
+ fix a + compa)
where comp a is the maximal number of disjoint triples (x, xa, x( 2 ) for x ~ X a. Also, Saito [109] obtained a complicated characterisation of when a E Tn is a product of 4 idempotents. Recently, Thomas [147] extended the 2 idempotent case to arbitrary sets, and Sullivan ([148] Chapter 3) has done the same for the 3 idempotent case. Finally, we note that in [4] the authors have used a computer to find all the subsemigroups of T(X) generated by sets of idempotents with defect 1 when IXI =4.
3.6 Nilpotents
In [120] Schwarz suggested that the role of "nilpotents" in semigroups should be investigated further. Taking the lead from Howie's work in 1966, I therefore initiated the study of N(X), the semigroup generated by all the nilpotents in P(X), in [139]. Actually, this work began much earlier, during one of my sabbaticals with Otto Steinfeld at the Bolyai Mathematics Institute in Budapest: the cases when X contains an odd finite or a singular infinite number of elements were the most difficult (and were resolved in part while sitting on a park bench on 'Margit Sziget', an attractive island in the Danube River near the centre of Budapest). Recall that P(X) contains a zero (namely, the empty mapping 0): we say E P(X) is nilpotent with index r if a' = 0 and a r - I f- 0. The following two results summarise Corollary 3 and Theorem 4 in [139] and Lemmas 2.5 and 3.2 in [142] .
a
Theorem 11. Let IXI = k be regular and a E P(X). Then a E N(X) if and only if g(a) f- 0, d(a) = k and g(a) = k or Iya-Il = k for some y E X.
208
Moreover, when this occurs, N(X) is a regular semigroup and each a E N(X) is a product of 3 or fewer nilpotents with index at most 3.
Theorem 12. Let IXI = k be singular and a E P(X) . Then a E N(X) if and only if g(a) =1= 0, d(a) = k and either g(a) 2: r(a) or a is spread over its rank. Moreover, when this occurs, N(X) is a regular semigroup and each a E N(X) is a product of 4 or fewer nilpotents with index at most 4. While showing in [142] that the numbers 3 and 4 above are best possible, Sullivan considered another two nilpotent-generated semigroups: namely, N(X, 2) generated by all nilpotents in P(X) with index 2, and N(X,3) generated by all nilpotents in P(X) with index at most 3. The following two results are Theorems 2.3, 3.3 and 5.1 in [142] .
Theorem 13. Let IXI = k be an arbitrary infinite cardinal and a E P(X). Then a is a product of nilpotents in P(X) with index 2 if and only if d(a) = k and g(a) 2: r(a). Moreover, when this occurs, N(X,2) is a regular semigroup and each a E N(X, 2) is a product of 3 or fewer nilpotents with index 2. Theorem 14. Let IXI = k be singular and a E P(X). Then a is a product of nilpotents in P(X) with index at most 3 if and only if g(a) =1= 0, d(a) = k and (a) g(a) 2: r(a)', or (b)
lya-11 2: r(a)
for some y E X, or
(c) g(a) 2: cf(k) and a is spread over its rank. Moreover, when this occurs, N(X,3) is a regular semigroup and each a E N(X,3) is a product of 3 or fewer nilpotents with index at most 3. Of course, J(X) also contains 0 as a zero and hence it also contains nilpotents: we let N J(X) denote the semigroup generated by the nilpotents of J(X) and recall the following result (see [139], Corollary 4) .
Theorem 15. Let IXI = k be an arbitrary infinite cardinal and a E J(X) . Then a is a product of nilpotents in J(X) if and only if d(a) = g(a) = k. Moreover, when this occurs, N J(X) is an inverse semigroup and each a E N J(X) is a product of 3 or fewer nilpotents with index 2. Actually, a few years earlier, Howie went close to using N J(X) to construct a congruence-free semigroup. That is, he let X be an infinite set and let No ::; n < m = IXI. He showed in [37] Lemma 1 that
U(n)
=
{a E J(X) : g(a)
=
d(a)
=
n}
209 is an inverse semigroup and that if
U(n,n)
= {a
E
U(n): s(a):S: n}
then U(n,n)/on is a congruence-free bisimple inverse semigroup (compare [29] Exercise 1.2.11 ). That U(n, n)/on is congruence-free can be deduced from a more general result (see [141] pp 577-578) . Likewise, in [42] the authors were led to the verge of N I(X) by assuming IXI = m ~ No and considering the semigroup generated by the set
{,B-1a: a, {3
E
B}
where B = BL(m, m) is the Baer-Levi semigroup of type (m , m). That is, they showed in [42] Theorem 3.2 that this semigroup equals the subsemigroup of I(X) generated by the nilpotents of I(X) with index 2 and hence, by Theorem 15 above, it equals NI(X). Next they in effect observe that
N1dX)
= {a
E
NI(X) : r(a) <
is an ideal of N I(X) for each
~
S
=
0 = NI(X) n IE(X)
:S: m and that the Rees factor semigroup NI(X)/Nlm(X)
is a O-bisimple inverse semigroup which is generated by its nilpotents with index 2. Moreover, each element of S is a product of 3 nilpotents with index 2, and 3 is best possible [42] Theorem 4.3. In fact , S has a homomorphic image - namely, N I(X)/om - which inherits all of these properties and is also congruence-free [42] Theorem 4.10. Again, this latter fact follows from a more general result (see [141] pp 578-579). In fact, in [89] the authors show that the ideals of N I(X) are precisely the sets N h.(X) for 1 :S: ~ :S: m', and that each Rees factor semigroup
NI2(X) = N1dX)/NIE(X) = {a
E
NI(X) : r(a) =
0
U
{O}
is O- bisimple and inverse, and is generated by its nilpotents with index 2. Moreover, they show that every non-identity, non-universal congruence on NI2(X) is the 'reduction' of a Malcev congruence to NI2(X); and hence that NI2(X)/OE is congruence-free. In addition, just as Howie proved that any semigroup is embeddable in a congruence-free bisimple semigroup which
210
is generated by its idempotents (see Theorem 29 below), the authors prove the following result.
Theorem 16. Every inverse semigroup can be embedded in a congruence-free O-bisimple inverse semigroup which is generated by its nilpotents with index 2.
In [89] the authors describe the ideals and Green's relations on each of the afore-mentioned semigroups: N J(X)
c
N(X,2) C N(X, 3) ~ N(X)
(11)
when IXI = k is infinite. This will enable us to determine the congruences on each of these semigroups [87] . Before proceeding to the finite case, we note that it is possible to study nilpotents in semigroups of total transformations. For example, let X be an infinite set with cardinal m and let
Km = {o E Rm : Iyo-Il = m for some y E X} U {O} where Rm = H(m, m')/ H(m, m) (see (8) for the notation) . Clearly, Rm has a zero and non-trivial nilpotents. In [43] the authors showed that if m is regular then Km is the subsemigroup of Rm generated by the nilpotents of Rm . In fact, Km is a regular semigroup and each element of Km is a product of 3 or fewer nilpotents with index 2, and 3 is best possible. Subsequently, in [86] the authors showed that if m is singular (that is, non-regular) then the set
Lm
= {o E
Rm: for eachp < m, there existsy E X such that Iyo-Il > p}U{O}
is a O-bisimple regular subsemigroup of Rm and consists of all elements of Rm which are products of nilpotents in Rm. In fact, yet again each element of Lm is a product of 3 or fewer nilpotents in Rm with index 2, and 3 is best possible. The correlation between these results for total transformations and those already cited for partial transformations is discussed in [142] p 455.
In passing we note that, just as for idempotents, a simple result ([43] Lemma 2.5) helps to show that certain numbers are 'best possible' and it is worth restating it here (compare Theorem 28 below) .
Theorem 17. Let S be a regular semigroup with a zero. If a = px where a, x -=I 0 and pES is a nilpotent with index 2 then a = qx for some nilpotent q E S with index 2 such that q R a .
211
We now turn to finite sets and suppose IXI = n < ~o. Lipscomb states in [70] section 6 that Cauchy introduced the notion of an even permutation in 1815 and "more than one and a half centuries later R P Sullivan [139] p 330 was evidently the first to advance a concept of an even transformation". To explain this terminology, and the clarification provided by Gomes and Howie in [25], we write
Dln- I Note that any
0
E
Dln -
I
= {o E I(X) : r(o) = n -I} .
has a unique completion a E G(X) defined by:
_ {xo,
xo =
if .
b,
x
E domo,
If x
=
a,
where X \ domo = {a} and X \ Xo = {b} ([25] p 388). We say 0 E Dln an even transformation if a is an even permutation of X and write
E n - I = {o
E
I
is
DIn-I : a is an even permutation}.
In addition, we recall that the ideals of I(X) are the sets Ir(X) = {o E I(X) : r(o) < r} where 1 :::; r :::; n result.
+ 1. In
[25] Theorem 3.18, the authors proved the following
Theorem 18. Suppose IXI = n 2: 3. (a) If n is even then NI(X) = In(X). (b) If n is odd then NI(X) = In-I(X) U En-I. Moreover, in each case, each non-zero 0 E N I(X) is a product of n - 1 or fewer nilpotents, each with index n (and rank n - 1). The corresponding result for P(X) is as follows (see [139] Theorems 1 and 2): to state it, we recall that the ideals of P(X) are
Pr(X) = {o where 1 :::; r :::; n
E
P(X) : r(o) < r}
+ 1.
Theorem 19. Suppose IXI = n 2: 3 and
0
E
P(X).
212
(a) If n is even then (b) Ifn is odd then
0
0
E N(X) if and only if g(o)
E N(X) if and only if g(o)
-I- O.
-I- 0 and 0
E Pn - 1 (X)uEn -
1.
In [89] the authors describe the ideals and Green's relations on N J(X) and N(X) for finite X . Since in this case both these semigroups are finite and G(X)-normal, their congruences are known [59].
3.7 Morphisms If S is a semigroup, any bijection ¢ : S ---7 S such that (xy)¢ = (x¢)(y¢) for all x, yES is called an automorphism of S, and we write Aut(S) for the group (under composition) of all automorphisms of S. In particular, if S <;;: P(X), we say ¢ E Aut(S) is inner if there exists 9 E G(X) such that o¢ = gog-l for all 0 E S (note that we do not assume 9 E S). We also say S <;;: P(X) has the inner automorphism property (lAP) if every automorphism of S is inner. In [133] Sullivan observed that different authors had shown that the non-zero ideals of P(X), T(X) and J(X) have the lAP; and that their automorphisms groups were isomorphic to G(X). To unify and extend these results, he introduced two simple ideas. We say a subsemigroup S of P(X) covers X if for each x EX, there exists some idempotent constant map in S with range {x}. And we say S is G(X)-normal if gog-l E S for all 0 E Sand 9 E G(X) (note that we do not assume G(X) <;;: S). The main result of [133] can be stated as follows. Theorem 20. If S is a subsemigroup of P(X) which covers X then every automorphism of S is inner. Moreover, if S is also G(X)-normal then Aut(X) is isomorphic to G(X) .
The aim of this result was to provide a simple test for a transformation semigroup to have the lAP and to have the symmetric group as its automorphism group: namely, it must contain 'sufficient' constant maps and be 'normal' in the usual group-theoretic sense. In particular, this was true for all the transformation semigroups that had been considered previously, as well as for Howie's semigroup E(X) generated by the idempotents of T(X) where X is arbitrary and for the semigroup P(X,O of all partial transformations which shift less than ~ elements where ~o ::; ~ ::; IXI. At the end of [133] I noted that the above result could also be applied to : J(X,O = {o E I(X) . s(o)
<0
213
where No ::; ~ ::; lXI, and that this should be compared with work on describing the automorphism groups of the normal subgroups of G(X) (see [121] Theorem 11.4.8 for an account of the latter) . However, this left open the possibility of there being transformation semigroups which have the lAP but do not have any constant maps (and are not permutation groups). In [21], two of my students, Fitzpatrick and Symons, verified this possibility by showing that if X is infinite and S is any subsemigroup of T(X) containing G(X) then S has the lAP (at about the same time, Chantip and Wood [8] proved the less general result: the semigroup of all mappings of an infinite set X onto itself has the lAP). This was extended in [135] to subsemigroups of J(X) containing Alt(X), the alternating group on an infinite set X, and from this the earlier work on the automorphism groups of the normal subgroups of an infinite symmetric group can be deduced . However, more importantly for subsequent work in this area, the following result was also proved. Theorem 21. Suppose X is an arbitrary set and S is any non-zero G(X)normal subsemigroup of P(X). Then either S contains a constant map or for all non-zero a E S,r(a) = IXI,g(a) < lXI , and if d(a) = IXI then Ixa-Il < IXI for all x E X. It can be readily shown that the Baer-Levi semigroups BL(p, q) are G(X)normal, and they contain neither constants nor permutations. Thus , they became a test case for Sullivan's Conjecture in the early 1980s that every G(X)- normal transformation semigroup has the lAP (this was hinted at in [135] and discussed with co-workers at the time). As shown in [61] every BL(p, q) does have the lAP. However, the most important feature of this result was the method of proof: namely, in a natural way, every automorphism of S = BL(p, q) determines an order-automorphism of Ran(S)
= {Xa
: a E S}
which is called the range of S (here 'order-automorphism' means a permutation of Ran(S) which preserves ~). Moreover, each order-automorphism of Ran(S) is induced by a permutation of X; and it is this permutation which ensures that 1; is inner. This approach was first introduced by Inessa Levi in her PhD thesis written under the supervision of G R Wood, and she has refined it considerably since then. Her first major result [51] was a positive answer to Sullivan's Conjecture.
214
Levi's Theorem. If X is infinite then every G(X)-normal subsemigroup of
T(X) has the lAP. In passing we note that the reviewer of [51] (see MR 86m: 20079) asserts that a more general result for P(X) was announced at a symposium in the Soviet Union in 1978. This result was also proven by Levi in [53]. After this major breakthrough, work progressed in a number of directions. For example, in [52] Levi provides necessary and sufficient conditions for certain families F ~ Sub(X) to be the range of a subsemigroup of P(X). Namely, we say F is normal if, whenever A E F and B ~ X satisfy IBI = IAI and IX \ BI = IX \ AI, then B E F. In addition, we write
Fe = {X \ A : A
E
F}
and for any Q ~ Sub(X), we put card(Q) = {IAI : A E Q}. Then we say F is hereditary if card(F) has the property: n E card(F) and IXI imply m E card(F). And F is finitely additive if card(Fe) has the property:
o< m < n <
nl, .. . , nr E card(Fe)
===>
nl
+ .. . + nr
E card(Fe) .
Theorem 22. A normal family F of subsets of X is the range of a transformation semigroup defined on X if and only if
(1) F is hereditary or it consists entirely of sets A
(2) F is finitely additive or it contains no sets A
~
~
X with IAI = lXI, and
X with IAI
=
IX\AI = IXI .
This dealt with one aspect of the proof used in [61] : the other was that all order-automorphisms of some F ~ Sub(X) are induced by permutations of X. In [54] Levi characterised the normal F which have this latter property. A related phenomenon was observed by Howie [36] while constructing a congruencefree bisimple semigroup which is generated by its idempotents. To explain this, suppose IXI = m is infinite and let Equ(X) denote the set of equivalences on X. For each p E Equ(X), let K(p) denote the union of all the non-singleton p-classes and put
K S
= {p E Equ(X) : IK(p)1 = m and Ixpl < m for all x EX}, = {A ~ X: IAI = IX \ AI = m} .
215
The next result and its proof are implicit in the proof of Lemma 3.15 in [36] .
Theorem 23. If m is regular then for all p, a E JC and all A, B E S (a) there exists
E JC such that
T
p n T E JC and
Tn a
E JC, and
(b) there exists C E S such that Au C E Sand CuB E S. To see why this is remarkable, we let
Sm
= {ex
E
H: d(ex)
= r(ex) = m and lyex-11 < m for all
y E X}
which was shown by Howie [36] to be a bisimple idempotent-generated semigroup provided m is regular. It can be shown that if ex E Sm then equ ex E JC and ran ex E S. In other words, Theorem 23 says that the equivalences and the ranges determined by the elements of Sm satisfy a lattice-theoretic property. This and Levi's work suggests that more attention should be devoted to Equ(S) and Ran(S) when investigating a transformation semigroup S: for example, can we characterise these sets for specific types of S7 Another major direction of research has been the description of all G(X)normal subsemigroups of P(X), T(X) and J(X) . For finite X, Symons did this for T(X) in [146]' Sullivan anG Symons obtained partial results for P(X) in [145], and Lipscomb completed the picture for P(X) as well as for J(X) (see [69] as well as [68] for a summary of related work in this area). Moreover, in [59] Levi described the congruences on all G(X)- normal semigroups when X is finite ; and in [70] Lipscomb found the congruences on the infinite alternating semigroup A(X) consisting of all the even transformations in J(X) where X = N, the set of positive integers: in this context, we sayan element of J(X) is even if it is a product of an even number of 2-cycles (i, j) and I-chains (i, j] (where (i,j] denotes the restriction of (i,j) to the set N\ {j}). Consequently, research in this area is reasonably complete. The infinite case has been more troublesome. In [55] Levi writes a G(X)normal subsemigroup of M(X) as a disjoint union of two semigroups A and B, where all elements of A have finite defect and those of B have infinite defect. She then studies A and B separately. On the other hand, for the class of partial one-to-one transformations, Levi and Williams [65] let p, q, r, s be infinite cardinals and put
S(p,q,r,s)
= {ex E J(X): s(ex):S p,d(ex) = q,g(ex) = r,r(ex):S s}
216
which they show is the smallest G(X)-normal subsemigroup of J(X) containing a given a E J(X) with shift p, defect q, gap r and rank s . In [64] they extend this to the case where the given a E J(X) has infinite shift and either its defect or its gap is finite . And later in [58] Levi investigated Green's .I-relation on such semigroups. In passing we note that early writers on semigroup theory ([72] as cited in [82] Translation p 250, and [151] p 107) call a subset N of a semigroup S normal if there exists a homomorphism S ----> T under which N is the inverse image of some element of T. It is not clear how this notion is related to that of G(X)-normality in general. After automorphisms, the next step is to describe more general types of morphism for a transformation semigroup, but surprisingly little progress has been made in this area. In the 1940s and 1950s, there was some interest by Kaplansky, Scott and others in generalising the notion of 'automorphism' for rings and groups, and in 1983 I took the same approach for transformation semigroups S. We say a bijection c/J : S -> S is a semi-automorphism if (aba)c/J = (ac/J)(bc/J)(ac/J) for all a, bE S; and it is a half-automorphism if (ab)c/J equals (ac/J)(bc/J) or (bc/J)(ac/J) for all a, b E S. The aim is to find transformation semigroups S for which each semi-automorphism is an automorphism or an anti-automorphism (and to describe it explicitly in each case). For example, in [136] I showed that if S is 2-transitive and extremally covers X (that is, S contains all total constants and all injective constants) then every semi-automorphism of S is an inner automorphism induced by a permutation of X. On the other hand, if S is a 2-transitive inverse subsemigroup of J(X) which covers X then each semi-automorphism of S is an automorphism (and thus inner) or an anti-automorphism (namely, the composition of an inner automorphism with the map S ----> S,a -> a-I) . Analogous results were obtained in [137] for half-automorphisms. As for the 'inner automorphism problem ' discussed above, thp. question remains as to whether the existence of 'sufficient' constant maps in S can be removed, and similar results obtained for a wider class of transformation semigroups. There are more standard ways to generalise the notion of 'automorphism' and Magill [74] was one of the first to describe the 'natural' injective endomorphisms of certain total transformation semigroups. This was generalised in [138] to certain partial transformation semigroups as follows. For convenience, we call any 'injective endomorphism ' of a semigroup S a monomorphism of S and let Mon(S) denote the semigroup (under composition) of all monomorphisms of
217
S. If S contains id x , we sayan endomorphism ¢> of S absorbs constants if, whenever Ax(id x ¢» = Ax for an idempotent constant Ax E S, then there is some idempotent constant Bx E S (that is, having the same range as Ax). The next result (see [138] Theorem 1) provides conditions under which every monomorphism can be easily described . Theorem 24. Suppose S is transitive , covers X and contains id x . If ¢> E Mon(S) absorbs constants and fixes 0 if 0 E S, then there exist A, p, E P(X) with p,A = id x such that a¢> = Aap, for all a E S. Conversely, any ¢> : S ----> S of this form is a monomorphism that absorbs constants. As usual, we let End(S) denote the semi group of all endomorphisms of S. The next result is also from [138].
Theorem 25. Suppose S is 2-transitive, covers X and contains idx . If ¢> E End( S) absorbs constants and fixes 0 if 0 E S then ¢> is either injective or constant. Unfortunately, these results do little to determine the monomorphisms of any important transformation semigroup, simply because we do not know whether their monomorphisms absorb constants (and this applies even more to 'endamorphisms'). Magill returned to this problem in [78] and provided topological examples in which every element cf Mon(S) is induced by some A, p, E S(X) with p,A = idx , as well as examples where some elements of Mon( S) are not of this form. Much more substantial progress on this problem was made by Levi in [56] and [57] . For example, suppose X is infinite and let S be any G(X)-normal subsemigroup of M(X) such that No :::; d(a) < IXI for all a E S. Choose Y ~ X with WI = IXI; and a family F of injections from X into Y such that Ran(F) forms a partition of Y; and also choose a homomorphism B from S into G(X \ Y); and do all this so that Y, F and B satisfy a complicated set of conditions. Then we can produce a monomorphism of S; and conversely any monomorphism of S can be obtained in this way [56]. In particular, this means the monomorphisms of the Baer-Levi semigroup BL(p, q) for q < pare now known. In [75] Magill continued his search for semigroups having only "natural" endomorphisms of a prescribed type . For example, if 8 is an idempotent in T(X) and 7f is a bijection from X onto X 8 then
B: T(X)
---->
T(X) , a
---->
87l'- l a7f
218
is an endomorphism of T(X) defined in a natural way using 8 and 7r. In Magill's setting, the aim is to determine the topological spaces for which every endomorphism of S(X) is natural in the above sense. But we could equally ask for a class of transformation semigroups whose endomorphisms are all of this type. Likewise, in [81] the authors produce a family of topological spaces X for which every epimorphism (that is, surjective endomorphism) of S(X) is natural in some sense. It seems that the corresponding question for transformation semigroups is much easier to answer. For, in [112] p 34, Schein calls a subsemigroup S of T(X) sufficient if (1) S contains all constant (total) transformations, and (2) S is 2-transitive (in our terminology). In [112] Corollary 5, he concludes that if S is a sufficient total transformation semigroup then every epimorphism of S is an automorphism (which must be inner). However , due to his approach, this cannot be easily extended to partial transformations: indeed, even to describe the automorphisms of P(X) ([112] p 46), Schein must first embed it in T(XO) in the standard way (see end of section 2). But in [138] Theorem 4, I noted that the following more general result can be proved using simple methods of checking what an epimorphism does to constant (partial) transformations: here , Epi(S) denotes the semigroup of all epimorphisms of S. Theorem 26. If S is any 2-transitive transformation semigroup covering X then Epi(S) = Aut(S).
Note that this provides a simple test for deciding whether a transformation semigroup has "proper" epimorphisms (that is, ones that are not automorphisms); we do not know if it can be extended (as happened for the lAP) to a wider class of semigroups . The ultimate goal is to describe all endomorphisms of some well-known transformation semigroup. However, almost nothing was known about this until Schein and Teclezghi [118] described all endomorphisms of I(X) for finite X. Their result is long and complicated, and is best read in the original version. The most complete result before this was Shutov's work on T(X, No) in [125]. He let P2 = {a,,8} where a,,8 E T(X , No) and
a,8 = ,8a = a 2 = a , ,82 = ,8, and let P3 = {a,,8, I'} where a,,8, I' E T (X, No) and
a,8 = ,8a = a')' = I'a = a 2 = a,
219
We say 7r E G(X, ~o) is even if its restriction to S(7r), the shift of 7r, is even; and we say it is odd otherwise. Define ¢2 : T(X, ~o) ----* P2 so that if A E G(X,
~o),
if d(A) ::::: 1. and define ¢3 : T(X , ~o)
A¢3 =
{
----*
P3 so that
(3,
if A E G(X,
~o)
and A is even,
"
if A E G(X,
~o)
and A is odd,
a,
if d(A) ::::: 1.
Shutov [125] Theorem 2 states that ¢2 and ¢3 are endomorphisms of T(X, ~o) and that the only endomorphisms of T(X, ~o) different from isomorphisms are ones like ¢2 and ¢3, plus those that map T(X, ~o) onto a single idempotent. In passing, we mention some work by Borisov [5] [6]. For any semigroup S, let ~(S) denote the poset of all subsemigroups of S and, for each family r ~ ~(S), let vr denote the smallest subsemigroup of S containing r. In this way, ~(S) becomes a semilattice and we say the semigroups S, Tare structurally isomorphic if the semilattices ~(S) , ~(T) are isomorphic. In [5] [6] Borisov determines when each structural automorphism of a subsemigroup S of P(X) is induced by an automorphism of S: that is, for any automorphism B : ~(S) ----* ~(T), there exists B* E Aut( S) such that AB = AB* for all A E ~(S) . For example, this occurs for S ~ T(X) provided that (1) S contains all non-identity idempotents of T(X), and (2) if id x E S then for each x E X there exist a, (3 E S such that a(3 = id x and xa = x(3 = x. For comparison, if S is any inverse semigroup, Johnston [47] lets L(S) denote the lattice of inverse subsemigroups of S and says a lattice isomorphism B : L(S) ---. L(T) is induced by B* : S --+ T if AB = AB* for all A E L(S). The inverse semi group S is said to be strongly determined by L(S) if every lattice isomorphism of S onto T is induced by an isomorphism of S onto T. She shows that if S is a simple inverse semigroup which is not a group and if its lattice of full inverse subsemigroups is modular, then S is strongly determined by L(S) . It would be interesting to know the extent to which this result is applicable to J(X), or to some other inverse semigroup of transformations .
3.8 Congruences
220
All work concerning congruences on transformation semigroups has its origin in Malcev's epic paper in 1952 [82] describing the congruences on T(X). As noted in [141] p 568, it is likely that Malcev began with the observation that two permutations a, (3 of X differ in ~ places if and only if 8(0.(3-1) = ~. Presumably, he also knew Baer's Theorem (see [121] section 11.3 for a proof): if X is infinite, the non-trivial proper normal subgroups of G(X) are the alternating group on X plus the groups G(X,n) = {a E G(X): 8(0.)
< n}
where No ::; n::; IXI (in fact, Malcev 's paper refers only to Schreier and Ulam's work in 1933 on G(X) for countably infinite X). Of course, the congruences on a group are completely determined by the normal subgroups of the group. Thus, Malcev could have interpreted all except one of the congruences on G(X) as relations p for which there exists an infinite n such that (a, (3) E P if and only if I{x EX: xa i- x{3} I < n . Since the latter formulation makes sense within T(X), it may have led Malcev to the notion of difference rank dr(a, (3) for a , (3 E T(X) (see below for the definition). However, Clifford and Preston [10] vol 2, p 237, "never succeeded in evolving a satisfactory presentation of Malcev's method" of proof and were obliged to use "a sequence of preliminary lemmas", rather than Malcev's process of reducing the problem to earlier work on the normal subgroups of G(X). In fact, if I recall correctly, the publication of the second volume of [10] was delayed while Clifford and Preston attempted to complete a proof of Malcev 's Theorem (after they had already spent 2-3 years, off and on, trying to do so) . Soon after I commenced my PhD in 1965, and they had obtained a valid proof, Gordon Preston asked me to find a simpler one and suggested I should start by determining all the congruences on the Rees factor semigroups Tf;' ITf.. for No ::; ~ ::; IXI (he also suggested they would all be so-called "Malcev congruences" ). To briefly indicate the achievements of those early days, as well as current work in the area, we need some notation and terminology from [10] vol 2, section 10.8. But, since the subject now covers partial transformations as well, we include a little more generality. To simplify matters in what follows, we sometimes write xa i- ((J or xa = ((J to signify that x is or is not in dom a. In addition, for each Y ~ X, we may write Yo. or Ya- 1 and take these to mean (doma n Y)a or (ran a n Y)a- 1 within context. For each n satisfying No ::; n :::; lXI, we let D(a,{3)
=
{x EX: xa
i-
x{3} dr (a,{3)
=
max (ID(a,{3)al, ID(a ,{3){3I)
221 ~n =
{(a,,8) E P(X) x P(X) : dr(a,j3) < n}
(12)
and note that, by [141] Theorem 3.1, each ~n is a congruence on P(X). We use the notation (13) for the restriction of ~n to a subsemigroup S of P(X), and the same notation (in a given context) for its reduction to a semigroup of transformations in which the operation is not exactly composition (compare [141] pp 558-559). The starting point for Malcev 's Theorem is the fact that if p is a non-identity congruence on T(X) then all the constant maps belong to a single p-class which must then be an ideal ([10] vol 2, Lemma 10.64). But, as Malcev showed in [82], this ideal must equal Tf}(p) for some cardinal T)(p) satisfying 2:::; T)(p) :::; IXI'. We call T)(p) the primary rank of p, and the problem of describing the congruences on T(X) divides into two cases depending on whether T)(p) is finite or infinite. The former case is the easier one to resolve and can be stated as follows (see [10] vol 2, Theorem 10.68). Theorem 27. Suppose n is a positive integer and (J is a non-universal congruence on Tn+dTn . Then the relation (J+ defined on T(X) by:
is a congruence on T(X) whose primary rank equals n. Conversely, if p is a non-trivial congruence on T(X) for which T)(p) = n is finite then p = (J+ for some congruence (J on Tn+dTn. Since Tn+dTn is completely O-simple for each finite n and arbitrary X ([10] vol 2, Lemma 10.54), its non-universal congruences are known ([10] vol 2, Theorem 10.58). An account of Tn+dTn as a Rees matrix semigroup when X is finite was provided by Hewitt and Zuckerman in 1957 (see [10] vol 1, p 95). The case when T)(p) is infinite is much more difficult. In fact, its resolution depends on a key result ([10] vol 2, Theorem 10.69) which involves two parts, both of which are problematical in themselves. But another of Preston's research students in 1965, Richard Buckdale, was able to apply Zorn's Lemma to "greatly shorten" the argument required for the first of the two parts. This result ([10] vol 2, Lemma 10.73) has been gradually extended over several years to the following form in [141] Lemma 3.4. Buckdale's Lemma. If a,,8 E P(X) and dr(a,,8) = ~:::: ~o then there exists Y ~ D(a ,,8) such that Ya n Y,8 = 0 and max (IYal, IY!3I) = ~ .
222
The second part of the key result was more troublesome: it asserts that if p is a congruence on T(X) containing a pair (a,j3) where a,j3 have infinite rank 7J but finite non-zero difference rank, then all elements of T(X) with rank at most 7J which differ in a finite number of places are p--equivalent. A proof of this that requires just one Lemma, rather than "a sequence of three tedious lemmas" ([10] vol 2, p 237) was provided in [141] Corollary 2.4. With all this in mind, we can now state the ' hard' part of Malcev's Theorem (compare [10] vol 2, Theorem 10.72). But before doing that, we need one more piece of notation: namely, T; denotes the Rees congruence on T(X) corresponding to the ideal T." of T(X).
Malcev's Theorem. If p is a non-identity, non-universal congruence on T(X) for which 7J(p) is infinite then (14) where 7J1 = 7J(p) and the cardinals ~r
~i '
7Ji form a sequence:
< ... < 6 ::; 7J1 < ... < 7Jr ::; lXI,
(15)
in which every term is infinite, except possibly ~r which equals 1 if it is finite. Conversely, given any sequence of cardinals as described in (15), the relation defined on the right-hand side of (14) is a congruence on T(X) whose primary rank equals 7J1. While Gordon Preston was my supervisor, he observed that every endomorphism of T(X) corresponds to a congruence on T(X) and hence , by Malcev's Theorem, to some sequence of cardinals. But, he asked: which ones? - we are still none the wiser today. Soon after Malcev's paper appeared, Liber [66] published a similar result for J(X) (see [111] for an alternative account: in fact , one using a more 'algebraic' approach to the topic). In 1961 Shutov [126] did the same for P(X) and, almost 20 years later, Mogilevskii [95] generalised this work to stable quasiorder relations on T(X) and P(X) (that is, relations which are reflexive and transitive, and are compatible with respect to composition) . Also in 1961, Shutov determined the congruences on the Baer-Levi semigroups of type (p ,p): in [127] he states without proof that the only non-identity congruences on BL(p,p) are restrictions of the Malcev congruences 6.€ for No ::; ~ ::; pi (proofs of this and other statements about B L(p, p) appeared later
223
in [129]). He then indicates [127] Theorem 10) how to construct a congruencefree right simple semigroup without idempotents: namely, BL(p,p)jop (this was discovered independently by Trotter [151] Theorem 2.4); and he concludes that every right cancellative, right simple semigroup without idempotents can be embedded in a congruence-free right simple semigroup without idempotents (compare [10] vol 2, Theorem 8.5). Later Lindsey and Madison [67] found the congruences on Baer-Levi semigroups of any type (p, q); and more recently Hotzel [31] has described the congruences on dual Baer-Levi semigroups DBL(p, q): namely, for No ::::: q ::::: p = lXI, DBL(p, q) = {a E T(X) : Ixa-Il = q for all x E X}
and the congruences on this semi group form a chain:
where
IXI
=
N7r and or = {(a,.8) : I{x : xa- I -=f. x.8- I }1 < r}.
Returning to 1961 : as already mentioned at (5), Shutov described the ideals of the semigroup M(X) and he states ([127] Theorem 7) that each non-identity congruence on M(X) is a finite combination of Malcev congruences, Rees congruences and normal subgroups of 'J(X), as in Malcev's Theorem above . And from this he deduces Baer's Theorem on the normal subgroups of G(X) for infinite X. More recently, Levi and Schein [60] have investigated the congruences on the semigroup: S = {a E
M(X) : 0 < d(a) < No}
where X is infinite. As usual , Malcev and Rees congruences are relevant, but there is another one defined by:
"V
=
{(a,.8)
E
S x S : d(a)
=
d(.8)} .
In addition, Levi and Seif [62] have described the congruences on any G(X)normal subsemigroup S of M(X) whose elements have infinite defect. It turns out that the lattice Con(S) of all congruences on any such semigroup S has the following properties: (1) semidistributive: for all p, <>, T E Con( S) , p /\ <> = P /\ p /\ (<> V T), and
T
implies p /\ <> =
(2) distributive-over-distributive: for some congruence D on Con(S), Con(S)jD is a distributive lattice and each D-class is also a distributive lattice.
224
By contrast, Clifford and Preston showed ([10] vol 2, Theorem 10.77) that the lattice of congruences on T(X) is distributive. Again returning to 1960: Shutov [125] also investigated the semigroup T(X , No) of all "almost identical" transformations of an infinite set X : that is , all a E T(X) with s(a) < No . As already noted at (4), Vorobev determined its ideals in 1953, and Shutov states without proof that all non-identity, non-universal congruences on T(X , No) can be expressed in terms of Rees congruences and normal subgroups of G(X, No), the group of permutations with finite shift (the proof is possibly contained in [128]). Thus Malcev's Theorem provided a method for describing the congruences on a wide variety of transformation semigroups and, of course, once all congrUtnces are known it is a simple matter to produce congruence-free semigroups. But in 1981 Howie discovered a way of using his semigroup H to construct congruencefree semigroups with desirable algebraic properties, and to do this directly. We now briefly indicate his approach, and start with his notion of 'depth'. In [32] Howie proved that every a E H can be expressed as a product of 4 idempotents, and in [38] he showed the number 4 is best possible: that is, there are a E H which cannot be written as a product of less than 4 idempotents. In fact, he formalised this by saying that, if 5 is an idempotent-generated semigroup and E(5) is its set of idempotents , then the depth .6.(5) of 5 is the least integer n such that 5 = E(5) n (if there is no such n, we write .6.(5) = 00). Thus, he had shown .6.(H) = 4. In fact , he did much more: he also showed in [38] that V is a regular , idempotentgenerated semigroup and .6.(V) = 00; and if we put
Hq
= {u
E
H : d(a)
= q}
where No ~ q ~ IXI then Hq is a regular, idempotent-generated semigroup with depth 4. A key step in proving the latter was a simple result [38] Lemma 3.8 that has been used by other people, so it may be useful to restate it here.
Theorem 28. Let 5 be a regular semigroup. If a E 5 and a = ex for some e 2 = e E 5 and x E 5 then a = f x for some f2 = f E 5 such that f R a. Next, in [36] he recalls an old result of Preston in which regular cardinals are characterised in terms of a set of total transformations being closed under composition ([104] Theorem 1 and [10] vol 2, Theorem 8.51).
225 Preston's Theorem. If X is infinite then following set is a subsemigroup of T(X):
{a ET(X) : r(a)
=
IXI
and
IXI
is regular if and only if the
Ixa-11 < IXI
for all x
EX}.
(16)
In [36] Howie assumes IXI = m (say) is infinite and regular, lets Sm denote the intersection of H and Preston's semigroup in (16), and considers the quotient semigroup Sm/8m (compare after Theorem 23 above) . The main result of [36] can be stated as follows . Theorem 29. Let X be a set with regular infinite cardinal m. Then Sm/om is a congruence-free, bisimple, idempotent-generated semigroup with depth 4. Moreover, it contains an isomorphic copy of every semigroup with order less than m.
In [141] Sullivan generalised most of the results published during the previous three decades concerning congruences on certain transformation semigroups and their use in constructing congruence-free semigroups. However, since the above result depends on X being an infinite set whose cardinal is regular, it could not be included in the generalisation. Therefore, in a separate paper [140] he determined all congruences on Sm: as expected, the only non-identity congruences on Sm are those induced by Malcev congruences. Howie proves the second statement in Theorem 29 (see [36] section 4) by first showing that if S is any semigroup containing at least two elements then there is an embedding S --> Sm where m is an infinite regular cardinal greater than lSI. Therefore, from [140] Theorem 2.6, we know that any semigroup can be embedded in a bisimple idempotent-generated semigroup, all of whose congruences are known: although this requires more effort to prove, it provides more information than Higgins' method of embedding an arbitrary semigroup in a bisimple monoid (see [book] [29] p 22). Next, Marques [85] removed the dependence on IXI = m being regular by first noting that in general {a E H : r(a) < m} = H(m,m) is the largest ideal in Hm = H(m,m') (compare (8)) and that hence the Rees factor semigroup
Rm
=
H(m, m')/ H(m, m)
(17)
is O-simple. In fact, it is O-bisimple [85] Lemma 3.3 and she proves the following result [85] Theorem 3.9: here, Om now denotes the 'reduction' of Malcev's congruence to Rm.
226 Theorem 30. Let X be an arbitrary infinite set and write IXI = m. Then Rm/8m is a congruence-free, O-bisimple, idempotent-generated semigroup with depth 4. Once again, in [86] it was possible to determine all non-identity congruences on Rm as being reductions of Malcev congruences to Rm· Thus , from the 1960s to the 1980s, various authors constructed different examples of congruence-free semigroups based on semigroups of transformations, all of which had the same (infinite) rank (except the zero element if it existed) : their approach was to take a specific semigroup of transformations and factor it out by an appropriate form of the Malcev congruence. From a review (MR90a: 20129), it appears that Feizullaev [19] has found conditions which guarantee that a subsemigroup S of the set {o: E T(X) : r(o:) = 0 is congruence-free whenever ~ < IXI. By this stage, it was clear that it would be useful to know all the congruences on E(X) . Actually, I had already started working on this problem in 1984, during a sabbatical leave with Graham Wood at the University of Canterbury, New Zealand. Of course, the first step was to find the ideals of E(X) and this was done in [106] (see (7) above). But further progress was not made until another sabbatical leave in 1996 and Paula Marques-Smith provided excellent support at the University of Minho, Portugal. To briefly describe some of the non-trivial congruences p on H, we proceed as follows. As with Malcev's Theorem for T(X) , the starting point is the ideal consisting of all 0: E H which are p-related to some constant map. However, if IXI = k ~ No, we know from (7) above that this ideal must equal H(8(p), 'I](p)) for some cardinals 8(p) and 'I](p) satisfying No ~ 8(p) ~ k and 2 ~ 'I](p) ~ k' : we call them the primary defect and the primary rank of p, respectively. The description of the congruences on H depends on the relative size of these cardinals: for simplicity, we state a result from [88] only for the case when No ~ 'I](p) ~ k (and hence 8(p) = k).
It happens that in this case (as for T(X): see [10] vol 2, p 234) to each nonuniversal congruence p on H there corresponds a sequence of cardinals: ~r
< . .. < 6
~
'I](p)
= '1]1 < . .. < 'l]r < 'l]r+l = k'
(18)
which is called the sequence of cardinals associated with p. It can be shown that ~r must equal 1 if it is finite , and all other cardinals in the sequence are infinite.
227
The next result looks very much like Malcev's Theorem. But, it does not depend on Malcev's Theorem, even though analogues of Clifford and Preston's results are used in its proof: in fact, in [88] the authors deduce Malcev's Theorem as a consequence of it. As before, we let H(8, 'r/)* denote the Rees congruence on H determined by the ideal H(8, 'r/).
Theorem 31. Suppose X is infinite and and ~i ' 'r/i be cardinals such that ~T
< ... < 6
:s: 'r/l
IXI =
k. Let r be a positive integer
< ... < 'r/T
:s: k,
(19)
where all the ~,,'r/i are infinite except possibly ~T ' and if it is finite then it equals 1. Then the relation e on H defined by
is a congruence on Hand (19) is its sequence of cardinals. Conversely, if p is a non-universal congruence on H for which 8(p) = k and No :s: 'r/(p) :s: k and if (19) is its sequence of cardinals with 'r/l = 'r/(p) then p = e. The proof of the above result (and the other cases when 'r/(p) = k' and when 'r/(p) < No) depends on the determination of all congruences on every Rees quotient semigroup of consecutive ideals in (8) . Fortunately, however, part of this is already available. For, as noted in [106] p 324, if 2 :s: 'r/ :s: k then H (8, 'r/) equals T." = {a E T(X) : r(a) < 'r/} and the congruences on T.", IT." (= DT~, say) are known: if 'r/ is finite, DT~ is completely a-simple [10] vol 2, Lemma 10.54 and so its non-universal congruences are given by [10] vol 2, Theorem 10.58; and if'r/ is infinite, each congruence on DT~ is induced by a Malcev congruence on T(X) [141] Corollary 2.8. To describe the congruences on the other quotient semigroups provided by (8), we first note that by [32] Lemma 6, Ho
= {a E H : d(a) = 8}
is a semigroup whenever No :s: 8 :s: k, and hence for 8 < k, H(8, k')1 H(8', k') is essentially Ho with a zero adjoined. A large part of [88] is devoted to finding the congruences on these semigroups. That leaves just one Rees quotient semigroup determined by (8) whose congruences must still be considered: namely,
Rk
=
H(k, k')1 H(k, k)
228 which was discussed just after (11) and, as already noted, its congruences were found in [86]. However, it is worth noting that this result can be deduced from a more general one in [141]. That is because Rk can be regarded as the semigroup:
{a E H: d(a) = r(a) = k} U {O} in which the product of two elements is 0 if its rank is less than k (recall that Hk is a semigroup). With this in mind, Rk is a proper subsemigroup of DT~ = Tk,/Tk . In fact, it is replete in DT~ in the sense of [141]: that is, for all a, {J E Rk and "( E Dn, a R "( L {J implies "( E R k · For, as noted in [141], Green's Rand L relations on DT~ are entirely similar to those on T(X), and so aoa- 1 = "(0"(-1 implies k = c(a) = cb), and X"( = X{J implies db) = d({J) = k (hence sb) = k since Db) <;;; S("()). Moreover, Rk contains the set
{a E H : d(a) = r(a) = k and
lya- 1 1= k for
some y E X} U {O} .
Consequently, the congruences on Rk are found from [141] Theorem 2.7 which we state as follows. Theorem 32. Suppose l'{o :S n :S k = DT~ containing the set: Kn
= {a
E DT~
: d(a) = k and
IXI
and S is a replete subsemigroup of
lya- 1 1= k
for some y E X} U {O}.
If p is a non-identity, non-universal congruence on S then p = 0<. for some satisfying l'{o :S ~ :S n.
~
The congruences on the whole of E(X) will be revealed in [144]. For major surveys on the lattice of congruences on arbitrary semigroups, see [93] and [94]. 4. The Future Recall that semigroup theory began with Suschkewitsch using T(X) to illustrate the structure of a finite simple semigroup and that he used an early form of Clifford and Preston's "eggbox diagram" in doing so ([10] vol 1, p 207). In other words, right at the start, transformation semigroups were recognised as important both to illustrate the more abstract side of the subject and to clarify the basic structure of special types of semigroups. Moreover, this was done initially in a finite, almost combinatorial, context and in the last decade
229 the interest in the general area of combinatorial mathematics has grown enormously. Consequently, in this section, I begin by suggesting there should be more interaction between the "abstract" and the "concrete", between algebraic semigroup theory and transformation semigroup theory. Then I list a few connections with combinatorics that are worth exploring further. And finally there are some ideas for research on transformation semigroups themselves, as well as their relation with other areas of mathematics. Once again, these reflect the personal interests of the author: there are many other possibilities, some of which have been mentioned in section 3.
4.1 Structures Nambooripad [98] has described the structure of all regular idempotent-generated semigroups in terms of biordered sets and inductive groupoids. It would be useful to know how Howie's semigroup E(X) = V U H, and various subsemigroups of it, can be described using Nambooripad's ideas. Likewise, in [90] Meakin explains how so-called "structure mappings" can be used to describe all possible inverse semigroups. Some authors have studied particular examples of inverse transformation semigroups, so it would be interesting to know how these and their special properties fall within Meakin 's theory. More generally, it would be helpful to determine some naturally occurring classes of transformation semigroups with prescribed properties (for example, orthodox) so they can be used as exemplars for all the algebraic work being done on such semigroups.
4.2 Combinatorics Given a family F of non-empty sets, we can form the intersection graph IG(F) on F by regarding the elements of F as vertices and saying two vertices A, B E F are joined if AnB =f. 0. In particular , if S is a subsemigroup ofT(X) and F is the family of all principal right ideals of S, then Levi and Seif [63] have related properties of IG(F) to properties of the range of S (that is, Ran(S) = {Xa : a E S}) and they have used this relation to investigate left zero congruences on S. Conversely, it would be interesting to study families F of non-empty subsets of X and transformation semigroups S which preserve the connectedness of IG(F): that is,
An B
=f. 0 ===>
Aa n Ba
=f. 0
230 for all A, B E F and a E S.
In particular, if S(X, F) denotes the semigroup of all intersection-preserving transformations of the graph IG(F), under what conditions is an isomorphism S(X, F) ----> S(Y, Q) induced by a bijection X ----> Y? When is S(X, F) idempotent-generated, and how is this reflected in IG(F)? Many other questions can be raised along the lines reported in section 3. See [96] for a survey of "endomorphism semigroups of graphs" . Suppose D = (V, B) is a block design. Following Margolis and Dinitz [S4] we say a E P(X) is continuous if, for each B E B , Ba- 1 is empty or belongs to B, and we let C(D) be the semigr()up of all continuous transformations of D. It is then possible to explore the relation between combinatorial properties of D and algebraic ones of C(D). For example, it is shown in [S4] that if D is a BIBD satisfying some restrictions on its parameters then C(D) is the union of a group of units and a completely a- simple ideal. 4.3 Congruences
The semigroup S(X) of all continuous mappings of a topological space X into itself is best left to recognised experts like Ken Magill and his co-workers. However, often they introduce structures of a purely set-theoretic nature that deserve further study in the context of transformation semigroups. For example, to generalise [SO] we let 2: be a family of non-empty subsets of a set X and let S be a (total) transformation semigroup defined on X. We say 2: is a unifying family for S if, whenever A E 2: and a E S is injective on A, then Aa E 2:. We now define a relation 7r(2:) on S via: (a, (3) E 7r(2:) if and only if, whenever one of a, (3 is injective on some A E 2:, then a , (3 coincide on A. Clearly, 7r(2:) = 7r (say) is reflexive and symmetric. Suppose (a, (3) E 7r and ((3,,,/) E 7r and a is one-to-one on A. Then a,(3 agree on A , hence (3 is oneto-one on A, so (3,,,/ agree on A: that is , (a,,,/) E 7r and 7r is transitive. Also, if ap, is one-to-one on A then a is one-to-one on A , so a, (3 agree on A, hence ap,,(3p, agree on A: that is , (ap,,(3p,) E 7r. And if >.a is one-to-one on A then a is one-to-one on A>' , so a, (3 agree on A>' , hence >'a, >.(3 agree on A : that is, (>.a, >.(3) E 7r. Therefore, 7r(2:) is a so-called unifying congruence on S, and Magill [SO] determines the largest unifying congruence on S(R), the semigroup of all continuous self-maps of R, the set of all real numbers. But, in the context of transformation semigroups, we can ask: for which families 2: , does 7r(2:) equal a Malcev congruence or Rees congruence on T(X)? Can we use Malcev 's Theorem to decide which congruences p on T(X) equal
231 7r(~) for some family ~? Does the collection of all families ~ for which 7r(~) equals some fixed P have some special features? For a given transformation semigroup S, can we always deterr.line the largest 7r(~) on S? When does the collection of all unifying congruences on S form a complete lattice? - compare [76] for topological spaces .
Actually in [79] Magill had already observed that to each unifying congruence 7r = 7r(~) there is a corresponding ideal J (7r), and he described such ideals in terms of the unifying family~. Then he studied the poset of all such ideals and investigated when one of them is prime. How is this related to our knowledge of the ideal structure of certain transformation semigroups? Likewise, in [77] Magill found a class of topological spaces X for which the family of all unifying congruences on S(X) is finite. Are there infinite transformation semigroups S ~ T(X) for which this same family is finite?
In a similar vein, Petrich and Rankin [102] investigated the congruences on an inverse semigroup S that are induced by transitive representations of S by oneto-one partial transformations. From the work of Liber [66] and Scheiblich [111] we know the congruences on I(X), and other people have produced congruencefree inverse semigroups. How is all of that work related to the results of Petrich and Rankin? Suppose S is a semigroup and K ~ S has the property: aK = K a for all E S. In [150] the authors define a congruence PK on S via: (a, b) E PK if and only if aK = bK . Depending on Sand K, PK could be trivial. Are there transformation semigroups S with subsets K for which PK is non-trivial? and for which every congruence equals PK for some K?
a
Trotter [152] says a subsemigroup A of a regular semigroup S is self-conjugate if xax' E A for all xES, a E A and x' E V (x), the set of all inverses of xES. And he describes certain congruences on some (abstract) self-conjugate semigroups. Is this related in some way to G(X)-normality for transformation semigroups? and can we determine all self-conjugate subsemigroups of some transformation semigroups?
4.4 Cardinals We have already mentioned Presto:,'s result stating that IXI is a regular infinite cardinal if and only if
{a E T(X) : r(a) = IXI and Ixa-Il < IXI for all x E X}
232 is a subsemigroup of T(X). We say a regular cardinal IXI is inaccessible if for any sets Y, Z with WI and IZlless than lXI, we have IT(Z, Y)I < IXI
where T(Z, Y) is the set of all mappings from Z into Y In [104] Theorem 3, Preston characterises inaccessible cardinals in terms of sets Y with 1 < WI < IXI and the semigroups consisting of all transformations of the set T(X, Y) into itself. It would be more useful to have a characterisation in terms of T(X) itself, possibly a subsemigroup consisting of transformations that preserve some structure on X . Given that there have been major advances in axiomatic set theory over the last two decades, it would also be interesting to relate the existence of other "large" cardinals to properties of T(X) [13] . In particular, is GCH equivalent to some statement concerning a subsemigroup of T(X)? For example, remarks in [107] p 137 suggest that if GCH is assumed and if H is an arbitrary Hilbert space then the set
{a E B(H) : 8(a)
=
v(a)
=
O"(a) 2: No}
is an idempotent-generated subsemigroup of B(H) , the semigroup of all bounded linear operators on H (here, the cardinals 8(a), v(a) and O"(a) are the Hilbert space analogues of defect, collapse and shift for a E T(X) : see [107] p 135 for the details). If this is correct, is the converse also valid? This would reveal the limit of any extension to Dawlings work [11] on the semigroup generated by the idempotent operators on a separable Hilbert space. 4 .5 Ideals
In [134] Sullivan determines the ideals I of T (X), P(X) and I(X) of a prescribed type: semiprime (that is, a 2 E I implies a E I: see [10] vol 1, section 4.1) and reflective (that is, ab E I implies ba E I : see [71]). But there are others to consider. For example, in [91] Miccoli says a subset B of a semigroup S is a bi-ideal if BSB ~ B (also see [10] vol 1, p 84, Exercise 15). And Steinfeld [131] has popularised the notion of a quasi-ideal Q of S: that is, Q satisfies QS n SQ ~ Q. Can we determine all such ideals in certain transformation semigroups? In [101] Pastijn shows that if S is a regular semigroup then the regular semigroup Q(S) of all quasi-ideals of S is locally testable (that is , AQ(S)A is a semilattice for all idempotent quasi-ideals A = A2 of S). Moreover, if S is
233 both regular and locally testable then the mapping S ---> Q(S) , a ---> aSa, is an embedding. More generally, can we characterise the semigroup Q(S(X)) of all quasi-ideals of a given transformation semigroup S(X)? And when is every isomorphism Q(S(X)) ---> Q(S(y)) induced in a natural way by a bijection X ---> Y?
4.6 M orphisms As a first step towards finding all endomorphisms of a given transformation semigroup S, we could determine all endomorphisms of S that have a specific property. For example, if the congruences on S are known, we could follow [105] and sayan endomorphism
4. 7 Tolerances A reflexive and symmetric binary relation T on a set X is called a tolerance on X [14] and we let T(X) denote the semigroup of all total transformations of X which preserve T. Can we determine the automorphisms, ideals and congruences on T(X) or some subsemigroup thereof? In [103] Pondelicek adopted a different approach and considered the set Tol (S) of all stable tolerances on a semigroup S (that is, congruences without the transitivity condition). It is easy to see that Tol (S) is a lattice under nand U, and it would be interesting to find its elements when S is a given transformation semigroup. References
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THE FINITE BASIS PROBLEM FOR FINITE SEMIGROUPS: A SURVEY M.V.VOLKOV Department of Mathematics and Mechanics, Ural State University 630083 Ekaterinburg, RUSSIA E-mail: [email protected] We provide an overview of recent research on the natural question what makes .. finite semigroup have finite or infinite identity basis. An emphasis is placed on results published since 1985 when the previous survey of the area had appeared. We also present a few unpublished results and formulate several open problems.
Introduction In his 1966 Ph.D. thesis ([48], see also [49]) Peter Perkins proved that the 6-element Brandt monoid B~ formed by the 2 x 2-matrix units
together with the zero and the identity 2 x 2-matrices admits no finite set of laws to axiomatize all identities holding in it. His striking discovery strongly contrasted with another fundamental achievement of the equational theory of finite algebras which had appeared shortly before: we mean Sheila Oates and Michael Powell's theorem [47] that the identities of each finite group are finitely axiomatizable. It was this contrast that gave rise to numerous investigations whose final aim was to classify all finite semigroups with respect to the property of having/having no finite identity basis. Even though those investigations have not yet led to a solution to this major problem, they have resulted in extremely interesting and often surprising developments. From the points of view of both the intensity and the depth of investigations, a definite peak was reached in the mid-80s. The achievements of that period were cumulated in the survey [65] by Lev Shevrin and the author; many of them had been first announced in the survey and only then appeared in journals in a full form. There are some indications of a new peak that we are approaching at the moment due to contributions of the next generation of researchers. Therefore the time seems to be appropriate for another attempt to survey the area, to say nothing of the millennium edge which naturally provokes one to compile an account of what has been already achieved and what is still to be done. 244
245
The present survey is however not a mere continuation of [65]. First of all, it is less ambitious concentrating entirely on the finite basis problem for finite semigroups, while [65] intended to cover the whole area "Identities of semigroups". Further, since the English version of [65] is not easily accessible and the quality of the translation is rather bad, we have decided to make the present survey, to a reasonable extent, self-contained even though this decision has caused a few overlaps with [65]. The survey is structured as follows. Section 1 gives an overview of necessary prerequisites. In Section 2 we recall the main open problems of the area; here we closely follow [65], Section 8. In order to create a feeling as to why the problems are so difficult to handle, we collect in Section 3 a few facts that demonstrate the extremely irregular behaviour of the class of finite semigroups with a finite identity basis with respect to almost all standard constructions and operators of semigroup theory. This section is based on [65], Section 11, but we provide several references which were not available when [65] appeared. The core of the survey is Section 4. There we analyze the methods developed for finding finite semigroups without a finite identity basis. It is the subarea that advanced most over the last couple of years. We give a classification of the methods and then present several recent results (of which many are not yet published). Section 5 is devoted to the opposite question: how to prove that a given finite semi group has a finite basis. In contrast to the previous section, it is quite short due to the rather modest progress in this direction. Finally, in Section 6 we list three series of finite semigroups for which the finite basis problem resists the known methods though it appears to be of great importance for further developments. 1
Preliminaries
As far as semi groups are concerned, we adopt the standard terminology and notation from [13] and [19]. Our main sources for universal algebra notions are [11] and [18]. We recall some of those notions adapting them to the semigroup environment. Let A be a count ably infinite set called an alphabet. We will assume that A contains the letters x, y, z with and without indices. As usual, we denote by A+ the free semigroup over A, that is, the set of all words over the alphabet A with word concatenation as the multiplication operation. Sometimes it is convenient to adjoin the empty word 1 to A+ thus obtaining the free monoid A* . By == we denote the equality relation on A*. A non-trivial semigroup identity over A is merely a 2-element subset { u, v} c A + usually written as u = v . A semigroup S satisfies the identity
246 u = v if the equality u
identities it satisfies. Given any collection E of non-trivial seroigroup identities (an identity system, for short), we say that a non-trivial identity u = v follows from E or is a consequence of E if every seroigroup satisfying all identities of E satisfies the identity u = v as well. The following well-known completeness theorem of equational logic (first discovered in Garrett Birkhoff's pioneering paper [9]) provides a syntactic counterpart to this important notion: Proposition 1.1. A non-trivial semigroup identity u = v follows from an identity system E if and only if there exists a sequence WQ, Wl,··· , Wk E A+ such that
• u
== WQ and v == Wk,
• for every i = 0,1, . . . , k - 1, there are words ai, bi E A*, Si, ti E A+ and an endomorphism (i : A+ -4 A+ such that Wi == ai(si(i)bi , Wi+l == ai(ti(i)bi and the identity Si = ti belongs to the system E. For an identity system E , we denote by Id E the set of all consequences of E. Given a semigroup S, an identity basis for S is any set E ~ Id S such that Id E = Id S or, in other words, such that every identity of Id S follows from E. A seroigroup S is said to be finitely based if it possesses a finite identity basis; otherwise S is called nonfinitely based. Let us briefly discuss an interesting subtlety which arises here. Since we are going to focus on identities of finite seroigroups, it appears to be rather natural to restrict the definitions above to the class 6 of all finite seroigroups. Thus, we could say that an identity u = v follows within 6 from a system E if u = v holds in every finite seroigroup satisfying E, and we could then call a finite seroigroup S finitely based within 6 if every identity of Id S follows within 6 from a finite subsystem E ~ Id S. Fortunately, this modified version of the finite basability of finite seroigroups turns out to be equivalent to the standard one as was shown by Mark Sapir [57]. It does not mean, however, that the two notions of a consequence of an identity system ("absolute" and within 6) are equivalent! The following example from [12] illustrates this: Example 1.1. The identity x 3yz3x 2 = (yx)3 zx3(yx)2 does not follow from the identity system E = {x 5 = y5, x 3yz3x = (yx)3 zx 3yx }, but follows from E within the class of all finite seroigroups. A similar remark can be made about the relationship between the finite basis properties of a finite monoid M as an algebra of type (2,0) and as a seroigroup: M is finitely based within the class of all monoids if and only if it
247 is finitely based in the standard sense, that is, in the class of all semigroups a . As above, this does not mean the equivalence between the "monoid" and the "semigroup" notions of a consequence of an identity system: for instance, xy = xz implies y = z in any monoid, but not within the class of all semigroups. Given a semi group S, the class of all semigroups satisfying all identities from Id S is the variety generated by S; we denote this variety by Var S. By the classic HSP-theorem by Birkhoff [9], VarS = JH[§JP(S) where !HI, §, JP are respectively the operators of taking homomorphic images, subsemigroups, and direct products. We call a variety finitely generated if it is generated by a finite semigroup. We will encounter also the operator JP fin of taking finitary direct products. Recall that a semigroup pseudovariety is a class of finite semigroups closed under !HI, §, and JP fin. The theory of pseudovarieties has its own variant of the finite basis problem based on the notion of a pseudoidentity, see [3). Fortunately again, when applied to a single finite semigroup, this version of the finite basability also reduces to the standard one: a finite semigroup S possesses a finite pseudoidentity basis if and only if S has a finite identity basis (Jorge Almeida [1], see also [3], Corollary 4.3.8). 2
General problems
As was said in the introduction, an ultimate solution to the finite basis problem for finite semi groups would consist in a method to distinguish between finitely based and nonfinitely based finite semigroups. In more precise terms, since any finite semigroup S is an object that can be given in a constructive way (by its Cayley table, say), what we seek is an algorithm which when presented with an effective description of S, would determine whether S has a finite identity basis. This formulation of the finite basis problem as a decision problem is due to Alfred Tarski (see [66)) who suggested it in the early 60's in the most general setting, that is, for the class of all finite algebras. We will refer to the restrictions of that general problem to various concrete classes of finite algebras (say, groupoids, semigroups, etc) as Tarski's problems for groupoids, semigroups, etc. With this convention, we may say that the research reported in the present survey groups around Tarski's problem for semigroups. Let us formulate the latter problem explicitly: Problem 2.1. ([65], Question 8.3; [63), Question 3.51) Is there an algorithm a We are not sure that this claim has been explicitly made in the literature, but it can be easily verified.
248
that when given an effective description of a finite semigroup S decides if S is finitely based OT' not?
Problem 2.1 is still open. We mention that, in contrast, Tarski's problem for groupoids has been recently solved in the negative by Ralph McKenzie [42]. An algorithm is known to decide whether the semigroup identities of a finite inverse semi group S possess a finite basis [76]. It is based on the fact that S is finitely based if and only if the Brandt monoid B~ does not belong to the variety Var S (it follows from the proof of the HSP-theorem that the latter condition can be algorithmically tested when given the Cayley table of S). The "if' part was established in [76], while the "only if' part is a consequence of Mark Sapir's results [55] which we discuss in Subsection 4.4. Here is the appropriate place for an important warning: the above algorithm does not yet provide a solution to the Tarski problem for inverse semigroups as algebras of type (2,1) . Even though it follows from a comparison between [76] and [27] that the "inverse" (that is, of type (2,1)) identities of a finite inverse semigroup are finitely based whenever its "plain" (of type (2) ) identities are , as yet we do not know whether the converse holds true b • Question 8.2 in [65] asks if the algorithm from [76] extends to finite orthodox semigroups. Though this question still remains open, very recently Marcel Jackson ([21], see also [22]) has shown that the algorithm can be used to decide whether a given finite orthodox monoid is finitely based. In variety language, Tarski's problem is the problem of an algorithmic selection of finitely based varieties among finitely generated ones. It is also very natural to ask a "reverse" question in which one looks for an algorithm to select finitely generated varieties among finitely based ones. This problem was also proposed by Tarski in [66], again for general algebras. It was solved in the negative by Murskii [45] who proved that even in type (2) there is no algorithm to determine if a given finitely based variety is generated by a finite groupoid. When further specialized to semigroups, the problem however remains open. Here is its explicit formulation: Problem 2.2. ([65], Question 8.4) Is there an algorithm that, given a finite identity system L: , decides if Id L: = Id S foT' some finite semigroup S? An algorithm is known when the system L: contains the commutative law b It was claimed in [65], p.19 of the English translation, that the "inverse" and the ·"plain" identities of every finite inverse semigroup are simultaneously finitely based. This claim, based on an announcement by Mark Sapir, had spread out and had even penetrated into the handbook [64]. Later Sapir [59] discovered that the announced result was wrong, and therefore, the question of the equivalence between the two versions of the finite basability of finite inverse semi groups should be treated as open.
249
= yx (Olga Sapir, [60]), and even this case is far from being trivial. It would be interesting to know if this algorithm extends to the case when E contains a permutation identity, that is, an identity of the form xy
where
7r
is a non-identical permutation on the set {I, ... ,n}.
Returning to the problem of distinguishing between finitely based and nonfinitely based finite seroigroups, we may ask what happens "on average" if one picks a random finite seroigroup 8. It turns out that such a seroigroup is very likely to be finitely based. To formulate this claim in precise terms, we denote by FBs{n) and NFBs{n) the numbers of respectively finitely based . . . . NFBs{n) and nonfinitely based senngroups wIth n elements. Then the ratIo FBs{n) tends to 0 as n tends to infinity. The reason for that is rather simple: it is known (see [30]) that the ratio of the number of 3-nilpotent seroigroups with n elements to the number of all seroigroups with n elements tends to 1 as n tends to infinity, and it is easy to see that every 3-nilpotent seroigroup is finitely based. One may want to exclude the trivial case of nilpotent seroigroups by switching to monoids, but it makes no real difference: if FBM{n) and NFB M (n) denote respectively the numbers of finitely based and nonfinitely based monoids with n elements, then again lim
n--+oo
N::({~) M n
= O. The
reason for that is similar to the seroigroup case: on one hand, as was shown in [31], almost all monoids with n elements are of the form 8 1 where 8 is a 3-nilpotent seroigrouPi on the other hand, each monoid of this form satisfies the identity xyx = x 2 y, whence it is finitely based by a result from [51]. NFBs{n) NFBM{n) . Since we know that both FBs{n) and FBM{n) are infinitesimals as n
tends to infinity, the next natural step is to estimate the order of these in-
finitesimals. For groupoids, Murskii [46] has proved that the ratio
NFB({~)
FBc n is asymptotically equal to n- 6 (where, clearly, FBc(n) and NFBc{n) denote respectively the numbers of finitely based and nonfinitely based n-element groupoids). Having in roind an answer of a siroilar flavour, we formulate Problem 2.3. ([65], Question 8.5) What is the asymptotic behaviour of the NFBs{n) NFBM{n) ratios FBs{n) and FBM(n) as n tends to infinity?
The last of the general problems which we want to recall is related to the notion of an irredundant identity basis. We say that an identity system E is irredundant if Id E' ~ Id E for each proper subsystem E' ~ E. Clearly,
250
if a semigroup S has a finite identity basis, then S also has an irredundant basis. The notion was invented soon after the first examples of nonfinitely based semigroups had arisen, in a hope that those "bad" semigroups could retain at least this property of their "good" (that is, finitely based) relatives. Unfortunately, the hope has proved to be too optimistic: not only are there various examples of finite semigroups without an irredundant identity basis (see [34,58,25]), but moreover no finite semigroup with an infinite irredundant basis is known so far. Now it rather appears that the answer to the following question might be negative: Problem 2.4. ([62], Question 2.51a; [65], Question 8.6) Is there a finite semigroup with an infinite irredundant identity basis '? With respect to this problem, a result by Avraam Trahtman [72] is worth mentioning. Namely, he has proved that a 6-element semigroup possesses an infinite irredundant identity basis within a certain variety of semigroups. The 6-element semigroup that appears in this result is A~, where A2
= (a, b I aba = a 2 = a, bab = b, b2 = 0)
is the 5-element idempotent-generated O-simple semigroup which can be alternatively described as the semigroup formed by the following 2 x 2-matrices:
The semigroup A2 as well as the 5-element Brandt semigroup B2 plays a distinguished role in the theory of semigroup varieties, and we will meet it again in this survey. 3
Irregularities
Let ~~ denote the class of all finitely based finite semigroups. As we already mentioned, this class is rather irregular in the sense that-up to only two exceptions-;J~ is closed under no standard operator or construction. In Table 1 (on the next page) we have collected a few references to results revealing such irregular behaviour. Similarly, ~~ is not closed under taking ideals or Rees quotients, forming O-direct unions or ordinal sums, building power semigroups, etc (cf. [65], Section 11, for a detailed discussion). The first of the two exceptional constructions which do preserve the finite basis property is the mere adjoining 0 to a (not necessarily finite) semigroup S. The fact that the semigroup Sa is finitely based whenever S is immediately follows from a combination of two observations. The first one is that S =1= Sa implies Var Sa = Var SVSC, where SC stands for the variety of all semilattices
251 Table 1. Semi group constructions and operators vs. the finite basis property
An example of a nonfinitely based finite semigroup being:
a subsemigroup or a homomorphic image of a finitely based finite semigroup the direct product of two finitely based semigroups the semidirect product or the wreath product of two finitely based semigroups
follows from [58] , Corollary 2.4; can be found in [77,58,60,21]; can be found in [20,2 ,67] ;
a right zero band or a left zero band of finitely based semigroups
follows from [35] , see [65] , Section 11 ;
a semilattice of finitely based groups an ideal extension of a finitely semigroup by another finitely semigroup the monoid Sl for some finitely semigroup S
follows from [49], see [65]' Section II.
semibased based based
and V denotes the varietal join c; this is easy and well-known. The second observation is that , given a finite identity basis for S, one can construct a finite basis for the join Var S V S.c ; this is due to Igor Mel'nik [43) . The second "finite basis-friendly" construction is inflation defined as follows: given any family {Q s} sES of disjoint sets indexed by elements of a semigroup S and such that Q s n S = {s}, the inflation is the semigroup on the carrier set U Qs and with the multiplication defined by a· b = st for sES
all a E Qs , bE Qt , s, t E S. Any inflation T with IQsl > 1 for at least one s E S is easily seen (and well-known) to be a subdirect product of S with a zero multiplication semi group whence Var T = Var S V ZM, where ZM denotes the variety of all zero multiplication semigroups. A specialization of another result by Mel'nik [44] (formulated in [44) in the universal algebra setting) shows how to construct a finite identity basis for the join Var S V ZM provided a finite basis for S . Therefore T is finitely based whenever Sis. Since the class ~~ fails to be lHI- , §- or IP fin -closed , it is quite natural to ask for a description of the closures of ~~ under the operators lHI, §, IP fin C By the join X V Y of two semigroup varieties X and Y we mean the least variety containing both X and Y; in other words, X V Y = IHISIP'(X u Y) .
252
and their combinations. The following problem is especially intriguing: Problem 3.1. What is the 1Hl§lP' fin -closure of the class ~23, that is, the pseudovariety generated by ~23 ? In particular, is this pseudovariety finitely based? In connection with Problem 3.1, the following new result (due to the author) seems to be worth mentioning: Proposition 3.1. For each n 2: 5, the variety generated by all finitely based semigroups with n elements is nonfinitely based. We will prove Proposition 3.1 in Subsection 4.3 below. 4 4.1
How to prove that a finite semigroup is nonfinitely based A classification of methods
We start with a rough description of the main ideas which underlie the known results showing the absence of a finite basis for the identities of a finite semigroup. Browsing through the literature, one may observe that in spite of the apparent diversity of the methods in use, they clearly group around four basic approaches. Here we attempt to present these approaches in the general form, while the rest of the section surveys their concrete incarnations. 1. Syntactic analysis. These methods directly appeal to the syntactic characterization of the deducibility of semigroup identities provided by Proposition 1.1. In order to show that Id S has no finite basis we first find a specific infinite series E of identities from Id S and then verify that due to the constraints caused by certain peculiarities of S, "long" identities of E cannot be formally deduced from any set of "short" identities in Id S. Successful implementations of this scheme include Perkins's method [48,49] that brought the very first examples of nonfinite1y based finite semigroups, the methods applied by Cristine Irastorza [20] and Jorge Almeida [2] to study the finite basis problem for certain semidirect products, Avraam Trahtman's method from [72], as well as the methods of the recent investigations by Marcel Jackson and Olga Sapir [21,24,60,61]. We briefly describe Perkins's method and its major applications and survey some results from [21,24,60,61] in Subsection 4.2. 2. Critical semigroups. Let V = Var S. For each positive integer n, we denote by v(n) the variety defined by all identities in no more than n variables that hold in V. Alternatively, v(n) can be described as the class of all semigroups whose subsemigroups with no more than n generators lie in V. It is clear that v(n ) ~ V for every n and that V = v(n) for some n if V is finitely based. On the other hand, it is well-known (see, for example, Section
253 27 of [18]) that the fact that V is finitely generated implies that every variety v(n) is finitely based. Thus, the equality V = v(n) for some n is not only necessary but also sufficient for S to be finitely based. Therefore showing that S is nonfinitely based is equivalent to proving that, for any n, the containment v(n) ;;:? V is strict, that is, there exists a semigroup Sn E v(n) \ V. We call semi groups Sn obeying the latter requirement critical with respect to S. To build a series of critical semigroups with respect to a given semigroup S, one needs a construction which is highly sensitive to removing a generator. Surprisingly enough, the classical construction of a Rees matrix semigroup over a group with zero has proved to be extremely appropriate for such a role. We present several methods based on the use of Rees matrix semigroups as critical semigroups in Subsection 4.3. These methods originated in a trick from Grigory Mashevitzky's note [34]; since then they have been essentially developed in [77,40,41]. "Rees matrix" methods have also been extended to the unary semigroup environment [4,6,79], to pseudovarieties [78,79]' and to so-called collective identities of finite semigroups [38]. A different way to mastering a series of critical semigroups was utilized by Evgeny Kleiman in [28] in order to show that the Brandt monoid B~ is nonfinitely based as an inverse semigToup. His critical semigroups arose as specific transformation semigroups; a similar approach has proved to be effective for solving the finite basis problem for several important varieties and pseudovarieties, see [29,14,53,80]. Yet another trick was used by Peter Trotter and the author [74] in the pseudovariety setting; in fact, the series of critical semigroups from [74] can be also applied to prove that certain finite semigroups are nonfinitely based (unpublished). For instance, the direct product J15 x G is nonfinitely based where G is an arbitrary non-abelian finite group and J 15 is the I5-element .J -trivial semigroup generated by the elements eo, . .. ,e4 subject to the relations e;=ei, eiej=O(i,j=O, . . . ,4, j=j:.i,i+I (mod 5)), e4eOel=O.
3. Finitely inexpressible properties of finitely generated varieties. Let e be a property of semigroup varieties such that 1) every finitely generated variety obeys
e;
2) any finitely based variety satisfying e must fulfil a certain additional restriction (say, possess an identity of a specific form). Then any finite semigroup S such that Var S violates the restriction e is nonfinitely based. Thus, every such property e may be a powerful source of examples of nonfinitely based finite semigroups.
254
Of course, it is very far from being obvious that there exists any e satisfying the requirements 1) and 2) above. A striking discovery by Mark Sapir [55) was that such a property does exist; namely, he proved that if all nil-semigroups from a finitely based variety V are locally finite , then V satisfies a non-trivial identity of the form Zn = w , where the sequence {Zn} of Zimin words d is defined by: Zl
== Xl, and Zn+l
= ZnXn+1Zn·
Since in every finitely generated variety all semigroups are locally finite (a well-known corollary of the standard proof of Birkhoff's HSP-theorem) , the property of varieties to contain only locally finite nil-semigroups satisfies both 1) and 2) . Therefore if a finite semigroup S has no non-trivial identity of the form Zn = w , not only is it nonfinitely based but it belongs to no locally finite finitely based variety. Semi groups obeying the latter condition are called inherently nonfinitely based. The inevitable Brandt monoid B~ serves as a concrete example of an inherently nonfinitely based finite semigroup. We discuss the basic facts about inherently nonfinitely based finite semigroups as well as the recent developments around them in Subsection 4.4. 4. Interpretation methods. Interpretation is a fundamental tool in the study of algorithmic problems and in complexity theory: in order to prove that a problem A is undecidable (hard) , we usually interpret in terms of A another problem B which is already known to be undecidable (or, respectively, hard). A similar idea may be applied to the finite basis problem which- in view of the completeness theorem of equational logic (see Proposition 1.1 above )- may be thought of as the finite axiomatizability problem for a specific deduction system. In order to show that the collection Id S of all identities of a given semigroup S has no finite basis, we may try to interpret within Id S another deduction system of which we know that it is not finitely axiomatized. Making this fuzzy idea more precise, recall that a deduction system is any set Q equipped with an inference relation f- between Q and the set of finite subsets of Q. Let F stand for the "transitive closure" of f-: for a subset p ~ Q , P F q if and only if either q E P or there is a finite subset R ~ Q such that R f- q and P F r for all r E R. The deduction system (Q, f- ) is said to be .finitely axiomatized if there is a finite subset F ~ Q such that F F q for all q E Q. By a dense interpretation of the deduction system (Q, f-) within Id S we mean a mapping; : Q -+ Id S such that d Named so after Anatoly Zimin whose crucial paper [81J has revealed the role these words play in the Burnside-type problems. See Section 3.3 of the survey [26] for an enthusiastic discussion of the history of Zimin words and of their importance.
255
• {ql, . . . , qn}
Fq
if and only if the identity
q follows from the identities
ql,··· ,qn ; • the identity system Q forms a basis of Id S. With these definitions we immediately obtain Proposition 4.1. If a deduction system (Q, f-) admits a dense interpretation within the set Id S of all identities of a semigroup S , then S is finitely based if and only if the system (Q, f-) is finitely axiomatized. Thus, to show that S is nonfinitely based, it indeed suffices to densely interpret within Id S a suitable non-finitely axiomatized deduction system (or, vice versa, to densely interpret Id S within such a system). So far the interpretation approach appears to be rather underexploited; there are however two very important applications of this idea. Crigory Mashevitzky [35) has found an example of a nonfinitely based finite simple semigroup by interpreting the identities of a Rees matrix semigroup over a group G in the identities of G with a distinguished subset, and Mark Sapir [58) has constructed several surprising examples of nonfinitely based finite semigroups by interpreting in their identities some "weak" deduction systems defined on relatively free periodic groups. We will discuss these two interpretation methods in Subsection 4.6. 4. 2
Syntactic methods
We have chosen Perkins's method to play the role of a representative of the group of syntactic methods: not only was it the very first tool developed for proving that a finite semigroup is nonfinitely based, but being simple enough, it nevertheless demonstrates two technical notions which are crucial for all such methods. The first of those notions is that of an isoterm: a word u is said to be an isoterm relative to a semigroup S if S satisfies no non-trivial identity of the form u = v ; more formally, if u r/:. U Id S . The second one is closedness under deletion. Denote by c( u) the content of u, that is, the set of all letters occurring in u . An identity system E is closed under deletion if for any consequence u = v of E, c(u) = c(v) and if Ic(u)1 > 1 and all occurrences of some letter in u and in v are deleted the resulting identity either is trivial or follows from E . It is easy to see that if S = Sl and S is not a group , then Id S is always closed under deletion. Now we can formulate Perkins's result. Theorem 4.1. ([49), Theorem 7) Suppose that a semigroup S possesses the following four properties:
256
1) the words xyzyx and xzyxy are isoterms relative to 8;
2) 8 satisfies neither of the identities x 2y = y 2x and (xy)2 = xy 2x; 3) the identity system Id 8 is closed under deletion; 4) for n = 1, 2, . .. , 8 satisfies the identity XYI ... YnXYn .. • Yl = XYn .•. YI X YI . • • Yn'
Then 8 is nonfinitely based. Perkins has then verified that the Brandt monoid B~ satisfies the conditions 1)-4). Another application of Theorem 4.1 in the same paper [49] is the result to which we referred in Table 1: there is a finitely based-finite semigroup 8 such that 8 1 is nonfinitely based. Nowadays it is clear that the Brandt semigroup B2 might have served as such an example since Ttahtman [69] has proved that it is finitely based. Fortunately the latter fact was not known in 1966--0therwise the following construction might not have appeared at all! We say that a word u E A * is a factor of another word w E A * if w == vuv' for some v, v' E A*. Let W be a finite set of words from A+. We denote by S(W) the set of all factors of words in W together with a new symbol 0 and equip this set with the following multiplication: u. v = {uv if uv is.a factor of a word in W , o otherWise. More formally, S(W) can be defined as the Rees quotient of the free monoid A * over the ideal
I(W)
=
{u
E A*
I u is not a factor of any w
E W}.
It is clear that S(W) is a finite monoid, while S(W) \ {1} is a nilpotent semigroup (and so it always finitely based). Perkins has observed that if W = {xyzyx, xzyxy, xyxy, x 2z} then S(W) satisfies the conditions of Theorem 4.1 and thus is nonfinitely based. This gave the example he was looking for. As a possible approach to Tarski 's problem for semigroups, Mark Sapir has suggested investigating the following question: Problem 4.1. ([65], Question 7.11) Is there an algorithm that when given a finite set W of words decides if the monoid S(W) is finitely based or not? Clearly, answering Problem 4.1 in the negative will mean a negative answer to Problem 2.1 as well. Though Problem 4.1 still remains open, it has motivated Olga Sapir and Marcel Jackson's profound studies of the equational properties of the monoids S(W). First of all, they have discovered several sufficient conditions for a finite monoid 8 to be nonfinitely based. Each of
257 the conditions says that S is nonfinitely based whenever certain words are isoterms relative to S , while an infinite series of words contains no isoterms relative to S. e Applying some of these conditions, Olga Sapir [60,61] has described all words w in two letters such that the monoid S( {w}) is finitely based: Theorem 4.2. Let w be a word with Ic(w)1 = 2. The monoid S({w}) is finitely based if and only if w up to a change of letter names coincides with one of the words x"ym or xnyxn (n and m are positive integers). In particular, the monoid S({xyxy}) is nonfinitely based. It consists of 9 elements and is, as verified by Jackson [21], the smallest nonfinitely based monoid of the form S(W). We note as a comparison that the monoid S({xyzyx,xzyxy , xyxy,x 2z}) in the above example by Perkins has 25 elenlents. In contrast with the clear description of "finitely based words" in two letters, the general picture seems to be rather complicated. In their joint paper [24], Marcel Jackson and Olga Sapir have exhibited many strange examples showing that, in a sense, the class of all semi groups of the form S(W) behaves with respect to the finite basis property as irregularly as the class of all finite semigroups. We collect some of their results in the following theorem: Theorem 4.3. a) Every word w is a factor of a word Wi at most four letters longer than w so that the monoid S( {Wi}) is nonfinitely based. If Ic( w) I > 1, then Wi can be chosen such that c( Wi) = c( w) . b) For every finite set W C A+ , there exist finite sets WI) W 2 ) ... of words with W S;; WI C W2 C . .. such that the monoid S(W2k) is finitely based and the monoid S(W2k-l) is nonfinitely based for each k = 1,2, ... c) There are finite sets VI, V2 C A+ such that the monoids S(VI) and S(V2) are no~finitely based [finitely based], while their direct product S (VI) X S (V2) is finitely based [respectively, nonfinitely based] . Jackson [21] has also shown that in some natural sense almost all monoids of the form S(W) are nonfinitely based--compare this with the discussion preceding the formulation of Problem 2.3. Thus, Perkins's unawareness of a finite identity basis for the Brandt semigroup B2 has indeed given rise to a very powerful source of nonfinitely based finite semigroups!
e The idea (of course, inspired by Theorem 4.1) to express conditions of the non-finite basability in the language of isoterms is very well suited for analyzing the finite basis problem for the monoids S(W) because isoterms relative to such monoids are easy to control; in particular, each factor of a word from W is an isoterm relative to S(W).
258
4·3
"Rees matrix" methods
Recall that these methods use Rees matrix semigroups over a group with zero so to speak behind the scene (that is, as critical semigroups) , while semigroups to which the methods apply may be of fairly general nature as in the following theorem from the author's paper [77]. As usual, the core C(S) of a semi group S is the subsemigroup of S generated by all idempotents of S. Theorem 4.4. Let S be a finite semigroup such that the 5-element idempotent-generated O-simple semigroup A2 belongs to the variety Var S . If there exists a group G E Var S \ Var C(S), then S is nonfinitely based. Theorem 4.4 has many applications. For example, it easily implies that the semigroup T" of all transformations of an n-element set is nonfinitely based if n 2: 3. Indeed, A2 E Var Tn since the representation of A2 by the right translations of either of its 3-element right ideals is faithful whence A2 embeds into T 3 . Further, the group §n of all permutations of an nelement set clearly belongs to Var Tn , and since all subgroups of the core C(Tn) embed into §n-l , it is easy to check that §n rt VarC(Tn) (see [77] for details) . Similar reasoning applies to other important types of transformation semigroups. For Rees matrix semigroups, Theorem 4.4 ensures that for any finite group G , the semigroup MO(G,I, A; P) is nonfinitely based whenever there exist A, p- E A , i,j E I such that P)..i,P)..j,Pj.tj i= 0, Pj.ti = 0 and the group G does not belong to the variety Var H, where H is the subgroup of G generated by all non-zero entries of the sandwich-matrix P. As a new application of Theorem 4.4, we deduce from it a proof for Proposition 3.1. We will need the following rough estimation of the size of subgroups of an idempotent-generated finite semigroup: Lemma 4.1. Let S be an idempotent-generated finite semigroup, H a subgroup of S. If lSI> I , then 21HI ::; lSI .
Proof. We induct on lSI. If lSI = 2 the claim is obvious. Now let lSI> 2. Fix a subgroup H in S. Denote by D the V-class of S that contains H and by J the union of all V-classes D' such that DiD' . Then J (if non-empty) is easily seen to be an ideal of S. If 111 > 1 , then the Rees quotient S/ J is an idempotent-generated semigroup of lesser size than S and H is isomorphic to a subgroup of S/ J . By the induction assumption 21HI ::; IS/ JI < lSI . Consider the situation when 111 ::; 1. If J is empty, then D is the least ideal of S ; if IJI = 1 , then J = {O} and DO = D U {O} is the least non-zero ideal of S. Both the cases are similar, and therefore, we restrict ourselves to the second one (which is a bit more complicated). Suppose that 21HI > lSI. If G is the }i-class containing H, then 21GI 2: 21HI > lSI 2: IDI. This enforces G = D (otherwise D 2: 21GI since all }i-classes of the same V-
259
class have equal size). Then the identity e of the subgroup H is the identity for the whole ideal DO and the mapping 8 ~ DO defined by s 1-+ es is a homomorphism. (Indeed, es· et = (es . e)t = est since es E DO and e is the identity of DO .) As the image of an idempotent is an idempotent, eI E {e, O} for any idempotent I E 8. Take an element h E H \ {e} (such an element exists because 21HI > 181 > 2 implies IHI > 1). Since 8 is generated by its idempotents, h = h ... In for some idempotents h, ... ,In E 8 . Multiplying the latter equality through on the left bye, we obtain h = eh = eh · ·· In = (eh)··· (eln ) E {e,O} , a contradiction. 0 Now we can prove Proposition 3.1. Recall that it claims that for each n ~ 5, the variety generated by all finitely based semigroups with n elements 8 i the direct is nonfinitely based. Thus, fix n ~ 5 and denote by 8 = product of all non-isomorphic finitely based n-element semigroups8i . Then the variety in question coincides with Var 8 . We are going to check the conditions of Theorem 4.4. First observe that A2 E Var 8. Indeed, the semigroup A2 is finitely based [70], and so is any of its inflations (see Section 3). Since n ~ 5 , an inflation of A2 appears among the 8 i 's. Now consider T = C(8). It is clear that T ~ C(8i ). By Lemma 4.1 any subgroup of each idempotent-generated semigroup C(8i ) has at most L~ J elements. Therefore each semigroup C(8i ) satisfies the identity
n
n
(1) where k is the least common multiple of the numbers 1,2, ... , l ~J . Then T also satisfies the identity (1). By Bertrand's postulate! there exists a prime number p such that ~ < p S; n. Consider the cyclic group C p of order p . It is finitely based and some suitable inflation of it appears among the 8 i 's whence C p E Var8. On the other hand, C p does not satisfy the identity (1) (since p does not divide k) whence C p ~ VarT. Now Theorem 4.4 applies. Since every 5-element semigroup is finitely based [73], Proposition 3.1 contains the previously known result ([77], Proposition 6) that the variety generated by all 5-element semigroups is nonfinitely based. The variety generated by all n-element semigroups is easily seen to be finitely based if n = 2 or 3, but we still do not know what happens if n = 4. f Bertrand's postulate is the claim that for each n > 7, there exists a prime number between ~ and n - 2. Joseph Bertrand [8] formulated it in 1845 without proof, and Pafnuty Tschebycheff [75] proved it in 1850. An elementary proof of Bertrand's postulate which one can find in many textbooks of number theory is due to Paul Erdos [55].
260
The unary semigroup version of Theorem 4.4 is due to Karl Auinger and the author [6] . For a unary semigroup (S, .,*) we denote by He(S) the Hermitian subsemigroup of S, that is, the unary subsemigroup of S which is generated by all elements of the form xx*. Furthermore, let C3 be the regular *-semigroup MO(3, E, 3; P) where 3 = {I, 2, 3}, E = {e} is the trivial group and
P=
eeeO ee) , (eOe
the unary operation * on C3 being defined by (i,e,j)* = (j,e,i), 0* =0. With this notation we have Theorem 4.5. Let S be a finite unary semigroup such that the semigroup C 3 belongs to the unary semigroup variety Var S. If there exists a group (0,., -1) such that 0 E Var S \ Var He(S) , then S is nonfinitely based. Theorem 4.5 applies to many important finite unary semigroups including: • the semi group (Bn, 0,-1) of all binary relations on an n-element set, 1 < n < 00 , endowed with the unary operation of taking the dual relation; • the semigroup (M2 (K),.,t) of all 2 x 2-matrices over a finite field K having more t.han two elements, endowed with transposition; • the semigroup (M2 (Zp) , ., t) of all 2 x 2-matrices over the field Zp where p == 3 (mod 4) , endowed with Moore-Penrose inverse 9 . In the recent papers [40,41] devoted to the finite basis problem for completely O-simple semigroups, Grigory Mashevitzky has used as critical semigroups certain Rees matrix semigroups which are more complicated than those involved in the proofs of Theorems 4.4 and 4.5. It has enabled him to prove the following result: Theorem 4.6. For each m ::::: 3, the semigroup Rrn = MO (m, ((:2, m ; Pm), 9 See [16] for the definition and a discussion of the concept of Moore-Penrose inverse in an involution semigroup.
261
where m = {I , 2 . .. ,m}, C2 = {e, a} is the 2-element group and eeO ... OO Oee ... OO OOe . .. OO
OOO ... ee aOO ... Oe is nonfinitely based.
We note that the sernigroups Rm are idempotent-generated, and therefore , Theorem 4.4 cannot be used to show that they are nonfinitely based. An interesting application of the technique from the proof of Theorem 4.6 is the result from [41] that the sernigroup T2(3) of all non-surjective transformations of a 3-element set is nonfinitely based. Theorem 4.6 may seem rather special, and its proof is quite bulky. However, it is worth recalling that Mashevitzky's paper [34] (in which the idea of using Rees matrix sernigroups as critical sernigroups first appeared) was also devoted to the identities of a very specific finite sernigroup and these identities were studied in [34] via direct calculations. After a structural substitute for those calculations was found in [77] , the method has become flexible enough to be successfully applied in many interesting situations. Now a challenging problem is to reveal the hidden structural reasons which stay behind the calculations in [40,41], thus mastering a new powerful general condition for the non-finite basability of a finite sernigroup.
4.4 Inherently nonfinitely based finite semigroups Let us start with presenting the definition of an inherently nonfinitely based finite sernigroup in a more explicit form. A nonfinitely based sernigroup S is said to be inherently nonfinitely based if every locally finite variety V for which S E V is also nonfinitely based h Mark Sapir [55] has proved Theorem 4.7. A finite semigroup S is inherently nonfinitely based if and only if all the Zimin words Zn are isoterms relative to S . In [56] , Sapir has given a description of inherently nonfinitely based finite sernigroups in structural terms. Recall that the upper hypercentre r( G) of a group G is the last term in the upper central series of that group. h The t erm "inherently nonfinitely based" was suggested by Peter Perkins [50J, while the very first example of an inherently nonfinitely based finite algebra (in fact, a 3-element groupoid) was exhibited by Murskii [46J .
262 Theorem 4.8. a) A finite semigroup S is inherently nonfinitely based if and only if there exists an idempotent f E S such that the sub monoid f Sf is inherently nonfinitely based. b) A monoid M with n elements is inherently nonfinitely based if and only if there exist b E M and an idempotent e E MbM such that if the elements ebe and ebn!+l e belong to the maximal subgroup He of M containing e, then they lie in different cosets of He with respect to its upper hypercentre r(He) . We note that Theorem 4.8 obviously yields an algorithm that when given the Cayley table of a finite semigroup S decides if S is inherently nonfinitely based or not. In contrast, it follows from Ralph McKenzie 's results (see [42]) that no algorithm can recognize if a given finite groupoid is inherently nonfinitely based. As mentioned in Subsection 4.1, the 6-element Brandt monoid B~ is inherently nonfinitely based. Of course, this implies that every finite semigroup S such that B~ E Var S is inherently nonfinitely based as well. Moreover, from Theorem 4.8 it easily follows that if all subgroups of a finite semigroup S are nilpotent, then the presence of the 6-element Brandt monoid in the variety Var S is not only sufficient but also necessary for S to be inherently nonfinitely based, see [56) , Theorem 2. Further classes of finite semigroups whose inherently nonfinitely based members can be characterized in the same way have recently been found by Marcel Jackson [21,23): Proposition 4.2. If S is a finite regular semigroup with n elements, then the following are equivalent:
(i) S is inherently nonfinitely based; (ii) B~ E VarS ; (iii) S does not satisfy the identity xyx = (xy)n!+lx. Proposition 4.3. If the idempotents of a finite semigroup S form a subsemigroup, then S is inherently nonfinitely based if and only if B~ E Var S . On the other hand, Sapir [56) has constructed an example of an inherently nonfinitely based finite semigroup T such that B~ t/:. VarT. Jackson [21,22) has shown that any such T must consist of at least 56 elements and contain at least 9 non-nilpotent subgroups. He has described all 56-element inherently nonfinitely based semigroups T such that B~ t/:. VarT ; moreover, he has deduced from Theorem 4.8 a description of all minimal with respect to division inherently nonfinitely based finite semigroups i Since an inher' Recall that a semigroup S is said to divide a semigroup T (or to be a divisor of T) if S is a homomorphic image of a subsemigroup of T Clearly, the division relation when restricted to the class 6 of all finite semigroups is a partial order.
263 ently nonfinitely based finite semigroup has at least one minimal inherently nonfinitely based divisor, the latter result provides another algorithmically effective characterization of inherently nonfinitely based finite semigroups. In contrast, the following problem still remains open: Problem 4.2. ([65], Problem 9.1) Describe all minimal (with respect to class inclusion) finitely generated inherently nonfinitely based semigroup varieties, that is, varieties V such that V = VarS for some inherently nonfinitely based finite semigroup S , but no proper subvariety of V has this property. Even though every minimal finitely generated inherently nonfinitely based variety must be generated by a minimal inherently nonfinitely based divisor, the converse is not true: for instance, the 6-element semigroups B~ and A~ both are minimal inherently nonfinitely based divisors, but Var B~ S;; Var A~ . Jackson [22] has observed that there are infinitely many minimal finitely generated inherently nonfinitely based varieties. We call a finite semigroup S weakly finitely based if S is not inherently nonfinitely based, that is, if S belongs to a locally finite finitely based variety. Clearly, the class !2t1J1l3 of all weakly finitely based finite semigroups strictly contains the class ~1l3 of all finitely based finite semigroups. LFrom the definition, the class !ID~1l3 is IHI- and §-closed, and it easily follows from Theorem 4.7 that !2t1J1l3 is also IP fin -closed. Thus, !ID~1l3 is a pseudovariety. We are going to show that it is finitely based. Let us briefly recall some basic facts concerning implicit operations and pseudoidentities referring to [3], Section 3.4, for details. Let m 2: 1 be an integer. An m-ary implicit operation on the class (5 of all finite semigroups is a family 7r = {7rs} SEG of m-ary functions trs : --> S which commute with homomorphisms between finite semigroups in the sense that, for every homomorphism 'IjJ : S --> T with Sand T being finite ,
sm
for all Sl, . . . ) Sm E S. We denote the set of all m-ary implicit operations on (5 by m . Given two implicit operations 7r, p E m , we define their point-wise product 7r. P by letting
n
n
nm
for each finite semigroup S . Clearly, becomes a semigroup under this multiplication; is also a complete metric space under the following metric
nm
264
d(rr, p): 2-r(1r ,p)
d(rr, p) =
{
o
where r(rr, p) is the minimum cardinality of a finite semigroup S such that rrs i= Ps if rr i= p if rr = p.
We list some well-known implicit operations which are important for the sequel: • projections Xi defined on each finite semigroup S by
(Xi)S(Sl ,. ·· ,sm) =
Si
for all S1, ... ,Sm E S; for aesthetic reasons, we write X, y, z, t rather than X1,X2,X3, X4 respectively; • the unary function X t-t XW which, for any finite semi group S, associates with each element S E S the idempotent of the cyclic subsemigroup generated by S; note that if lSI = n , then SW = sn! for every s E S; • the unary function x t-t x w+1 = XW • x ; observe that in any finite semigroup S and for each s E S the element sw+1 belongs to the maximal subgroup Hsw containing the idempotent sW ; • the unary function x t-t x w - 1 which, for any finite semigroup S , associates with each element s E S the group inverse of the element sw+1 in the subgroup Hsw . We also need quite a natural implicit operation which, to the best of our knowledge , has not yet appeared in the literature j . To introduce it, we define the following sequence of implicit operations: [x , yh = xw-lyw-lxy, [x, y]n+l = [[x, Y]n, yhIt can then b~verified in a straightforward way that its subsequence {[x, Y]n!} converges in D2 ; we denote the limit of this subsequence by [x, Y]oo.
From the above implicit operations we can build many others using the composition of implicit operations defined as follows . Suppose that rr E TIm and p(l ), p(2), ... ,p(m) are implicit operations with arities kl' k2' . .. , km , respectively. We then construct a new k-ary implicit operation 7 = rr(p(1 ), p(2), . . . , p(m») where k = kl + k2 + ... + k m by putting 7S(Sl ,. ·· , Sk) =
7fS(p~l\Sl ' . .. , Sk,) , p~)(Skl+l ' .. . , Sk 1 +k2 ), . •• ,p~m)(Sk_k>n +l' . . . , Sk)) j It is, however, a special instance of a general construction suggested by Jorge Almeida (unpublished) .
265
for each finite semigroup S. A pseudo identity is a formal identity of implicit operations, say, 7f = P, and a finite semigroup S is said to satisfy this pseudoidentity if tfs = PS· Jan Reiterman [52], see also [3], Section 3.5, has shown that every semigroup pseudovariety QJ is defined by some set I; of pseudoidentities as the class of all finite semi groups which satisfy all pseudoidentities from I;; in this case I; is called a pseudo identity basis of QJ. Proposition 4.4. A monoid M is weakly finitely based if and only if M satisfies the pseudoidentities
((xyt(yx)W(xy)wt [eze,
(eye)W- 1 eyw+1e]oo =
=
(2)
(xy)W,
e where e
=
(xyzt)w.
(3)
Proof. We start by recalling some well-known characterizations of the pseudovariety :D6 of all finite semigroups whose regular V-classes are subsemigroups. They are taken partly from [3], Section 8.1, partly from [56], Lemma 1. Lemma 4.2. For a finite semigroup S, the following are equivalent:
(i) S E :D6; (ii) S satisfies the pseudoidentity (2); (iii) the 5-element Brandt semigroup B2 does not divide S x S; (iv) for each bE S and for each idempotent e E SbS the element ebe belongs to the maximal subgroup He of S containing e; (v) for each idempotent e E S, the set of all elements b E S such that e E SbS forms a subsemigroup .
We need also a group-theoretical lemma which follows, for example, from a result by Reinhold Baer [7] . To formulate it in a convenient form for us, we note that the function representing the implicit operation [x , yh in a finite group G is nothing but the usual group commutator. Therefore if for all g E G, [g, h]oo = e , the identity element of G, then the element h is what is called an Engel element in group theory, and vice versa. Lemma 4.3. The upper hypercentre nG) of a finite group G with the identity element e coincides with the set of all Engel elements of G, that is, with the set
{h E G I [g, h]oo
=
e for all g E G}.
266 Now suppose that M is a weakly finitely based monoid. Then the inherently nonfinitely based monoid B~ does not belong to Var M whence the semi group B2 does not divide M x M. By Lemma 4.2, M satisfies the pseudoidentity (2). In order to check that M also satisfies the pseudoidentity (3) , assign to the letters x , y, Z , t arbitrary elements a, b, e, d E M and let e = (abcd)w. Then e E MbM and e E MeM whence by Lemma 4.2 the elements ebe, ebw+1e and eee belong to the maximal subgroup He containing e. In view of Theorem 4.8b) , the elements ebe and ebw+1e lie in the same left coset of He with respect to its upper hypercentre r(He). Hence (ebe)W-1ebw+1e E r(He) , and by Lemma 4.3 [eee, (ebe)W- 1ebw+1e]oo = e.
Conversely, let M satisfy the pseudoidentities (2) and (3) . Take any bE M and any idempotent e E MbM . By Lemma 4.2 both ebe and ebw+1e belong to the maximal subgroup He. If e is an arbitrary element of the group He , then obviously e = eee and e E MeM. Hence by Lemma 4.2 e E MbeM , that is, e = abed = (abcd)W for some a, d EM. Substituting a,b,e,d for respectively x , y,z,t in the pseudoidentity (3) , we see that [e, (ebe)W-1ebw+1e] 00 = e
whence (ebe )W-1ebw+1e E r(He) by Lemma 4.3. This means that ebe and ebw+1 e lie in the same left coset of He with respect to r(He) , and by Theo0 rem 4.8b) the monoid M is weakly finitely based. From Proposition 4.4 and Theorem 4.1a) we immediately obtain Corollary 4.1. The pseudo variety ~1)3 is defined by the two pseudoidentities obtained from the pseudo identities (2) and (3) by substituting each x 2 , i=l, ... ,4 , bYX'5XiX'5 .
Mark Sapir posed the following question: Problem 4.3. ([65], Question 11.2) Is the pseudo variety !ID~1)3 generated by finitely based finite semigroups?
This problem is related to Problem 3.1 above as follows: if the pseudovariety generated by all finitely based finite semigroups is nonfinitely based, then it cannot coincide with the finitely based pseudovariety ~1)3 . If one focuses on the finite basis problem for finite semigroups (like we do in this survey), then the notion of an inherently nonfinitely based semigroup appears to be rather abundant. Why should we care about locally finite varieties which are not finitely generated when we are only interested in finitely generated ones? This question leads us to introduce the following notion: call
267
a finite semigroup S strongly nonfinitely based if S cannot be a member of any finitely based finitely generated variety. Clearly, every inherently nonfinitely based finite semigroup is strongly nonfinitely based, and the question if the converse is true is another intriguing open problem: Problem 4.4. Is there a strongly nonfinitely based finite semigroup which is not inherently nonfinitely based? As some evidence for a positive answer to Problem 4.4 being possible, we mention the situation with a similar question for quasiidentities. Recall that a semigroup quasiidentity is an expression of the form UI
=
VI
& U2 =
V2
& ... & Un =
Vn =?
u =
V,
where UI, VI, U2 , V2, .. . ,Un, V n , U, V E A+. A semigroup S satisfies such a quasi identity if for any homomorphism <{J: A+ --> S, U<{J = V<{J provided that UI<{J = VI<{J, U2<{J = V2<{J, ... , Un<{J = Vn<{J· After having defined satisfaction this way, one straightforwardly proceeds with the "quasi" -analogues for all notions of the theory of identities and varieties, including those considered in this subsection. A surprising fact discovered by Stuart Margolis and Mark Sapir [33] is that no finite semigroup can be inherently nonfinitely based with respect to quasiidentities k. On the other hand, Mark Sapir has proved in his Ph.D. thesis [54] that there exists a finite semigroup which is strongly nonfinitely based with respect to quasiidentities (in fact, every completely simple finite semi group which is not a rectangular group or which contains a non-abelian p-group has this property). Let us return to the realm of identities and varieties. Here we observe that the notion of strong non-finite basability is of special interest in the unary semigroup setting. The point is that Mark Sapir has proved in [59] that no finite inverse semigroup is inherently nonfinitely based as an algebra of type (2,1). This result can be easily extended to wider classes of unary semigroups, for instance, to regular *-semigroups. Having this in mind, one seeks a notion which, being weaker than inherent non-finite basability, could be nevertheless applied to the finite basis problem for finite unary semigroups with a similar effect. The notion of a strongly nonfinitely based semigroup is a reasonable candidate here. The crucial problem is: Problem 4.5. Is there a finite inverse semigroup which is strongly nonfinitely based as an inverse semigroup? In particular, is the Brandt monoid B~ strongly nonfinitely based as an inverse semigroup? k It is quite interesting that there exist finite unary semigroups which are inherently nonfinitely based with respect to quasiidentities. A method for constructing such unary semigroups(in fact, rectangular bands) has been developed in a recent paper by John Lawrence and Ross Willard [32]
268
The second question in this problem was first asked in 1979 by Evgeny Kleiman [28]. In Subsection 4.1 we tried to put the notion of inherent non-finite basability into a more general context having related it to a certain finitely inexpressible property of finitely generated varieties. Perhaps, one may also use this idea to approach the problems connected with the notion of strong non-finite basability. Thus, we encourage the continued study of finitely generated varieties in the hope of discovering another finitely inexpressible property. As an example of a fairly non-obvious property of finitely generated varieties we mention the following result due to Olga Sapir (unpublished): a finitely generated variety does not contain the variety of all bands. We do not know if the latter property is finitely inexpressible.
4.5 A comparison between the three "standard" methods The word "standard" in the title of this subsection is merely an abbreviation of the expression "most frequently used so far" No doubt, the three groups of methods which we presented in Subsections 4.2- 4.4 above have the right to be called standard in such sense. After having discussed each of them individually, we want to compare them from the point of view of their ranges of applicability. These ranges are represented by the three boxes on Figure 1.
Seroigroups with trivial subgroups
Seroigroups with non-trivial subgroups I I
S( {xyxy})
I
A2
X
Cp
I
B~ , A~
Tn,
I
n~3
I I
Syntactic methods
On,
n~3
I
Tn(k),
n>k~3
The Rees matrix methods
I
Inherently nonfinitely based seroigroups Figure 1. The ranges of applicability of the three standard methods
269 We put in the boxes certain nonfinitely based finite semigroups. When a semi group S appears in the box corresponding to one of the three standard methods, it means that the fact that S possesses no finite identity basis may be obtained by the method. Of course, if S stays in the intersection of two boxes, then each of the corresponding methods applies. The reader is already acquainted with a majority of semi groups in Figure 1. The only new objects are the semigroups On and Tn (k) which are semigroups of transformations of the set {I, 2, .. . ,n} : the semigroup On consists of all transformations that preserve the standard order of this set, while the semigroup Tn (k) (k S n) is the ideal of the full transformation semigroup Tn consisting of all transformations whose images have at most k elements. The dashed line in Figure 1 symbolizes the border between aperiodic finite semigroups (that is, finite semi groups having only one-element subgroups) and all other finite semigroups. We see that so far the syntactic methods have been applied only to aperiodic finite semigroups, while any application of the Rees matrix methods has required the presence of a non-trivial subgroup. Of course, these constraints are caused by the very nature of the methods. In contrast, the method of inherently nonfinitely based semigroups can be applied to semigroups with and without non-trivial subgroups. A further important constraint on the syntactic methods is that they only apply to monoids (or, more precisely, to semigroups sharing identities with a monoid): all these methods heavily exploit the closure under deletion. To some extent, the method of inherently nonfinitely based semigroups is also of the monoidal nature- cf. Theorem 4.8a) above-€ven though it can be applied to certain semi groups with no identity element (as the semigroups Tn(k) with n > k ~ 3 , for instance). The Rees matrix methods do not depend on the presence of an identity element. In conclusion, we mention that although each of the three approaches has its own specificity, they are tightly connected with each other. For example, Mark Sapir's proof [57] that inherently nonfinitely based finite semigroups are nonfinitely based within the class of all finite semigroups uses the critical semi group method, and the critical semigroups that appear in his proof are of the form S ({w }) for suitable words w E A + 4.6
Interpretation methods
a) Mashevitzky's pointed group method. Following Roger Bryant [10], we call a pair (G,p) , where G is a group and pEG is a fixed element considered as an additional nullary operation, a pointed group. A striking result by Bryant [10] is that there exists a finite pointed group (G,p) which is nonfinitely based
270
(as an algebra of type (2,0)). Grigory Mashevitzky [35] has used Bryant's pointed group (G,p) in order to construct a nonfinitely based finite simple semigroup. In fact, Mashevitzky's semigroup is the Rees matrix semigroup over that group G with the sandwich matrix (: ~) where e is the identity element of G . This important example still remains the only known nonfinitely based finite simple semigroup; moreover, for 15 years it was the only example of a nonfinitely based variety of completely simple semigroups over a finitely based variety of groups. Only recently Karl Auinger and Maria Szendrei [5] have constructed another completely simple semigroup variety with this property (their variety is not finitely generated). The idea of relating the identities of Rees matrix semigroups with the type (2,0, .. . , 0) identities of their structure groups in which the entries of corresponding sandwich matrices play the role of distinguished constants is very natural and promising. Unfortunately, in spite of its successful debut in [35], it seems to have been abandoned. Mashevitzky mentioned this idea in his survey [36] where he announced that it could be used to prove that any completely simple semigroup over a nilpotent group of finite exponent is finitely based, but no detailed proof of this result has appeared so far. There is no doubt that this interesting direction deserves more attention.
b) Sapir's verbal subset method. We describe this method following [65], Section 11. Let 8 be an arbitrary semigroup, a and 0 two new symbols. Consider the semigroup (T(8),0) with the carrier set 8 u {a} x 8 1 U 8 1 x {a} U {a} x 8 1 x {a} U {O} and with the multiplication extending the multiplication in 8 and such that for all SI, S2 E 8 1 , t E 8 ,
(a, SI) 0 t = (a, SIt), to (S2, a) = (ts2' a) , (a, SI) 0 (S2' a) = (a , SIS2, a) , while all other products are equal to O. Given a set W ~ A+, W(8) denotes the set of all values of words from W in 8 , that is, the union of the sets W cp over all possible homomorphisms cp : A+ -> 8. We will call subsets of the form W(8) verbal subsets of 8 . By T(8, W) we denote the Rees quotient of the semigroup T(8) over the ideal {a} x W(8) x {a} U {O}. The main property of the semigroup T(8, W) is revealed by the following result due to Mark Sapir (d. [65], Proposition 11.1) :
271
Theorem 4.9. Let F be the free semigroup over the alphabet A in the variety Var S, W ~ A+ a set of words. If W(F) =I- V(F) for all finite subsets V c A+ , then the semigroup T(S, W) is nonfinitely based. The idea behind Theorem 4.9 is an interpretation of a natural deduction system (F, r) on the relatively free semigroup F . In this system, P f- q (where P C F and q E F) means that there exists pEP and an endomorphism ?jJ : F - t F such that q = p?jJ. Clearly, subsets of F closed under fare precisely verbal subsets of F. A comparison between the inference rule of (F, f-) and that of equational logic (see Proposition 1.1 above) shows that, informally speaking, the inference in (F, r) is much weaker: in equational logic, besides using endomorphisms, we may also multiply on both sides. The construction of the semigroup T(S, W) is designed to "wrap" each identity of S by a new letter. This excludes using the multiplication and basically reduces the deduction apparatus to endomorphisms only, in other words, to the inference rule of (F, f-). Because of the "weakness" of f- , usually it is pretty easy to find a verbal subset W(F) such that the subsystem (W(F), f-) is not finitely axiomatized, that is, W(F) =I- V(F) for all finite subsets V C A+ This makes Theorem 4.9 a very powerful source of interesting examples of nonfinitely based semigroups, including finite ones. Several concrete examples found this way have been collected in Sapir's paper [58]. For instance, let S be any finite non-abelian group of exponent n and denote the word xn-lyn-lxy by [x, y]. If
then W(F) is the commutator subgroup of the group F, and it is known to be not finitely generated as a verbal subset. The corresponding finite semigroup T(S, W) is then nonfinitely based. Taking here as S the group P P ab " = bae ea = ae eb = be) (a , bel , a = bP = e = 1'
of order p3 and exponent p (p is an odd prime), one obtains an example of a nonfinitely based finite semigroup T = T(S, W) which is minimal in the following sense: the variety Var T has only finitely many subvarieties, and each proper subvariety of Var T is finitely generated and finitely based. Again, it seems that the potential of this approach is underexploited, and it is worth looking for further applications of the method.
272 5
How to prove that a finite semigroup is finitely based
As mentioned in the introduction, this direction has progressed relatively slowly, and it still remains a collection of isolated results rather than a unified theory. Nevertheless, some of the results gathered in the subarea so far are interesting and worth discussing here. A vraam Trahtman [73] has published a proof of his result (first announced in 1983, see [71]) that every 5-element semigroup is finitely based. We refer to [65] , Section 10, for a detailed discussion of the history of the problem and the previous steps towards its solution. Of course, the paper [73] contains many clever tricks which may be useful for finding a finite identity basis for further classes of finite semigroups. Grigory Mashevitzky [37] has proved that the semi group Mn(1) of all n x n-matrices of rank ::; 1 over any field is finitely based. This completes a certain chapter in the study of the finite basis problem for semigroups of n x n-matrices because the answer to this problem for semigroups Mn(k) of all n x n-matrices of rank ::; k was already known for all k > 1 . Namely, if the ground field is finite , then the semigroup Mn(k) with k> 1 is nonfinitely based and even inherently nonfinitely based because it contains a subsemigroup isomorphic to the Brandt monoid B~ ; if the ground field is infinite, then the semi group Mn(k) with k> 1 satisfies no non-trivial identity; this follows from an observation due to Igor Golubchik and Alexander Mikhalev [171. In [39] Mashevitzky has introduced and studied the notion of a left hereditary system of semigroup identities. Let u and w be words. By uiw we denote the longest prefix of u not containing w as a factor. For any identity u = v , the identity uiw = v iw is called the left section of the identity u = v relative to w. A system of identities containing all left sections of each of its identities relative to all words of length n is said to be left n -hereditary. Left hereditary identity systems often arise as the identity systems of semigroups that are extensions of a left zero ideal. For example, if S is a subdirectly irreducible semigroup having a non-trivial left zero ideal, then Id S is left 1hereditary ([38], Proposition 1.1). Another example is the semigroup Tn(2) of all transformations of an n-element set whose images have at most 2 elements. If n :::: 5 , then the identity system Id Tn (2) has been shown by Nina Torlopova [68] to be left 2-hereditary. The main results of [381 are two theorems showing that , in certain situations , the condition of being ~ft hereditary suffices to ensure that the ideal extension of a left zero semigroup by a finitely based semi group is again finitely based. Theorem 5.1. Let T be a semigroup with 0 but without zero divisors satisfy-
273 ing the identity Xk = x for some k > 1. Suppose that S is an ideal extension of a left zero ideal by the semigroup T and that the identity system Id S is left I-hereditary . Then S is finitely based whenever Tis . Theorem 5.2. Let T = MO(G , I, A; P) be a Rees matrix semigroup over a group G of a finite exponent, and there exist A, f.-L E A, i, j E I such that P)" i, P)"j,PJ.Lj =I 0, PJ.Li = O. Suppose that S is an extension of a left zero ideal by the semigroup T and that the identity system Id S is 2-hereditary. Then S is finitely based whenever Tis. An important application of Theorem 5.2 is the fact that the semigroup Tn(2) is finitely based provided that n ;::: 5. This is a partial solution to Question 22.1 in [65]. Recall that the semigroup T3(2) is nonfinitely based (we mentioned this result from [41] in Subsection 4.3 above). Since the 4element semigroup T 2(2) = T2 is obviously finitely based, the only remaining member of the family {Tn (k)} for which the finite basis problem is still open is the 88-element semigroup T4(2). Mashevitzky once announced (in his abstract submitted at XVIII All-Union Algebra Conference held in Kishinev in September 1985) that T4(2) is finitely based but this announcement was never confirmed in a detailed publication.
6
Concrete problems
We conclude our survey by listing three concrete series of finite semigroups for which the finite basis problem still remains open. The most important among these concrete questions is the following problem suggested by Veniamin Rasin: Problem 6.1. ([65], Question 8.1) Let Cp = (a I a P = e) be the cyclic group
of prime order p, M
= M(2, C p , 2; P) where 2 = {l, 2}, P = (: :) Is the
semigroup Ml finitely based? A solution to Problem 6.1 would open the way to a description of finitely based finite completely regular semigroups. The next question deals with a natural class of semigroups of matrices: Problem 6.2. ([65], Question 22.2) Is the semigroup of all upper triangular n x n-matrices (n ;::: 2) over a finite field finitely based? Our final problem is perhaps the most challenging one: Problem 6.3. Is the symmetric inverse semigroup In, n ;::: 2 , that is, the semigroup of all partial injective transformations of an n-element set, finitely based as an inverse semigroup? Recall that In is even inherently nonfinitely based as a plain semigroup.
274
Acknowledgments
This paper is an expanded version of the invited lecture at the International Conference on Semigroups held in Braga in June 1999. The author is indebted to Prof. M. P. Smith and to all members of the Organizing Committee for the invitation, for great help during the conference, and for understanding and patience shown when the author was preparing this survey. The financial support of the INVOTAN is also gratefully acknowledged. The author thanks Jorge Almeida, Karl Auinger, Marcel Jackson, Grigory Mashevitzky, Olga Sapir who have given him permission to mention their yet unpublished results in the survey and who have provided copies of their preprints. The author also thahks very much the colleagues just listed as well as Mark Sapir, Peter Trotter and the anonymous referee for their valuable remarks and comments on the preliminary version of the survey. References 1. J. Almeida, Equations for pseudo varieties, in J.-E. Pin (ed.), Formal Properties of Finite Automata and Applications [Lect. Notes Comput. Sci. 386], Springer-Verlag, Berlin- Heidelberg- N. Y. , 1989, 148- 164. 2. J . Almeida, On iterated semidirect products of finite semilattices, J. Algebra 142 (1991) 239- 254. 3. J. Almeida, Finite Semigroups and Universal Algebra , World Scientific, Singapore, 1995. 4. K. Auinger, Strict regular * -semigroups, in J. M. Howie, W . D. Munn, and H.-J. Weinert (eds.) , Proc. Conf. on Semigroups with Applications, World Scientific, Singapore, 1992, 190-204. 5. K. Auinger and M. B. Szendrei, On identity bases of epigroup varieties, J. Algebra 220 (1999) 437- 448. 6. K. Auinger and M. V. Volkov, Non-finitely based varieties of unary semigroups, in preparation. 7. R. Baer, Engelsche Elemente in Noetherschen Gruppen, Math. Ann. 133 (1957) 256- 270 [German) . 8. J . Bertrand, M emoire sur Ie nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu 'elle renfreme, J. :Ecole Polytechn. 30 (1845) 123- 140 [French). 9. G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935) 433- 454. 10. R. M. Bryant, The laws of finite pointed groups, Bull. London Math. Soc. 14 (1982) 119- 123.
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List of Participants Jorge Almeida Departamento de Matematica Pura, Faculdade de CiEmcias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: [email protected] Jorge Andre Centro de Algebra da Universidade de Lisboa, Av. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal.
Prof.
e-mail: [email protected] Isabel Araujo Mathematical Institute, University of St Andrews , North Haugh, KY16 9SS , Scotland, U.K.. e-mail: [email protected]. uk Joao Araujo Centro de Algebra da Universidade de Lisboa, Av. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal.
Prof.
e-mail: [email protected] Karl Auinger Institut fur Mathematik, Universitiit Wien, Strudlhofgasse 4, A-1090 Wien, Austria. e-mail: [email protected] Assis Azevedo Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Bernd Billhardt Fachbereich Mathematik/lnformatik, Universitiit Gesamthochschule Kassel, Holliindische Strabe 36, 34127 Kassel, Germany. e-mail: [email protected] Tom Blyth Mathematical Institute, University of St Andrews, North Haugh, KY16 9SS, Scotland, U.K. . e-mail: [email protected]. uk Stojan Bogdanovic University of Nis, Faculty of Economics, Ttg VJ 11 , P.O. Box 121, 18 000 Nis, Yugoslavia e-mail: [email protected] 281
282
Manuel Branco Departamento de Matematica, Universidade de Evora, Apart. 94, 7001 Evora Codex, Portugal. e-mail: [email protected] Mario Branco Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Colin Campbell Mathematical Institute, University of St Andrews, North Haugh, KY16 9SS, Scotland, U.K.. e-mail: [email protected] Paula Catarino Secc;iio de Matematica, Univ. Douro, 5000 Vila Real, Portugal.
Tris-os-Montes e Alto
e-mail: [email protected] Miroslav Cirie University of Nis, Faculty of Philosophy, Cirila i Metodija 2, P.O. Box 91, 18000 Nis, Yugoslavia. e-mail: [email protected] Antonio Veloso da Costa Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Jose Carlos Costa Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Manuel Delgado Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: [email protected] Marie Demlova Department of Mathematics, Faculty of Electrical Engineering, Czech Thecnical University, Thechnicka 2, 16627 Prague 6, Czech Republic. e-mail: [email protected] Klaus Denecke Universitat Potsdam, Institut fur Mathematik, PF 60 15 53, 14415 Potsdam, Germany. e-mail: [email protected]
283
Luis Descalf,.o Departamento de Matematica, Universidade de Aveiro, Campus de Santiago, 3810 Aveiro, Portugal. e-mail: [email protected] Ana Paula Escada Departamento de Matematica, Universidade de Coimbra, Apartado 3008,3000 Coimbra, Portugal. e-mail: [email protected] Oscar Felgueiras Centro de Algebra da Universidade de Lisboa, A v. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Vitor Hugo Fernandes Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Juan Inacio Garcia-Garcia Departamento de Algebra, Universidad de Granada, E - 18071 Granada, Spain. e-mail: [email protected] Pedro Garcia-Sanchez Departamento de Granada, E - 18071 Granada, Spain.
Algebra,
Universidad
de
Ana Paula Garrao Departamento de Matematica, Universidade A<;ores, Rua Mae de Deus, 9502 Ponta Delgada Codex, Portugal.
dos
e-mail: [email protected]
e-mail: [email protected] Enulia Giraldes Sec<;ao de Matematica, Univ. Douro, 5000 Vila Real, Portugal.
Tras-os-Montes e Alto
e-mail: [email protected] Gracinda Gomes Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Marin Gutan Universite Blaise Pascal, MatMmatiques, Complexe Scientifique des Cezeaux, 63177 Aubiere Cedex, France. e-mail: [email protected]
284
Anthony Hayes Department of Mathematics, University of York, Hesligton, York, Y010 5DD, U.K.. e-mail: [email protected] Peter Higgins Department of Mathematics, University of Essex, Wivenhoe Park, Colchester C04 3SQ, U.K.. e-mail: [email protected]. uk John Howie Mathematical Institute, University of St Andrews, North Haugh, KY16 9SS, Scotland, U.K.. e-mail: [email protected]. uk Manuel Messias Jesus Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Mati Kilp Institute of Pure Mathematics, University of Tartu, Tartu, Estonia. e-mail: [email protected] Ulrich Knauer Carl von Ossietzky Universitiit Oldenburg, Fachbereich 6, Mathematik, D-26111 Oldenburg, Germany. e-mail: [email protected] Janusz Konieczny Department of Mathematics, Mary Washington College, Fredericksburg, VA 22401 , U.S.A .. e-mail: [email protected] Joerg Koppitz Universitiit Potsdam,Institut fur Mathematik, PF 60 15 53, 14415 Potsdam, Germany. e-mail: [email protected] J elena Kovacevic University of Nis, Faculty of Philosophy, Cirila i Metodija 2, P.O. Box 91, 18000 Nis, Yugoslavia. e-mail: [email protected] Valdis Laan University of Tartu, 50090 Tartu, Estonia. e-mail: [email protected]
285
Carlos Leandro Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Lucinda Lima Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: [email protected] Arminda de Azevedo Maia Instituto Politecnico de Portugal.
Bragan~a , Bragan~a ,
e-mail: [email protected] Antonio Jose Malheiro Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: malheiro@alfl .cii.fc. ul.pt Stuart Margolis Department of Mathematics and Computer Science, BarIlan University, Ramat-Gan, Israel. e-mail: [email protected] Paula Marques-Smith Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Paula M. Martins Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] .pt Don McAlister Department of Mathematical Sciences, Northen Illinois University, Dekalb, IL 60115, U.S.A .. e-mail: don@math .niu.edu John Meakin Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323, U.S.A .. e-mail: [email protected]. edu Paulo Jorge Medeiros Departamento de Matematica, Universidade dos A~ores , Rua Mae de Deus, 9502 Ponta Delgada Codex, Portugal. e-mail: paulo @alj.uac.pt
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Maria Claudia Mendes Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: clmendes @math.uminho.pt
Maria Suzana Mendes Departamento de Matenuitica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: sumendes @math.uminho.pt
Melanija Mitrovic University of NiS. Faculty of Mechanical Engineering, Beogradska 14, 18 000 NiS , Yugoslavia. e-mail: [email protected]
Heinz Mitsch Institut fur Mathematik, Universitiit Wien, Strudlhofgasse 4, A-1090 Wien, Austria. e-mail: sek:[email protected]
Douglas Munn Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, U.K.. e-mail: wdm @mailis.gla.ac. uk
Peeter Normak TPU, Karva mot 25, 10120 Tallinn, Estonia. e-mail: [email protected]
Jan Okninski Institute of Mathematics, Warsaw University, Banacha 2. 02791 Warsaw, Poland. e-mail: [email protected]
Ana Maria Oliveira Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: [email protected]
Luis Oliveira Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: loiiveir @jc.up.pt
Pedro Patricio Departamento de Matemcitica. Universidade do Minho, Campus de Gualtar. 4700-320 Braga, Portugal. e-mail: pedro @math.uminho .pt
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Tatjana Petkovic University of Nis, Faculty of Philosophy, Cirila i Metodija 2, P.O. Box 91 , 18000 Nis, Yugoslavia. e-mail: [email protected] Jean-Eric Pin LIAFA CNRS, Univ. Paris VII, Case 7014, 2 Place Jussieu , 75 251 Paris Cedex 05, France. e-mail: [email protected] Gonc;alo Pinto Universidade Lusiada, Rua da Junqueira, 188-198, 1300 Lisboa, Portugal. e-mail: [email protected] Luis Pinto Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Maria Fernanda Pinto Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Libor Polak Department of Mathematics, Masaryk University, Janackovo nam. 2a, 66295 Brno, Czech Republic. e-mail: [email protected] Zarko Popovic University of Nis, Faculty of Economics, Trg VJ 11, P.O. Box 121 , 18 000 Nis, Yugoslavia. e-mail: [email protected] Gordon Preston Mathematics Department, Monash University, Clayton, Victoria 3168, Australia. e-mail: gbp@gizmo .maths.monash.edu.au Peeter Puusemp Department of Mathematics, Tallinn Technical University, Ehitajate tee 5" Tallinn 19086, Estonia. e-mail: [email protected] Roland Puytjens Department of Pure Mathematics and Computeralgebra, University of Gent, Galglaan 2, 9000 Gent, Belgium. e-mail: [email protected]
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Norman Reilly Department of Mathematics, Simon Fraser University, Burnary, British Columbia, Canada V5A 1S6. e-mail: [email protected] Jim Renshaw Faculty of mathematical Studies, University of Southampton, England S017 lBJ, U.K.. e-mail: [email protected] Arkadiusz Salwa Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw , Poland. e-mail: [email protected] M. Helena A. Santos Departamento de Matematica, Faculdade de Ci€mcias e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825 -114 Costa da Caparica, Portugal.
e-mail: [email protected] Ricardo Severino Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Helena Sezinando Centro de Algebra da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649 - 003 Lisboa, Portugal. e-mail: [email protected] Lev Shevrin Department of Mathematics and Mechanics, Ural State University, Lenina 51, 620083 Ekaterinburg, Russia. e-mail: [email protected] Kunitaka Shoji Department of Mathematics, Shimane University, Matsue, Shimane 690-0823, Japan. e-mail: [email protected] Teresa Silva Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Pedro V. Silva Departamento de Matematica Pura, Faculdade de Ciencias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: pvsilva@!c.up.pt
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Andrew Solomon Mathematical Institute, University of St Andrews, North Haugh, KY16 9SS, Scotland, U.K.. e-mail: [email protected] Benjamin Steinberg Departamento de Matematica Pura, Faculdade de Ci€mcias, Universidade do Porto, 4099-002 Porto, Portugal. e-mail: [email protected] .up.pt Bob Sullivan Department of Mathematics and Statistics, University of Western Australia, Nedlands, 6907, Australia. e-mail: [email protected] Edmund Swylan 3 Vecsaules Street, Riga LV 1004, Latvia. e-mail: [email protected] Maria Szendrei J6zsef Attila University, Bolyai Institute, H-6720 Szeged, Aradi vertanUk tere 1, Hungary. e-mail: M [email protected] M. de Lurdes Teixeira Departamento de Matematica, Universidade do Minho, Campus de Gualtar, 4700-320 Braga, Portugal. e-mail: [email protected] Laila Tunsi Departments of Mathematics, Universiry of York, Hesligton, York , YOlO 5DD, U.K.. e-mail: [email protected]. uk Mikhail Volkov Department of Mathematics and Mechanics, Ural State University, Lenina 51,620083 Ekaterinburg, Russia. e-mail: [email protected] Barbara Weipoltshammer Gheleng. 34/5/11 , A-1130 Wien, Austria. e-mail: [email protected] Ilya Zhiltsov Department of Mathematics and Mechanics, Ural State University, Lenina 51, 620083 Ekaterinburg, Russia. e-mail: [email protected]
