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 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. as a consequence of Proposition 1.10 and Proposition 1.7, we have Part (l) of the following Theorem is a special case of [8, Theorem 1.10]. Corollary 1.11 If S is Idempotent-connected on which i is a congruence, then S/6 is type A. For the convenience of the reader we give a proof of the result. Theorem 2.3 Let 8 be a *-homomorphism from S onto a semigroup T. (1) If 5 is idempotent-connected, then also is T. 2. *-H0MOMORPHISMS (2) If S is a congruence on 5, then «(T) = p is also a congruence on T. proof (i) Let tgl and choose s e S so that s6 = t. For some s", s ; In this section certain properties of ldempotent-separating good s L*(S) s», s B"(S) E+ which implies seL*(T) S»8, S6E»(T) S+9. Denote homomorphisms on quasi-adequate semigroups are investigated. It has been s*B by t* and s+8 by t+. 5 is idempotent-connected. Then there is a noted recently [8], the idempotent-connected property Is preserved under connecting-isomorphism a: < s+ > * < s* > , that is, a is a bijection with good epimorphisms. We show (Theorem 2.3) that also the equivalence xs - s(xa) for any x €r < s+ > . Notice that for any y £ < t > , there relation 6 is a congruence on any idempotent-separating good homomorphic exists a unique x in B such that x8 = y. Since image of a quasi-adequate semigroup S whenever <5 is a congruence on S. (s* x s+)9 = s*8 x9 s+6 =t yt =y=x8 Throughout this section, S will denote a quasi-adequate semigroup with a band B of Idempotents. First we say, a semigroup homomorphism which implies s+ x s+ = x, then x 6 < s+ > . Define *f: S + T is a »-homomorphism if for any elements a and b in S, + ycf = (xa)e for any y 6 * t > , where x £ < s > and y = x8. Since for a L*IS) b if and only if a fL*(T} b1^ where as a R»(S)b if and only any x £ < s > : if af R*(T) b*f . In fact »-hojnoiisorphisins on S are Just the idempotent- sep&rating good homomorphisms as shown in the following Lemma: t*(xa)et» = (s»(xa)s»)8 = (xa)8 (xa £ < s* >) Lemma g.l Let -5- The congruence \i ia an idempotect-separating congruence on S then (xa)B6 < t* > and "? la a veil-defined map In fact Y= 0 I <++> a & vhich ia readily a tiijectfon. Finally, for any (Proposition 1.6). If S Is Idempotent-connected, ttie v is good {Lemma 1.8), that Is, v is a •—congruence and we have as a direct y £ < t > , x £ < s > with y = x6 we get; application of Theorem 2.3, the following corollary: yt = x6 s8 = (xs)B = (s(xa))9 = s6(xa)6 = t(y¥) . Corollary g.1* (i) If S is Idempotent-connected, then S/y is an Therefore, f is a connecting isomorphism and T is an idempotent-conneeted Idempotent-connected quasi-adequate se»igroup. (ii) If j Is a quasi-adequate semigroup. congruence and u is good on S, then i(£/u) is a congruence on S/p. (2) Let u, v tie in T such that (u, v) £ p, that is, u = gvh It has been noted in [h] that u Is not always good on S. for some gfe E(u+), hfe E(v») (Proposition 1.9 -8- -7- components used in this construction we: a fundamental quasi-a.dequa.te Lemma 3.2 Is a *-homomorphism from Q onto F/p. semigroup F on which & Is a. congruence, an adequate semigroup Q, and a •-homomorphlsm frcm Q onto F/4. Proof It is clear from the definition that * is an epimorphism. Let g be In Q and gijf • ap for some a £ F where there exists x In S such Recall from the previous sections that If S Is a weak type W that go ™ xS, aB = xu. It follows from the fact thati p, a, S, B, y are semieroup, the S/S is an adequate semigroup and S/y is a fundamental good and S/i, F/p are adequate: quasi-adequate semigroup' on vhlch 6{S/u) is a congruence. To achieve our (g\(i)* • a*p, g*a = x"S, a'B LxBy goal ve start vith the folloving proposition. Let e be an Idempotent in P such that efl = x*y. llien g*i(i • ep and Proposition 3.1 If S is a weak type W semigroup, then S can be • a* Le. Thus a*p = ep and g"i(i = aBp = Cg*)* which implies g** L"(F/p) B embedded in the direct product S/y * S/S. Similarly g% B«{F/p} g*. Pro°f Define the map that s1 = s2 and f is one-to-one. idempotent-separating and thus * is a *-homomorphism. Proposition 3.1 shows that any weak type W semigroup is Isomorphlc Vfe retain the above notations and put: to a subdirect product of a fundamental quasi-adequate semigroup on which T = {(a, g) 6 F * Q ; ap = gty) the equivalence relation 4 13 a congruence and an adequate semigroup. T Is readily a subsemigroup of F x Q. In fact we have: Our aim now is to locate the copy of S in a direct product of a more general form. Lemma 3.3 T is isonorphic to S. Let S be a weak type W semigroup. Let Q be an adequate semi- Proof Let (a, g} be an element in T. Let x be an element in S such group isomorphic to S/4 and F be a fundamental quasi-adequate semigroup that ga = x«, xu - aB. If also y is in S with fca = yfi, yu = aB, then isomoj-phic to S/y. Write p = 4(F). p Is a congruence on F. Let (x, y) & 4 r\ y and x •» y. Therefore we have a map >: T + S defined by a: Q •+ S/4 and B: F •* S/u be isomorphisms. If g & Q, go • xfi for (a, E)X = X for any (a, g) in T where x is in S and ga = xS, xy » aB. some x & S and xu fe S/y, then there exfsts an elenent a In F such It Is easy to check that X Is an isomorphism. that afl « xu and ap In F/p is well-determined by g. To justify this In particular, if Q • S/4 and F = S/u, then the •-homomorphism claim, let go = y{ for some y 6 S and yy = bfi for some b in F. t: S/S + F/4 is defined by (xfiH = (xu)p for any element x in S, and + Then (x, y} £ 6, that is, x = eyf for some e £ E(y ), f fe E(y*). T = {(xu, x&); x6 S} is the copy of S in the direct product S/u x S/*. Therefore xy = ey yy fy where ey £ E(y y), fy G E(y*y). Let g and h It can be noted that the following result has been proved. be idempotents in F such that gB = ey, he = fy. Then aB = (gbh)B and Proposition 3.it Let S be a weak Type W semigroup, Q be an adequate a = gbh. Since en £ E{y y), yy = 1)0, then y y |R b y which implies semigroup isomorphic to S/4 and F be a fundamental quasi-adequate semi- E(y+y) = E(b+6) and gB fe E(b+B), that is, g GE(b+). Similarly hft E(b«) group isomorphic to S/u. Then the equivalence relation 4(F) = p is a It follows that (a, b) 6 p and the claim is Justified. Therefore we have congruence on F and there is a •-homomorphisra « from Q onto F/p a map \|i: Q •* F/p defined by gt = ap for any g in Q, where ga = xS such that S is isomorphic to T = {(a, g) 6 F * Q; ap = g*). and xy = aB for some x in S. -9- -10- Our objective now is to prove a converse of Proposition 3.U , that Similarly, is, to describe a specific construction anil thus establish a structure C(a,. g), tb, p)) 6''P_— (a, b) £ n_ theorem for weak type W semigroups. Therefore Let Q be an adequate semigroup and F he a fundamental quasi- adequate semigroup on which S(F) = P is a congruence. Assume there ((a, g), (b, p))£ v ••> (a, b) £ UL(F) H UR^F) = u(F) is a *-hooomorphism lji from Q onto F/p . Define a subset S of the Hence a = b because F is fundamental. direct product F x Q by; On the other hand, if (a, g), (b, p) are elements in S such S = {(a, g) t F x Q; ap » et) that a = b, then ap = bp and g^i = pji. Let (e, f) be an idempotent S is readily a subsemigroup of F x Q. in S, that is, ep = f*. Then fi|/ gt|i = f* pg> vhich implies fg L*(ftjfp (t is a •-hamooorphism) and ea = eb. Therefore (ea, fg)L*(FxQ)(eb, fp) The following sequence of results provides considerably more and (e, f)(a, g)L«(S)(e, f)(b, p). information about S. By a similar argument we get (a, g)(e, f)R*(s)(b, p)(e, f). Lemma 3-5 S is a quasi-adequate semigroup. Hence ((a, g), (b, p))6 u. Proof For any eleiaent (a, K) in S; we get a*p * (ap)* • (gtlO" = g*i|i Corollary 3.8 U is good on S. and (a", g#) is an idempotent tn S. Since (a*, g") L*(F x Q}(a, g). Then (a*, g*) L»(S)(a, g). Similarly {a+, g+) R»(sKa, g}. It is Proof Let (a, g) be an element in S; (a, g)L"{S)(a*, g*), Then for any noticable that the set of idempotents in S forms a subsemigroup of 5 elements (b, p), {c, v) in S. and thus the result holds. ((a, g)(b, p), (a, g)(c, v))6 u «* ((ab, gp), (ac, pv)) £ p The following corollary is an inmediate consequence from the definition of S and the proof of Lemma 3.5- =? ab = ac (Lemma 3-7) Corollary 3.6 (i) For any idempotent f in Q{F) there exists an -£. a»b = a»c (Corollary 1.2) idempotent e in F(Q) such that (e, f)((f, e)) is an iaempotent in S. (ii) L»(S) = L»(F x ft) 0 (S x S) and R*(S} = R*(F X Q) C\ (S x S). => ((a*b, g»p), (a^c, g»v))6 u (I*mma 3.7) Lemma 3.7 For any elements (a, g), (b, p) in S. ((a, g), (b, p))£u => ((a», g»)(b, p), if and only if a = b. Similarly Proof Let e be an idempotent in F and choose an idempotent f in Q tCb, p){a, g), (c, a+, g+), (c, such that (e, f) in S. Then for any elements (a, g), {b, p) in Si Thus the result follows by Corollary 1.2 and Lemma l.ti. ((a, g), (b,p))6 uL=> (e, f](a, g) L»{S)(e, f)(b, p) (definition of wL} Lemma 3-9 For any elements (a, g), {t>, p) in S ({a, g), (b, g => (e, f)(a, g) L»(FxQ)(e, f)(b, g) (Corollary 3.6 (ii)) if and only if g = p. => ea L»(F) eb Proof Let (a, g), (b, p) be elements in S such that ((a, g), (b, p))fe 6. Then (a, g) = (e, f)(b, p}(q_, h) for some + + + -* (a, b) G uL(F) (e, f) £.E((b, p) ) = E(b , p ) and (4, h) G E((b, p}«) = E(b», p« -12- -11- It follows that f 6E(F+), h6 E( »). Since d is. adequate, then lP equivalence relation & is a congruence. In thta section we describe the f « P , h = P and g •= fph = p+p p* = p. least fundamental adequate good congruence on S, the largest superabundant On the other hand If g = p in the elements {a, g), Co, p) of full eubsemlgroup of S and the largest full snbsemigroup of S which is a S, then g* = p* and ap = hp, that is, a «= ebf for some e 6 E(b+), band of cancellatlve monoitis. These results generalize those of Venkatesan f€E(b»). Notice that: p** = b«p = fp and (f, p») IB an idempotent [12] for orthodox semigroups. in S. Moreover, (f, p*)£ E(b*, p») =. E{(.h, p)»). Let S be a type W semigroup with band B of idempotents. It Similarly; {e, p+)G E((b, p)+). Now from a = ebf we get follows from the results in the first and second sections that: u is a (a, s) = (e, p+)(b, p)(f, p*) vhich implies ((a, g), (b. p)) £ S, good congruence on S, S/u is a fundamental quasi-adequate semigroup whose band of idempotents iB {e u: e & B}, S is the minimum adequate good con- As an immediate consequence of Lemma 3.9 we have; gruence on S, S/S is a type A semigroup whose, semilattice of idempotents Corollary 3.10 S is a congruence on 5. is (e fi; e fe B}, and u(S/4) is a good congruence on S/6. These facts Moreover, are used in the discussion frequently without comment. Lemma 3.11 S/y is isomorphic to F and S/6 is isomorphic to Q. Consider the natural homomorphisms 6 : S + S/6, u(S/S) : S/* * (S/6)/u and define A to be the kernel of the composition 6 -u(S/6) . Proof Define y: S * F by (a. g)y - a for any element (a, g) in E. Clearly, y is a homomorphism from S onto F. If (a, g), (b, p) are Since the composition of good homomorphisms is good and a surjective elements in S, then homomorphism is good if and only if its kernel is a good congruence. We conclude that 1 is a good congruence on S and thus S/X is a quasi- (a, S)y = (b, p)y <=> a = b <—> ((a, g), (b, p)) €: V (Learos. 3.7) adequate semigroup whose band of idempotents is CeX; e € B}. Borne more properties of X will be investigated in order to achieve part of our Hence the kernel of y is u and S/D is isomorphic to F. objective. We start with the following Lemma which can be deduced directly By a similar argument it follows that S/S is isomorphic to Q. from the definition. Now a converse of Proposition 3.1* is evident and in conclusion ve Lemma l*.l For any elements x, y In S, the following statements are have equivalent: Theorem 3.12 Let Q be an adequate semigroup and F be a fundamental (1) (x, y) e x quasi-adequate semigroup on which fi(F) = p is a congruence and there is (2) (x«, yi) £ y(S/6) •-homomorphism i> from Q onto F/p. Then (3) (ex)»i = (ey}#6 and (ex)+6 = (ye)+S for any e €r B. S = {(a, g) £ F " Q; ap = gi(i} Corollary It.2 x|B = 6 |B Proof For any g, h in B with (g, h) & X, that is, in particular is a weak type W semigroup. egS » ehfi for any e €• B. Take in turn e = g and e = h to get Conversely, every such semigroup can be constructed in this way. g6 = gh« - hg6 = h« and (g, h) £ 6. Conversely, if g, h in B such that (g, h) £ S, then for any e fe B; (eg)*S = (eg)« = (eh)S = (eh)»S and (ge)+4 = (hel+6. Thus (g, h) & X. 1*. SUBSEMIGROUPS OF A TYPE W SEMIGROUP The following proposition establishes one of the main properties of X. As it has been mentioned In the previous section, a type W semi- group 3 is an idempotent-connected quasi-adequate semigroup on which the -lU- -13- Proposition U.3 x is the least fundamental good congruence on 5. Lemma it.5 If M is a superabundant full aubsonigroup of S, then for any t & H, Ct, t£) & X. Proof Let gX, hX be idempotents in S/X for which g, h in B. Then gS, hfi are idempotents in S/& so that ghS = hgfi and by Proof Let M be a superabundant full subsemigroup of S. Than for any Corollary k.S {gh, hg) fe X. Thus S/X is adequate and X 1B an eleaent t In H, there exlBts f in B such that t 6 HJ and thus adequate good congruence on S. To show that S/X is fundamental, let n f t € H» (Corollary 1.3). For any element e in B, let g and h fee e be in B and x, y be in S. Then In B where et 6 H«, etE 6 H». It follows that; g E»et B»ef, eX xX L*(S/X) eX yX -^> (ex)«X = (ey)»X h R»et2 R*ef and thus (et}»« = gS (et L#g) —*> (ex)"S = (ey)"S (Corollary It.2) Likewise Ig » ef} h« (ef B h) xX eX IR"(S/X) yX eX —> (xe)+S = (ye)+« Hence Similarly, (te)+4 = (t2e}+6. (xX, yX) fe n(S/x) —=> (ex)«6 = (ey)»S and (xe)+6 = (ye)+« Ve G B Therefore (t, t ) £ X. => (x, y) e X (Lemma h.l) Corollary t.6 If M is a superabundant full subsemigroup of S, then the that Is, X is a fundamental adequate good congruence on S. Now let it restriction of X on H is a aemilattice congruence on M. be a fundamental adequate good congruence on S. Then for any x, y in S; Proof By Lemma it.5 all the elements of M/X are idempotents and the (x, y) eX —> (ex)*S = (ey)*4, (xe)+6 = (ye)+fi e B idempotents commute by Corollary It.2. => (ex)"n = (ey)*?r, (xe) IT » (ye) ir (p £, IT) In fact the restriction of X on M is the minimum semilattice =* (ex)n L»(S/n)(ey)Ti, (xe)Ti fl»(S/iO{ve)* {ir is good) good congruence on M as a direct consequence of the following Lemma: —> (xtr, yit) fe u(S/it) (definition of y(S/ir)) Lemma ^4.7 If n is any aemilattice (band) good congruence on S, then =^ XTT = y¥ (S/it is fundamental) X Sc n(v £. n). Hence the result. Proof Let n be a semilattice good congruence on S, and a, b be elements in S. Then As an immediate consequence we have (a, b) fe X (ea)»S = (eb)»6 Ve 8B Corollary h.k S/X is type A and 6 £. X. (ea)»n = (eb)*n By a full subsemigroup of S we mean simply one which contains B. (6 C n) Recall from [2] that any full subsemigroup U of S satisfies the following (ea)n = (eb)n (n is good, S/n is a semtlattice) property: ea n = eb ri (ea. R* ea+, eb IR» eb+) L»(U) =L'(S)n(U*U) and E*( U ) = P"(S) H {U *U). +. + + + = abn.ban = (talcing m turn e = a+, e = b+} and it is abundant. (a b n = b a n) To begin the process of describing the largest full subsemigroup of S which is superabundant or a band of cancellative monoids, let us see the effect an of X on superabundant full subsemigroups of S through the following results. Hence X C i -15- -16- The eaay proof of the other part of the Lemma is. oottted. Proof Since for any idempotent xA In S/X, there exists, an Iderapotent The following Lemma is needed for further Investigation. e in S such that x & H*e (lemma It.6), then clearly A = {x fe S; {x, x ) & J,} is a superabundant full sub semi group of S. LeMma k'B If x* is- an Idempotent in S/X, then there exists an Idem- potent g in S such that x€-H». By Lemma It. 5, any superabundant full subsemlgroup of S Is a subset of A, Proof Let x\ be an idempotent In S/A. Choose e In B such thnt To show the other part of the Theorem, it Is readily (x, e)C X, that Is, (xp, ep)& u(S/6) (Lemma l+.l). In particular M • {x £ S; (x, x ) 6 y) la a superabundant full subsemlgroup of S, Since for any x fe M; xu is an Idempotent in S/w and there exists an (xp, ep)fe H»(S/5). Therefore, x+p = ep, x»p = ep and E(x+) * E(e) = E(x»). • + * idempotent e In 5 with (x, e) 6 u so it is easy to see that the Since x x x = x x x x = xx» = x and for any s, t in S; H*-classes of M are the u-classes and H* Is a congruence. Hence + + xs = xt x x s = x x t by Lemma U.10, M is a band of cancellative monoids. Let N be a full then x L» x x by Corollary 1.2. subsemigroup of S which is a band of cancellative monoids. Then H*(N) Is a congruence and S Is superabundant (Lemma U.10). Notice that A similar argument leads to x IR» x+x*. Hence (x, x+x*) & H». We may take g to be x+x . n 6-N •*> (n, e) £ H"{JJ) for some e In B We combine Lemma It, 8 with Corollary 1.3 to get » (n, e) t u(N> (H»(N) = uOO) Corollary It.9 If x\ is an Idempotent in S/A, then H« is a cancellative =» (n, e) & u(S) (L»(]f) = L*(S) O (N » If) and monoid. R*(N) = R»(S) H (N * N)) The following Lemma is mentioned in [l] and IB needed for the final ar> n fe M result. It is included here for the sake of completeness. proving N is a subset of M. Lemma It.10 A semigroup P is a band of cancellative monoids if and only if P is superabundant and H* is a congruence. Proof Let P = U T be a band of cancellative monoids T (a &X) and Qt m. A Ot Q let eQ be the identity of Ta> Let i t I , ! t I., t 6 I be such that as = at. Then 08 = ay = 6, say, and e a, e t fc T,, Hence ae, e s = ae, e t and since T, is cancellative, we have e s = e t. ACKHOWLETCEHEJiTS Thus a L*ea- Similarly a rfl»e and so P is superabundant. It is easy to see that T = H* for any a 6 T and it follows that H* is a a a a The author would like to thank Professor Abdus Salam, the congruence. International Atomic Energy Agency and UNESCO for hospitality at the On the other hand, if P Is superabundant then each H*-class is a International Centre for Theoretical Physics, Trieste. cancellative monoid and x H* x for any x € P. If H* is a congruence, then P/H* is a band and the natural homomorphism S1: P •* P/H* defined by xf = H* shows that P is a band of the H*-classes. Theorem U.ll Let A = {x t S; (x, x^ & I] (M=[i€ S; (x, X2) & v)). Then A(M) is the largest full subseninroup of 5 which is superabundant (a band of cancellative monoids), -18- -17- REFERENCES [l] A. El-Qallali, "L*-unipotent semigroups", preprint, (to appear}. [2] A. El-Qallali and J.B. Fountain, "Idempotent-connected abdundant semigroups", Proc. Roy. Soc. Edinburgh, Section A91 (l98l) 79-90. [3] A. El-Qallali and .T.B. Fountain, "yuasi-adequate semigroups", •, Proc Roy. Soc. Edinburgh, Section A£L (I98l) 91-99. ( [U] J.B. Fountain, "Adequate semigroups", Proc, Edinburgh Math. Soc. : 22. (1979) 113-125. [5] J-B. Fountain, "Abundant semigroups", Proc. London Soc (3) M*_ (1982) 103-129, [6] T.E. Hall, "Orthodox semigroups", Pacific J. Math. J£. (Wl) 677-686. [7l J.H. Howie, An Introduction to Semj-group Theory, (Academic Press, London, 1976). [8] M.V. Lawson, "The natural partial order on ;*n abundant semigroups, ' Proc. Edinburgh Math. Soc. 30. (1987) 169-186. (. [9] B.D. McAlister, "One-to-one partial right translations of a right cancellation semigroups", J. Algebra 1*3. (1976) 231-251- [10] F. Pastijn, "A representation of a semigroup by a semigroup of matrices over a group with zero", Semigroup Forum 10_ (1975) [ll] P.S. Venkatesan, "Orthodox semigroups", Semigroup Forum 2p_ (I9o0) 227-231. [12] P.S. Venkatesan, "Certain subsemigroups of orthodox semigroups", Semigroup Forum 3£ (1985) 183-187. -19-