IC/87/359 /§/ KBTH INTERNATIONAL CENTRE FOR cf. i THEORETICAL PHYSICS QUASI-ADEQUATE SEMIGROUPS ATDdulsalam El-Qallali INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/87/359 INTRODUCTION There Is a structure theorem for type W semigroups [3] - a International Atomic Energy Agency certain class of quasi-adequate semigroups - which generalizes that of and Hall [6] for orthodox semigroups. The Ball construction of the orthodox United Nations Educational Scientific and Cultural Organisation semigroup W (see [7]} has been used in the structure theorem of [3]. Recently, Venkatesan 111] gave a structure theorem for orthodox semigroup IHTERHATIONAL CENTRE FOE THEORETICAL PHYSICS without the use of the Hall construction W_, One of the two objectives of this paper is to extend the structure theorem of Venkatesan [11] to a certain class of quasi-adequate semigroups. The other objective is to describe for a type W semigroup S, the least fundamental adequate good QUASI-ADEQUATE SEMIGROUPS • congruence on S, the largest superabundant full subsemigroup of 5 and the largest full subsemigroup of S which is a band of cencellative monoids. Abduls alam El-Qallali •• After the preliminaries in which we collect some basic information International Centre for Theoretical Physics, Trieste, Italy. from the literature, comments are made in Section 2 on the idempotent- separating good homomorphic images of particular quasi-adequate semigroups. This is a preparation for the last two sections where in the third one we investigate a class of quasi-adequate semigroups using fundamental ABSTRACT quasi-adequate semigroups and adequate semigroups. The main The least fundamental adequate good congruence on an arbitrary result of this section is similar to that of Venkatesan [ll] and close to type W semigroup S is described as well as the largest superabundant El-Qallali and Fountain's work [3]. However, we give a direct proof of full subsemigroup of S and the largest full subsemigroup of S which the result Independent of the Kail construction V£. The class of quasi- is a band of cancell&tive monoids. Weak type W semigroups are defined adequate semigroups in the present approach may properly contain the class by replacing the idempotent-connected property in type W by one of its of type W semigroups. In the last section we extend Venkatesan's consequences and a structure theorem is obtained for such semigroups. result [12] to type W semigroups. We use the notation and terminology of [l] and assume some familiarity with the contents of [3] and [5j. MIRAMARE - TRIESTE 1. PRELIMINARIES November 1987 We begin by recalling some of the basic facts about the relations L* and P*. I*t a, b be elements of a semigroup S. We say that aL*b if and only If a and b are related by Green's relation L in • To be submitted for publication. some oversemigroup of S. The relation P* is defined dually and the *• Permanent address: Department of Mathematics, Al-Fateh University, P.O. Box 13211, Tripoli, Libya, S.P.L.A.J. -2- relation H* is the Intersection of L* and E*. Evidently L* is & right congruence and ft* is a left congruence on S. For any result about L* there is a dual result for R". The following Leama from [9] On quasi-adequate semigroups we have from [3] a generalization of and [103 gives an alternative description of L». the main part of Lallementa's Lemma [7, Lemma II.U.6] as follows: Lemma 1.1 Let a, b be elements of a semigroup 3. Then the following Proposition 1.5 If S Is a quasi-adequate semigroup with set E of conditions are equivalent: idempotente and p Is a good congruence on S (a semigroup good homomorphism from S onto T), then 5/p IT) is a quasi-adequate semi- (1) a L» b group whose set of idempotenta is {ep; e fe B}. (2) for all x, y fc S': ax = ay if and only if bx - by. Let S be an abundant semigroup and E be Its set of Idempotents. As an easy but useful consequence of Lemma 1.1 we have The relations; Uy. Up and u have been defined in [2] by the following Corollary 1.2 Let a be an element of a semigroup S • and e be an rules: idempotent of S, Then the following conditions are equivalent: (a, b) £ uL if and only if {ea, eb) € L» for all e € E, (1) a L* e (a, b) £ wR if and only if (ae, be) £ IR* for all e 6- E {2} ae * a and for »11 x, y£s'; ax = ay Implies ex . ey. and It follows from Corollary 1.2 and its dual that: S is called fundamental If u is the Identity relation on S. Corollary 1.3 If e is an idempotent in a semigroup S, then Hj| is a Tne main property of u is stated in the following proposition. cancellative monoid. Proposition [g] 1.6 y is the largest congruence on S contained in H*. In particular, we emphasize that on any semigroup S, we have If a is an element of S, then a* and a denote typical idem- L £L". It Is well known and easy to show that for regular elements potents in L* and B* respectively. S Is said to be 1dempotent- a, b in S; a L* b if and only if a L b, thus if 3 Is a regular semi- connected if for each element a in S and for some (hence for all) a+, a*, group, then L* = L. there exists a bisection a: < a > + < a* > satisfying xa = a(xa) for all Recall that, an abundant semigroup Is one in which each L"-C1RES und x £ < a > . A type A semigroup is an Idempotent-conneetetl adequate semi- each F*-class contains an Idempotent, and it is superabundant if each group. In this case a is defined by xa = (xa)* ft]. H*-clas3 contains an idempotent. An abundant semigroup is said to be Proposition [3] 1.7 Let S be an idempotent-connected quasi-adequate semi- quasi-adequate if its idempotents form a subsemigroup and it is adequate group and S: S •* T be a good homoraorphism onto an adequate semigroup T. if the idempotents commute {see [3], [k] and [55). Then T is a type A semigroup. A semigroup homomorphism p: S •+ T is said to be a good homomorphism Moreover, we conclude from [2] the following result. if for all elements a, b of S; a L#(S}b implies ap L»(T) bp and that Lerona 1.8 If S is an idempotent-eonnected abundant semigroup, then u is a ft*(S) b implies ap R*(T) bp. A congruence T on a semigroup S Is good on S. said to be a good congruence if the natural homomorphism from S onto S/T is a good homomorphism. From [2] we quote: For the rest of the section, S Is a quasi-adequate semigroup with band B of idempotents. Following [7], we denote the e-elass in B of Lemma 1•k Let S be an abundant semigroup. If p is a semigroup an idempotent e by E{e). The relation & Is defined In [3] on S by homomorphism from S into T (a congruence on 5), then the following the rule: conditions are equivalent: a 6 b if and only iff E(a+) a E(a») = E(b+) b E(b») (1) p is good + + „ (2) for any element a of S, there are Idempotents e, f with e L# a, for some a , a", b , D*. f P» a in S such that ep L» ap and fp fi" ap in T(S/p). -It- -3- It is clear that £ ia en equivalence relation. Same of Its (1) Y is a *-homomorphism properties have been examined in [3], in particular «e have (2) *-f Is an idempotent-separatlng good homomorphism. Proposition 1.9 (1) For any elements a and a In S; t(b If and Proof If {lj holds and e, f are In B such that e^P - f °f , then only if a = ebf for some e € E{b+), f £ E(b"). (ii)H»A« = l, e L f and e R f which implies e H f. Therefore e * f and (2) holds. (ill) S ia a congruence on S if and only If for any elesents a and b . On the other hand, If (2) holds and a, b are in S such that in S; a<fL*(T) b<?, a* L*(S) a, b* L»(S) b for some a», b*; then a»f L»(T) a*P, b»TL»(T) b=P and a'^Lb*5? ,-that ia, (a'b*)'? = a* <P «id (U»a»)f =li«f. a E(a») E(b+) C,E((ab)+ ) sb EC(ab)*). But a»b», b»a» are also In B, then a«b» = a*, t»a* = b» and ft L»(S) b. A congruence relation p on S is called an adequate congruence By a similar argument; a^PP»(T) bf implies a P»(S) b. The if S/p is an adequate semigroup. We have from [3]j reverse implications are guaranteed by the hypothesis. Hence (l) holds. As an immediate consequence of Lemma 2.1 and Proposition 1.5 we have: Proposition 1.10 If S Is a congruence, then it is the minimum adequate good congruence on S. Corollary 2.2 If 6 Is a "-homomorphism from S onto a semigroup T, then T Is a quasi-adequate semigroup whose band of idempotents is If S is idempotent-connected on which 6 Is a congruence, then {e8; e G B} and isomorphie to B.