Rend. Sem. Mat. Univ. Poi. Torino Voi. 50, 3 (1992)

P. Aglianò

ON C O M B I N A T O R I A ! , I N V E R S E SEMIGROUPS^)

A b s t r a c t . A characterization of ali the simple (in the universal algebraic

sense) combinatorial inverse semigroups, as well as of the monolith of the

subdirectly irreducible ones, is obtained. Then the ideal of varieties of

combinatorial inverse semigroups is studied and a theorem is proved, which

states that the variety of semilattices has a unique upper cover.

This paper originates from the author's investigation on varieties of algebras whose congruence lattices are semimodular [1]. To make the contents of the paper more understandable to the reader we'll start with some hi story. Semimodularity assumes particular relevance in varieties of semigroups since any other "familiar" congruence property is too strong to work with. For instanee Freese and Nation proved that any variety of semigroups in which all' the congruence lattices of ali the semigroups satisfy some nontrivial lattice identity consists entirely of groups [7].

Inspired by such results the author started investigating congruence semimodular varieties of semigroups, knowing not much more than the defìnitions and being totally unàware of the investigatlons that were already been carried on by several authors. This lack of proper background payed off somehow, in that the author was able to prove a number of theorems that seemed very interesting to him. Later on the author started going through the

(*)This paper constitutes part of a dissertation submitted to the Graduate Division of the University of Hawaii in partial fulfìllment of the requirement for the PhD in Mathematics. The dissertation has been written under the supervision of Prof. J.B. Nation.

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literature and discovered to his dismay that many of the theorems he proved were already well-known. However even if many results were already been published, most of the author's proofs and tecniques were new: this justifles the existence of this paper that so can be considered mostly an expository one, even if some results are not published elsewhere.

The author has continued his investigations on congruence semimodular varieties and the many results obtained had since been collected in severa! papers ([2], [3], [4], [5]). This paper tells how it started.

Any nontrivial variety of combinatorial inverse semigroups is congruence semimodular and contains the variety of semilattices (which in many ways can be considered the prototype for congruence semimodular varieties). Moreover any variety of combinatorial inverse semigroups cannot contain nontrivial groups. Due to ali the above it seemcd worthy to ' take a closer look at the congruence structure of combinatorial inverse semigroups. Our analysis will lead to a characterization of ali the simple (in the universa! algebraic sense) combinatorial inverse semigroups, as well as of the monolith of the subdirectly irreducible ones. This wrill also give us some information on the ideal of varieties of combinatorial inverse semigroups.

Our notation is more or less standard. For general background the reader can consult [11] (for uni versai algebra and lattice theory) and [6], [8] and [13] (for semigroup theory).

1. Preliminaries and Definitions

For any algebra A, Con(A) will denote the congruence lattice A and 0̂ 1 and 1^ will denote its smallest and largest elements. For I C A we will denote by $(X) the minimal congruence collapsing ali the pairs of elements of X. If À' = {«,£>}, we will write t?(a,6). An a tom in Con(A) is any a E Con(A) such that a ^ 0^ and a is minimal with respect to the lattice ordering. For any lattice L and x,y 6 L we say that y covers x and we write x -<y if, whenever x < z < y,we have z — x.

DEFINITION 1.1.

(1) A lattice L is said to be semimodular if for ali xyy 6 L we have

x A y -< x implies yAxMy.

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(2) A class IC of aìgebras is said to be congruence semimodular (briefly CSM) iffor every algebra, A G IC, Con(A) is a semimodular lattice.

The proof of the nontrivial facts that semimodularity is preserved by subdirect products and by interval sublattices can be found in [1] or [9]. For congruence lattices of aìgebras we actually have something more.

LEMMA 1.2. Let K, be a class of aìgebras closed under homomorphic images. Then K, is CSM if and only iffor ali A G IC, for ali 6 e Con(A) with 6 y OA, we have 6 V (p y (p for ali (p G Con(A) with 6 ^ (p.

Proof Necessity is obviotis. On the other way let IC he such a class, let A G IC and let 6,(p E Con(A) with 6 y 6 A(p. Then 6/6 A (p is an atom of Con(A/6/\(p) and, since /C is closed under homomorphic images, A/6A<p G /C. Hence by hypothesis we get

(e V <p)/(9 A <p) = (6/(0 A v)) V ( v / ( « A v>)) >- <p/(0 A <p).

Therefore 6 V <p y <p in Con(A) and so Con(A) is semimodular. •

We recali that, for any class of aìgebras /C,- one defines certain operators:

(1) H (IC) denotes the aìgebras that are homomorphic images of some member of K.

(2) S(/C) denotes the class of aìgebras that are isomorphic to a subalgebra of some member of IC.

(3) P(IC) denotes the class of aìgebras that are isomorphic to a direct product of members of K.

Let us observe that if IC is a CSM class, then so is H(JC). In fact if B G H(K) then B = A/6, for some A G IC and 6 G Con(A), and, by the second homomorphism theorem, Confò) is isomorphic to the interval sublattice [0,1./J of Con(A). So, by Lemma 1.2, Confò) is semimodular. Let now V be a variety; it is well-known that, for any algebra A G V, A G H(F\;(X)) for a large enough X, where Fy(X) is the relatively free algebra on V generated by X. Therefore a variety is congruence semimodular if and only if each of its free aìgebras is.

The rest of this section collects some facts about combinat ori al inverse semigroups that will be used in the sequel. Inverse semigroups can be regarded

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as algebras with a binary and a unary operation (the inverse) satisfying the following identities [13]:

(xy)z « x{yz) xx~xx « x (x~1)~1 « x

(.T?/)_1 « y~1x~1 xx~1yy~1 « yy~xxx~x.

They can be characterized as regular semigroups in which the idempotents commute or as regular semigroups having exactly one inverse. Let us recali that in any semigroup S, the set E$ of idempotents of S can be given a partial order by setting e < / if ef = fé = e. If S is inverse, since idempotents commute, then it is easily seen that the above ordering is a semilattice ordering, i.e. S# is a meet-semilattice where the meet is the semigroup operation. An inverse semigroup is combinatorial if the Green's relation H is tri vi al on S. The proof of the following important fact can be found in [13], p. 519.

LEMMA 1.3. For a variety V of inverse semigroups the following are equivalent:

(1) V is combinatorial.

(2) There is a naturai n sudi that V J= xn~*~l « xn.

(3) No S G V has nontrivial suhgroups.

A certain inverse semigroup will be particularly useful in the sequel. It is cali ed the bicyclic semigroup and will be deuoted by C. It can be deiìned as the semigroup with identity e generated by a two element set {p, q}, subjected to the single relation pq = e. It is not hard to show that EQ = {qnpn : n > 0} and that the semilattice E^ i s an infinite descending chain where qlpl > qJpJ

if and only if i < j . AH this assertions (and more) are consequences of the following lemma, that characterizes C very efTectively. The Lemma is due to 0 . Anderson and it is contained in his unpublished thesis. For a proof the reader can consult [6].

LEMMA 1.4. Let S be a semigroup and let e,a,ò E S sudi that ea = ae — a, eh — be — b, ab = e and bei ̂ e. Then the subgroup generated by {a, 6} in S is isomorphic to the bicydie semigroup.

It is fairly clear that any periodic, and hence any combinatorial inverse semigroup, cannot contain the bicyclic semigroup as a subgroup. This gives us a very useful lemma.

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LEMMA 1.5. Let S be a periodic semigroup and let E be the set of idempotents ofS. Then, if e, / E E and e > f, we cannot have e E S1fS1, the two sided ideal generated by f.

Proof. Suppose that e E S1fS1. ; then there are x,y E S1 with e = xfy. Set now u = exf and v = fye. Then we have

uè = eu = uf = u fv = ve = ev = v uv = e.

Now let us set g = vu and note that

g = vuvu = veu = vu — g

gf = vuf = vu = g fg = fvu = vu = g.

Hence g E E and g < f < e. But then e,w,v satisfìes the hypotheses of 1.4, therefore they generate the bicyclìc semigroup. This is a contradiction, so the proof is completed. •

Let us now get back to congruence semimodularity; the fact that combìnatorial inverse semigroups are congruence semimodular is a consequence of a deep result by P. Jones [10]. In the case of combinatorial inverse semigroups, however, a shorter direct proof can be produced and we will proceed to do it now. We start quoting a famous lemma by Reilly and Scheiblich [14].

LEMMA 1.6. Let S be an inverse semigroup with E as the lattice of idempotents, and let L = Con(S) . Define 0 G L2 by setting (a,/?) G 0 iff O:\E = /3\fi. Then 0 G Con(L), hence the map a t-—> Q\E 1S a lattice homomorphism from Con(S) to Con(E).

We will be able to show that, if S is combinatorial, then the map a i—> a\jjj is one to one, after we have proved the following lemma.

LEMMA 1.7. Let S he an inverse semigroup, E as its semilattice of idempotents and let a, fi E Con(S). Then the following are equivalent:

(1) a</3 in Con(S).

(2) a\fi C /3\E and, for ali a £ S with a E Re for some e E E, if(a,e) E a then (a,e) E fi.

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Proof. One implication is obvious. Suppose that (2) holds and let e G E, a G S and (a, e) G a. Now a/a is an idempotent in S /a , therefore (a, a2) G a. Set now g = aa~2a ; from (e, a) , (a, a2) G a we get (e, a2) G a, hence (e,a~2) G a and therefore (e,#) G a. Moreover clearly g E E, so by hypothesis we get (e,g) G /?. Now (ga,g) G a and ga G -R ,̂ since {ga)(ga)~l = gaa~xg = aa~lg = aa~laa~2a = aa~2a = g. Therefore by hypothesis (ga,g) G (3. Néxt from (a,e) G a we get ( a a - 1 , e ) G a, therefore (aa*1 ,g) G a and so (a~1aìa~1ga) G a. But a~lga = a~1aa~2a2

ì so a~lga G i£ and hence (a _ 1 a,a _ 1 5fa) G (3. Therefore (aa _ 1a,aa - 1<7a) = (a,ga) G /? and, putting together ali the information, (a, e) G f3.

We have therefore proved that (a, e) G a implies (a, e) E /3 for ali e E E and a G 5 . Suppose now (a, 6) G a ; from (b~1a,b~1b) E a we get (b~1a,b~1b) E (3 and from (aa _ 1 , 66 _ 1 ) G a we get ( a a - 1 , ò ò _ 1 ) G /?. So

a = aa^apbb^apbb-^b = 6

and so (a, 6) E (3. Hence a < f3 and we terminate the proof. •

LEMMA 1.8. Let S he a combinatoria,! inverse semigroup with witness equation .TW + 1 = xn and let E he its lattice of idempotents. For a,ny a, {3 E Con(S) we ha.ve

a = P ifF <*\E = P\E-

Proof. Necessity is trivial. Suppose now a\^ = /3\E 5 then in particular aÌE < P\E* Let e E E, a E S and suppose that a E Re- If («5e) G a then (« n , e ) G a and, since S is combinatorial, an E E. Therefore (a n , e ) G /?. Since a E Re we have ea = a so

a = eaf3ana = an+1 = an/3e.

Hence (a, e) G /? and, by Lemma 1.7, a < (3. But we can repeat the entire argument starting from (3\E < a\j? to get (3 < a. Therefore a — f3 and the lemma is proved. •

The previous lemma basically says that , if S is combinatorial inverse, then the lattice Con(S) is isomorphic with the lattice L where L — {CI\E • a E Con(S)}. We will prove that L is semimodular and hence we will have our result.

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T H E O R E M 1.9. For any combinatoria! inverse semigroup S the lattice L is semi modular.

Proof. Let O:\EIP\E É Con(L) with a | ^ ^ /?|_E and a ^ A fi E -< a_g. By moding out by a A /? we can assume a E y 0^. Suppose, by way of contradiction, that O.\E V /3\E )/- P\E* Then there are e,f E E with (e, / ) G a V fi - fi and a g $(e, / ) V /?. Moreover we can pick e > / . Since (e, / ) G a V /?, by Lemma 1.6 we can fmd e i , . . . , en G E with

e (a U fi) ei . . . en (a U /?) / .

Among ali possible choices of e and / , we pick a "minimal" (e, / ) in the sense that, if ( e ' , / ' ) G a U fi — fi and ( e ' , / ' ) is connected by a shorter sequence, then a C tf(e',/') V /?. Clearly, for a minimal ( e , / ) , a and /? alternate in the sequence.

Look now at (e ,ei) ; if (e,ei) G fi, then ( e i , / ) G a V fi — fi, (e ' i , / ) ^ /3 and ( e i , / ) is connected by a shorter chain. So a C i9(ei,/) V•/? and since ${ei->f) ^ fl(eif) w e get. a contradiction. Hence (e ,e i) G a and similarly (en , / ) G cv. Moreover, by minimality, e > ej for i = 1 , . . . , n. Let now 7 be the congni enee on S generated by Àr-, where

X = {(c t;,ei+1) : (e ; ,e i + 1 ) G a } .

Clearly 7 < a and consequently 7 ^ = a|_g;.' Hence in particular (e, ei) G 7. But then, by the congruence generation theorem, we must have e G S1eiS1

for some e,-.'.But e > e{ for ali i and therefore this constitutes a contradiction, via Lemma 11.4. So 7 ^ < O:\E-, contradicting the fact that a\g was'àn atom. Hence a\E V/?]_£> /3\E and the theorem is proved. •

Before closing this section we need one more property of combinatoria! inverse semigroups.

A congruence 9 on a semigroup S is idempotent separating if for ali e , / G E 5 , ( e , / ) G 0 implies e .= / . It is well-known that, in a regular semigroup, the largest idempotent separating congruence is contained in the Green's relation?^. Since li = $s m a n y còmbinàtorial inverse semigroup S, it follows that no nontrivial congruence of S can be idempotent separating. Hence any nontrivial congruence has to collapse two distinct idempotents. This fact will be used without further comments throughout the paper.

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2. Simple and subdirectly irreducible combinatorial inverse semigroups

Most of our proofs in this section rely heavily on the so called "Munn representation" for inverse semigroups, that was discovered by W.D. Munn [12]. We will describe it briefly and we refer to [8], p. 142 ff or [13] p. 170 ff for details and proofs.

Let E be a semilattice and let Tej be the set of ali isomorphisms

between the principal ideals Ee and Ef, regarded as partial maps on E.

L e t T £ = | J e , / e £ T e , / . We have:

T H E O R E M 2.1. [12].

(1) For any semilattice E, T ^ is an inverse semigròup whose semilattice of idempotents is isomorphic to E via the map e \—> ìdj?je.

(2) IfS is an inverse semigròup and E is its semila,ttice of idempotents then there is a homomorphism (f : S i—• T ^ with ker (p C H. Moreover, for ali a E S, we have <p(a) = a where a is the isomorphism from Eaa~1 to Ea~1a defmed by à(e) = a~ìea.

If S is combinatorial, then Ti = O5, so the above theorem implies that the map <p is one-to-one, i.e. for ali a,b £ S, à = b implies a — b. Another consequence of this fact is the following :

LEMMA 2.2. Let S he a combinatorial inverse semigròup and let E be its semilattice of idempotents. Then, if<p is the homomorphism defmed in Theorem 2.1, we have (f(S) D Te>e — {idjpe} = {e}. In other words, the only automorphism of a principal ideal of E representing an element of S is the identity.

Proof. Let a £ ^(S) fi Tee ; then a is an isomorphism from Ee to Ee. But, by defìnition, it also an isomorphism from Eaa~l to Ea~1a, hence we must have aa~1 — a~xa — e. But then we get (a,e) E H and hence a = e since S is combinatorial. Therefore a = e. •

Let us recali that an algebra A is s imple if its congruence lattice contains only two congruences, 0y\ and 1^. Unfortunately the word simple has a different meaning in semigròup theory (where the word used for the corresponding concept is congruence-free). We will try to avoid confusion

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by saying that a semigroup S is a simple algebra whenever it has the above property. An algebra A is subdirectly irreducible if Con(A) has exactly one atom, that will be called the monolith of A. If K is a class of algebras, SI(K) denotes the class of ali the subdirectly irreducible algebras in /C.

In the sequel we will characterize the monolith fi of subdirectly irreducible combinatorial inverse semigroups and we will show that they always have a unique nontrivial block. First we characterize certain atoms of Con(S) for any combinatorial inverse semigroup S.

;»

T H E O R E M 2.3. Let S he a combinatoria! inverse semigroup and let E be its semilattice of idempotents. Suppose that e , / E E are sudi that

(!) ^EÌe^S) 1S an atom in Con(E) .

(2) For any a E S such that aci~l = e, we have that f is the unique lower cover ofa~1a in E.

Then $ ( e , / ) is an atom in Con(S) and it has exactly one nontrivial

block.

We prove the theorem through several lemmas. From now on we assume that e , / E E and satisfy (1) and (2) of Theorem 2.3 and we set X = {a E S : eia"1 = e}.

LEMMA 2.4. For any a E X, ,d^(a~1aìf) is an atom of Con-(E).

Proof. Since aa~l = e, the ideal Ee is isomorphic to Ea~1a via the map a. Hence it is clear that ^£-(fl -1a, / ) must be an atom of Corc(E) as well.

LEMMA 2.5. For any a, b E X - {e}, a / b and g E E, g < f, we have

(!) cr^ga = aga'1 = g

(2) a* = f

(3) àf = fa = f and a " 1 / = / a " 1 = /

(4) a3 = a2 = / and ab = f = a^b'1

(5) ab-1 = / .

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Proof. We start with (1). Since / -< e = aa~x and Eaa~l and Ea~la are isomorphic via a, it follows that a~l fa -< a~1aa~1a = a~1a. But the only lower cover of a~1a is / , so a~1fa = f. But then afa-1 = aa~1fa~1a = f. Hence the map «|j£/ is an isomorphism from Ef to Ef. But, by Lemma 2.2, the only isomorphism between principal ideals of E can be the identity, so à\Ef — ìd-Ef' But this implies a~1ga = g for ali g < e. Similarly we can see that aga~l = g. For (2) we compute

a2a~2 = aaa~1a~1 = aaa~1a~1aa~1 = aea~1aa~l = afa~l = /

a~2a2 = a~1a~1aa = a~1a~1aaa~1a = a~lfa = / .

Hence ( a 2 , / ) G Ti and so a2 — f. The proof of (3) is similar and (4) follows easily from (2) and (3). For (5) we compute again, using the fact that a - 1 a ^ 6_16,

(ab'^ab-1)-1 = ab^ba'1 = ab~l ba~l aa~l = afa'1 = f

(ab'1)-1 (ab'1) = ba^ab"1 = ba^ab^bb'1 = bfb'1 = / .

So again (ab"1, f) G Ti, implying a ò - 1 '= / . •

As a consequence of Lemmas 2.4 and 2.5 we have:

LEMMA 2.6. Lei A be the inverse subsemigroup generateci by X in S. Then

A = {e,fìa,a~1,a~1a,a~1b : a, 6 G X and a ^ b}

and its semilattice of idempotents is represented in Figure 2.3.

Proof. That the set displayed is actually the base set of A, it follows almost immediately from the previous two lemmas. The only thing that we have to check is that a~1bb~~1c G A whenever «,ò, e G X. But if 6, e G X then bb~x — ce-1 = e, so

a~1bb-1c = a - 1 e c = a _ 1 c c - 1 c = a~1c G A

Next if a - 1 6 were an idempo.tent for some a ^ 6, then we would have

a^b = a-Ha'H = a-1 fb = f

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e a^a b-'b o o • • • ^ °

o

/

Figure 2.1

and consequently

e = aa~lbb~1 = c t /6 _ 1 = / ,

a contradiction. So ali the idempotents of A are actually represented in Figure 2.1. •

Let us emphasize that another consequence of the previous lemmas is that A is "symmetric", in the sense that , if we start from any pair a~1arf and we construct the inverse subsemigroup of S according to the specifications of Theorem 2.3 referred to a~la and / , we again obtain A. We will use this fact without further mention.

LEMMA 2.7. A is a. simple algebra,

Proof. Since any congruence on A must collapse two distinct idempotents, by what we bave said above it is enough to prove that ^ ( e , / ) = 1A. But for any a G X, ea = aa~1a = a and fa = / , so ( a , / ) G ^ ( e , / ) . Hence ( a " 1 , / ) G tfA(e,/)'and ( a " 1 ^ / ) G ^ ( e , / ) . Therefore^A(e, f) = 1A

and A is simple. •

Let nów C be the crown.of A, i.e. C = {g G -£̂ 4 : <7 ^ / } • Clearly, for a G A, « a - 1 G C if and only a / / .

LEMMA 2.8. A (the base set of A) is the only nontrivial block of a

congruence on S.

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Proof. Let x,y E A, x ^ y and u E S. We can assume w.l.o.g. that xx~l E C. Suppose that uu-1 Jf x~1x,y~1y. Then we must have a^arMtt - 1 = y~1yau~1 < f. So we compute

(xu)(xu) 1 — xuu 1x * = #?n/ *£ 'xx 1x 1xx 1

— !«.—1^. u, . T~_~~~,* o c« ..: —1„ —1 — uu x x by Lemma 2.5,' since uu x x < f

= uu~1y~1y = yuu~1y~1yy~1

= yuu~ly~l = (yu^yu)'1,

and

(a;w) 1(.TW) = n *£ 1a;M = u 1x 1xuu lu

= u~1y~1yuu~1u = (yu)~1(yxi).

ITence (xu,yu) E W, implying xu = yu.

Assume now that uu~l > x~1x. Then clearly w7e must have / < uu~1y~1y < y~1y. Moreover

(xu)(xu)~l = xuu~1x~1 = xuu~1x~1xx~1 = xx~* E C,

hence (xu) E A. Ilence (xu)~1(xu) = u~1x~1xu >- / by assumption (2); but recali that Euu-1 and Eu~lu are isomorphic via fi. Ilence we must have t^-1a;~1x?i >- u~lfu as well. But since (xu) E 4̂ the only lower cover of (xu)-1(xu) = u~1x~1xu is / . ITence we must have u~lfu = f. Next if V~1y < uu"1 we can prove that yu E A as above ; if y~1yvu~1 = f instead, then we have

(yu)(yu)~l = yuu~ly~x = yuu~ly~lyy~l = yfy~x = /

(yu)~1(yu) = u~1y~1yu = u~1y~lyuu~1u = u~l fu — f.

Therefore (yu, f) E Ti and so yu = f.

Finally suppose that uu-1 > y~~1y. AH cases are proved by symmetry with uu"1 > x~lx except for the one in which y~ly £ C, i.e. y — / , and uu~x ^ x~1x. But in that case we compute

(xu)(xu)~* = xuu~lx~l = xuu~1x~1xx~1 = xfx~~x = /

(fu)(fu)-1 = fuu-1f = f

(xu)~1(xu) = u~rx~lxu = u~1uu~1x~1xu = u~l fu

(fu)-\fu) = u-iffu = u-'fu.

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Hence (xu, fu) E H and again xu = fu.

So far we have proved that, if u E S and x, y E A, then either xu,yu E A or #w = 2/w. The proof that either ux, uy E A or ux = uy is totally symmetric (we assume x~~xx E C instead). Therefore we conclude that A is the unique nontrivial block of a congruence in S. •

Proof of Theorem 2.3. Since A is a simple subalgebra of S, for any 0 E Con(S) we must have either 9 fi A2 = O5 or A2 C 0. But by the previous lemma (A x A) U O5 is a congruence of S, hence it is an atom in Con(S). Moreover it is clear that (A x A) U 05 = $ ( e , / ) . •

We are now able to give the promised characterization of the monolith of a subdirectly irreducible combinatoria! inverse semigroup.

T H E O R E M 2.9. Let S be a subdirectly irreducible combinatoria! inverse semigroup and let fi be its monolith. Then fi = ^ ( e , / ) where e , / are idempotents of S satisfying the hypotheses of Theorem 2.3.

Proof. Since any nontrivial congruence on S must collapse two distinct idempotents, we must have ^ = $ ( e , / ) for some idempotents e , / with e > f. If $ £ ( e , / ) were not an atom in Corc(E), then we could find e', / ' E E such that e' < e , / ' < / and ^ ( c ' , / ' ) < 0 £ ( e , / ) . We claim that % ' , / ' ) < 0 ( e , / ) as well. Infact we have that certainly (e, / ) ^ $(e', / ' ) , since otherwise, by the congruence generation theorem, we would have e E Se'S or e E Sf'S. But since e > e ' , / ' this is not possible, via Lemma 1.5. So ^ ( e ' , / ' ) < $ ( e , / ) and $ ( e , / ) were not an atom. Therefore we may safely assume that i?£(e , / ) is an atom in Con(E), and hence that e y f.

Let now a E S and a a _ 1 = e. Note that, since Eaa-1 and Ea~la are isomorphic via a, we must have, from aa~1 y / , that a~laa~1a = a - 1 a >- a~lfa and a~l fa is an idempotent. Moreover we cannot have a - 1 fa < / , otherwise EaaT1 and Ea~la would not be isomorphic via a. Next, if a~1 fa ^ / , then we would have fa~x fa < f -< e. But it is easy to see, using an argument analogous to the one above, that # ( e , / ) £ i ? ( / , a _ 1 / a / ) -Therefore # ( e , / ) fi $ ( / , / a _ 1 a / ) — O5 and S would not be subdirectly irreducible. Hence a-1 fa = f. Suppose now that a~1a y g / / . • Then again we can see that "0(gf,g) < $ ( e , / ) and so fi•= $ ( e , / ) would not be an atom. Therefore the proof is fìnished. •

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Moreover we can actually prove that ali the simple combinatoria! inverse semigroups are of this kind. In fact we have :

T H E O R E M 2.10. Let S be a combinatorial inverse semigroup. Then S is a simple algebra if and only ifS = TgK for some E K .

Proof. Clearly, by Lemmas 2.5 and 2.7, any TEK 1S simple and

combinatoria!. Let now S be a simple combinatorial inverse semigroup and let E be its lattice of idempotents. Theorem 2.1 shows that S can be embedded in T # via the map a \—• a. Moreover, from the proof of Lemma 1.5 it is clear that , for ali e , / , # E E with e > f > g, we have tì(f,g) < $ ( / , e). Hence, by the simplicity of S, weget that no chain in E can have length greater than 2. Therefore E = EK for some K.

Let now (p E T ^ , say cp : Ee — • Ef. Clearly Ee = {e,g} and Ef = {/ ,#}, where g is the smallest element in E. Hence we must have <p(9) — 9 a n d <£>(e) = / • Now the simplicity of S forces #(e,(j) = ^(/,<7), so in particular (f,g) E ^ ( e , / ) . By the congruence generation theorem we must have that either / E SeS or / E SgS. But / > g, so by Lemma 1.5, we cannot have / E SgS. Therefore / E SeS, so there are x,y E S with / = xey. But then, taking a — xyy-1, we have / ^ flea-1, as the reader can easily check. Moreover, since g is the smallest element in E and aga~l E E, we must have g < aga~x ; but strict inequality is again forbidden by Lemma 1.5, so we have g — aga~1. Hence <p = a and the function a i—> a is also onto. Therefore S = T^; and the theorem is proved. •

Let us mention that a different characterization of congruence-free combinatorial inverse can be found in [13].

3 . T w o c o u n t e r e x a m p l e s

The one-block property (OBP) has been defìned in [2]. An algebra A has the OBP if any atom in Con(A) has exactly one nontrivial block. A class of algebras K. has the OBP if ali the algebras in /C have the OBP. In [2] or [5] it is shown that if a variety has the OBP, then it is congruence semimodular. Combinatorial inverse semigroups do not have the OBP; it is not hard to verify that, if E is the semilattice in Figure 3.1 below, then T # does not have the one block property, where the atom of T^; failing the OBP is d(e,f).

269

e h

Figure 3.1.

This counterexample is very naturai: in the sequel we will see that it prevents any variety of combinatoria! inverse semigroups, other then semilattices, from having the OBP.

Gn the other hand it is clear from the previous section that any subdirectly irreducible combinari al inverse semigroup has the OBP. It is only naturai to wonder whether the fact that SI(V) has the OBP might imply congruence semimodularity for the whole variety. Unfortunately this is not the case even for regular varieties of semigroups, as the following example shows1.

Let now S be the semigroup whose base set is {0,1 , a} and whose miiltiplication table is the one below.

0 1 a

0 0 0a 1 1 l a a a aa

Figure 3.2

It is a nontrivial exercise to check that any algebra in (S) is a strong semilattice of left-zero semigroups. We need a lemma here.

This counterexample has ben suggested to me by K. Kearnes.

270

LEMMA 3 .1 . IfV consists entirely of strong semilattices of semigroups from W, (where V and W are classes of semigroups), then S 6 SI(V) if and only if one of the following holds:

(i) s E s/(w). (2) S is the two element semilattice.

(3) S is the semigroup obtained by adding a zero element to a subdirectly

irreducible in W not having already a zero element.

Proof. First it is easily seen that ali semigroups in the given list are subdirectly irreducible. Conversely suppose that S E V. By the defmition of strong semilattice of semigroups there is a congruence 6 E Con(S) such that S/6 is a semilattice E and each congruence class Sa is a semigroup in W. Moreover there are homomorphisms <pa p : Sa —> Sp such that

(1) <P<xa = idSa

(2) If a < (5 < 7 in E then <paìp<ppn = (pa,T

Let now Iia be equal to S^ with a zero element 0a adjoined, if tv is not the minimal element for E, and equal to Sa otherwise. We will show that in fact that S is a subdirect product of the Ka, a £ E.

We defìne a function fa : S —• Sa in the following way; if s 6 Sp then

1 0a otherwise.

The reader can easily check that fa is indeed a homomorphism from S to.Ka. Moreover

f | ker(fa) = {(6,0 : for ali a e E , / * ( s ) - fa(t)}. Q-GE

If s E Sp and / 6 Sy we then have

s = VP A") = M°) = hi*) = {If otherwtl

If P i 7 we get s = Qp; but this would imply that 0^ E Sp and therefore that j3 = m m ( E ) , which yields a contradiction. Hence we must conclude that fi < 7. Similarly we see that 7 < fi and thus fi = 7. Therefore we have

271

that s = t and f l^eE ^ e r ( / « ) = ^S- This *s e n o u g h to conclude that S is a sub direct product of the Ka.

Now if S is subdirectly irreducible, then we must have S = Sa for some a. If a = ram(E), then S £ Sa G SI(W)\ if a ^ min(E), then we get the two element semilattice, in case Sa = {5}, or a semigroup in W with a zero element adjoined otherwise. It is fairly obvious that if a semigroup with zero is subdirectly irreducible, then the semigroup obtained by taking the zero away is stili subdirectly irreducible (the congruence lattice does not change). •

Therefore there are precisely three subdirectly irreducibles in (S), i.e. the 2-element left-zero semigroup, the 2-element semilattice and S itself. Note that they ali have the OBP, hence SI((S)) has the OBP. The fact that (S) is not CSM follows from a general result by P. Jones [10], but the reader can easily check that the direct product of the 2-element left zero semigroup and the 2-element lattice is in (S) and does not have a semi modular congruence lattice.

4 . Two consequences

It is clear that , if we denote by Xn the variety of combinatoria! inverse semigroups defined by xn+1 « xn

ì we have obviously Xn C J n + i . Moreover, from Lemma 1.3, we get at once that any variety V of. combinatori al inverse semigroups is contained in Xn for some n. This clearly implies that the the varieties of combinatoria! inverse semigroups form an ideal in the lattice of varieties of inverse semigroups.

The question we ask here is: are there varieties of combinatorial inverse semigroups (besides the variety of semilattices) having the OBP? We will give a negative answer to that question, and, in the process, we will get some information on £(Z), the ideal of varieties contained in J . First we note that the counterexample in the previous section satishes z 3 = x2. Therefore Xn

does not have the OBP for n > 2, while clearly X\ — the variety of semilattices does. We have:

T H E O REM 4 .1 . Let V be a variety of combinatoria! inverse semigroups. IfV contains a simple algebra diflerent from the 2-element semilattice, then V does not have the OBP. In factV contains the counterexample Tj? of the previous section.

272

Proof. Suppose that S E V, S is simple and different from Z = ({0,1},-) the 2-element semilattice. Since S is simple S =. TgK the complete Munn semigroup over the semilattice EK, for some « > 2. It is easily seen that TgK contains ali T#A for A < K, as subalgebras. In particular it contains T £ 2 . TE2 has five elements and it is not hard to see that the lattice of idempotents of B = TE2 X Z is the one in Figure 4.1 below and that $(i,j) is an atom in Cor?(B), whose only nontrivial block is {i, j}.Checking that A = B/i?( i , j ) is isomorphic to T ^ is left to the reader. So the theorem is proved. •

Figure 4.1

LEMMA 4.2. Fj2(a:), the free monogenerated algebra in I2, is simple. Therefore any monogenerated combinatoria! inverse semigroup satisfying x3 « x2 is either trivial or it is isomorphic to Fy 0 (#) . In particular, if A £ J2 and a E A, the subalgebra generated by a in A is isomorphic to Fj2(x) if and only if a2 ^ a.

Proof. We only need to prove the first claim, ali the others being easy consequences of it. First let us observe that in F = F J 0 ( . T ) we must have x2 = x~2. In fact certainly x2 is an idempotent and so we get

Jb I"V ( * ' ) - ' * ( * * ) - 1 _ i _ i _ o

Jb Jb >**» Jb

From this and the fact that x3 « x2 it is easy to infer that x~n « xm « x2

for any m,n > 2 and hence that the base set of F is {x,x~1ìx

2ìxx~1,x~1x}.

The simple verifìcation that F =.-T^0 is left to the reader. •

273

LEMMA 4.3. The variety ( T ^ 3 ) is the unique upper cover of X\ in C(Z), the ideal ofvarieties of combinatoria] inverse semigroups. Consequently there are no varieties of combinatoria! inverse semigroups having the OBP, besides the variety of semilattices.

Proof. Suppose that V is a variety of combinatori al inverse semigroups. Then, by Lemma 1.3, V C ln for some n, i.e. xn « xn+ì holds in V. Let now F = Fy(ar) and 9 = tì(x2, x3). Then it is well known that F ' = F/d(x2,x3) is the relatively free monogenerated algebra in some subvariety >V of V, where x2 « x3 holds. But then, by the previous lemma, F ; = T # 2 , since it cannot be trivial. Therefore any variety of combinatoria! inverse semigroups contains T^Etì hence it cannot have the OBP, via Theorem 4.1. The fact that (T£ 2 ) is the only upper cover of Vi in C(l) is now obvious. •

We now turn our attention to another fact. It is clear that

( T J J J - I I C ^ C ^ J O . . .

On the other hand there are identities, sudi as x~1yx~1 tó' y~ìxy~1ì that

hold in any T ^ but fall to hold in X2. Hence \/K(TEK) < Ti- Somewhat surprisingly an even stronger statement holds:

T H E O R E M 4.4. For any cardinal n, T^2K G H\TE2)K. Hence ali the

T-EK generate the same variety.

Proof. Let us label the elements of T =. T_£2 according to the multiplication table in Figure 3.4 below.

f e 9 a a" 1

7 f f f f / e

- • / •

e f a /

9 / f 9 f a - 1

a / f a f e a'1

/ a" 1 f 9 f Figure 4.2

We defìne a binary relation 0 on TK by setting (a?i , . . . , xK) 9 (y\,..., yK) if and only if they are equal or there are i,j < K such that X{ = / = yj. That 9 is an equivalence relation is obvious. It also a congruence, since / is an

274

absorbing element in T. Note that TK/0 has exactly 2K -f 1 idempotents and that the only nontrivial block of 9 is ( / , . . . , f)/9. Moreover this block is the unique minimal idempotent in the semilattice of idempotents of TK/9. The easy verification that indeed TK/9 = T^2K is left to the reader. In Figure 4.4 below one can see this argument for the simplest case k = 2. The boxed idempotents are the one to be identiiied in 9. •

As a final remark we note that in J2 there are infinitely many nonisomorphic simple algebras that generate the same variety. Moreover, via Theorem 2.10, we conclude that I2 contains ali simple combinatorial inverse semigroups.

(e, e) o (e, 2) (g,e) s(9,g)

becomes

Figure 4.3

REFERENCES

[1] P. AGLIANÒ, Algebras whose congruence lattices are semimodular, PhD Thesis, University of Hawaii (1988).

[2] P. AGLIANÒ, The one-block property in varieties of semigroups, Semigroup Forum 42 (1991), 253-264.

[3] P. AGLIANÒ, Congruence semimodularity and identities, Algebra Universalis 27 (1990), 600-601.

[4] P. AGLIANÒ, K. KEARNES, Congruence semimodular varieties I: locally fìnite varieties, to appear in Algebra Universalis.

[5] P. AGLIANÒ, K. KEARNES, Congruence semimodular varieties II: regular varieties, to appear in Algebra Universalis.

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R. FREESE, J. B. NATION, Congruence lattices ofsemilattices, Pacific J. Math. 49 (1973), 51-58.

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Paolo AGLIANO

Dipartimento di Matematica, Università di Siena

53100 Siena, Italy.

Conferenza tenuta il 10.3.1992.