Math. Nachr. 128 (1984) 169-176

Perfect Complotoly Regular Semigroups

By HOWARD HAMILTON of Sacramento

(Received Juny 28, 1983)

Introduction

Let S be a semigroup, and L(S) the lattice of congruence relations on S. If A and B are subsets of S then A B = {ab I a E A , b E B} . For u E L(S) and a E S let au denote the a-class containing a. If u E L(S) then u is said to be perfect if for all a, b E S we have aubu = (ab) u. S is perfect if every congruence on S is perfect.

The concept of perfect congruence relations was introduced by VAQNER [9]. Some examples of perfect algebras are groups, quaaigroups, BooLean algebras, and CANTOR'S algebres (FORTUXATOV [5 ] ) . Thus perfect semigroups are a generalization of groups. FORTUNATOV [3], [4] has shown that completely (0-) simple semigroups are perfect, and has determined the commutative perfect semigroups, perfect bands, perfect inverse unions of groups, and perfect orthogroups. A further study of finite inverse perfect semigroups waa made by the present author and TAMURA [6].

In section I we discuss some properties that perfect semigroups have in general. Then in section 2 we characterize the perfect completely regular (unions of groups) semigroups. In section 3 we re-interpret the results of section 2 in terms of a structure theorem for completely regular semigroups due to CLIFFORD and PETRICH [l], and use this to obtain the known perfect semigroups as corollaries. Section 4 gives a charac- terization of perfect finite semigroups.

All undefined terms may be found in [2] or [7].

1. Preliminary Facts

Lemma 1. A homomorphic image of a perfect semigroup is perfect.

Lemma 2. Every non-zero deal of a perfect semigroup is completely prime ideal. Proof. Let I be a non-zero ideal of a perfect semigroup S. Let PI be the Rees con-

gruence module I . If a, b E S \ I then n o I . beI = {ab} + I . Therefore, ab c l I. Thus I is a completely prime ideal.

Lemma 3. A semilattice is perfect if and only if it is a chain.

Proof . The congruence relations on a chain are determined by the partitions of the chain into intervals. Thus chains are easily seen to be perfect.

To see the other direction suppose r is a (lower) semilattice and that I' is not a chain. Then there exist u, b E I' with u incomparable to B. Let A = d. A is an ideal of I' and PA, the REES congruence module A , is not perfect as u t$ Bur = . upA 5 upA.

170 Math. Nachr. 128 (1985)

Throughout this paper S = u S, will denote the greatest semilattice decomposition a E r

of S where r is a lower semilattice.

Lemma 4. Let S = u So be perfect. Z f a is not the least element of r then S, is simple. a c r

Z f a .is the least element of r then S, ti simple or 0-simple with zero diztiors.

S,ZS,

Therefore S is not perfect if 8, is not simple. If a is the least element of r, then we consider two cases. The first case is where S,

has no zero. In this case if I is a proper ideal of S, then S,ZS, is a proper non-completely prime non-zero ideal of S. Thus in this case 8, must be simple. The other case is where S o has a zero and zero-divisors (otherwise S, would not be semilattice indecomposable). Suppose (0) =+ I & S, and 1 is an ideal of S then by lemma 2 Z is a completely prime ideal of S. And therefore Z is a Completely prime i d a l of S,, but S, is semilattice in- decomposible, so I = s,. Hence S, is a o-minimal ideal of s. Therefore, * = (0) or So is o-simple (Theorem 2.29 121). If S: = (0) then S, is null and (unless S, is trivial) the REES congruence modulo a null ideal is clearly not perfect. Thus S, is o-simple.

Lemma 6. Let S = u So be perfect. If /3 > y and b E Sp then bS, = S, = Syb.

Proof. Let /I > y and let I = u S,. I is an idcal of S. Let u be the intersection of

the REEs congruence modulo I and the smallest semilattice congruence on 8. Then for b E S, and c E S, we have bu = {b) and cu = S, . Thus, since u is perfect, bS, = bun cu = (bc) u = S,. Similarly, S,b = S,.

Proof . First assume a is not minimal. Suppose I is a proper ideal of S, then. u S , is an noncompletely prime non-zero ideal of S. Z 5 8, and (S,ZS,) u

L a )

a c f

aBv

Lemma 6. (FOBTUNATOV [4]) A completely (0- ) simple semigrozbps i s perfect. Let S = u S,. For /3 < y and b S, we will denote the right (left) translation of S,

induced by b as e;,,,(;,.,A). That is @!,, = cb(~,,Ac = bc) for all c E s,. Tf for all /I < y and b E Sp the translations e!,, and !,oA are permutations of S, then we say S is per- mutation composed. Furthermore, if for all /? > y the sets Rp,, = {pi,, : b E Sp} and LB,, = (;,.,A : b E AS,} are groups then we say S is permulalion group composed.

nu-

Lemma 7. If S = u S, i s a perfect wmpk#ely regular semigroup or a finite perfect #cr

semigroup then 8 ti permulation group composed.

Proof. Ifit /3 > y and b E Sp By lemma 6 e!,, and !,,A are surjective maps from S, into S,.

If S is finite then S, is finite and so the surjective maps e!,, and !,?A are 1 - 1 , also. Furthermore, since R,, and L,., are finite semigroups of permutations, they are groups.

If S is completely regular then for all a E r S o is a completely simple semigroup ([2] theorem 4.6). Let Pp,? be the homomorphism of 8, into Rp,, given by P,,,(b) = e!.,. As the honiomorphic image of a completely simple semigroup is completely simple we have Rb,, = PB,,(S,q) is completely simple. Furthermore, if e2 = e E S , then e;,., is a surjective idenipotent map of S, into S, hence e$,.y is the identity map. Thus Rp,y is a cornplctely simple semigroup with an identity element. Hence Rp,, is a group and

Hamilton, Perfect Completely Regular Semigoups 171

t,hus the members of Rp,y are all permutations. Similarly, the left translations $,yA are all permutations. Thus S is permutation group composed.

For S = IJ S, let t denote the smallest semilattice congruence on S , i.e. t is the - a e r

congruence determined by the partition ( S , : u 6 r]. Lemma 8. If S = u S., r a chain, is permutation group wmped and S, is perfect

a c r for all a E F then ecery congruence catained in t is perfect.

Proof . Let u E L(S) and u 5 t. Let b E SB and c E S, with 5: y. Also, let

Let p = y then bu . ca = baa. cog = (bc) a, = (bc) Q where the next to the last

If > y then since L,., is a group, there exists b’ E S , such that ;::,A = ;.yl.-l. -

= b(cu,) E bas - coy = ba . cb E (bc) u. Therefore, bu - cu = (bc) u. Similarly, CQ . bu = (cb) u. Thus t is perfect.

0, = u I S, for u E r. equality holds because S , is perfect and the other equalities hold because u E t.

Thus (bc) (I = (bc) aY = !,.,A * ;:,J.[(bc) a,] = bb’(bc) u,. & b(b’a1 * h,) E b[(b’&) a,]

The next lemma is of primary importance to what follows.

Lemma 9 [8]. Zf I is an ideal of a semigroup S and a is a group congruence on Z then a has u unique extension, Z to a group congruence on S. If e E I and eu is the identity of Z/a then s y for x, y E S i f and only i f exaey.

Definition. Let a E L(S). A subset A of a semigroup S is a section of a if for all X E S x u n A +0.

Definition. Let S = u S,. A congruence qa c L(S,) is said to have the r-section

If S = IJ S, then for each a E r the a-filter of S is Fa = u S, and S, is an ideal

a e r property if for all a E S, and j3 > OL as, is a section of 7,.

of Fa. a e r P a

Lemma 10. Zf S b perfect then Fa is perfect for each OL E r. Proof. Let u E L(F,) then u u w ~ - ~ , E L(S) and u u w ~ - ~ ~ perfect impliea that

a is perfect.

Lemma 11. If S = u S., is perfect them for each a E r every group congruence on S, has

u8 the r-section property. Furthermore, i f S, hag a smallest group congruence 7, then ertry group congruence m S, has the r-section property i f and only i f 7, has the r-section pro- perty.

> a , and let Za denote the unique extension of a, to a group congruence on u S,. Then u = Z, u iTa

a S v S 8 E L(T,). By lemma 10 u is perfect. By lemma 9 every element of S , is a-related to some element of S,. Let e E S. with eu. the identity of S+,, then by the definition of 5, for b E S , and a E S, we have aca, be Z,,b for some c E S,. By perfectness we must have m. cu = ba. Hence aa n S , =+= 0. Let ea, denote the identity of S,,,,. Then for a E 5, there exists b E S , with uZ.bz,eb. Thus for all a E S,, aa,eb E eS, so eS, is a section of a,. And, since a, is a group congruence, we have aSp is also a section of a, for each a E 8,.

a c r

Proof. Let u E r and let u, be a group congruence on S,. Let

172 Math. n'achr. 128 (1985)

Furthermore, if T, is the sniallcst group congruence on S, and a, is any group con- gruence on s., then aa 2 q.. Assume that 7, has the r-section property. Then for all /? > a, a E S,, a s , is a section of 7,. The fact that aS, is also a section of a, follows since each a,-class is a union of q,-classes.

2. Perfect Completely Regular Semigroups

Theorem 1. S = U 8, i s a perfect completely regular semigroup if and only i f : ,€I-

(1.1) r is a chain, (1.2) S, is a completely simple semigroup /M each a E r, (1.3) S is permutation group composed, (1.4) For each u E r every group congruence on S, has the r-section property,

(1.5) For each a E r the smallest group congruence an S, has the F-section property.

To prove this theorem we will determine the structure of each congruence on a semi- group satisfying (1.1) thru (1.4) and then show its perfectness. We do this in a sequence of lemmas.

For the remainder of this section we assume S satisfies (1.1) thru (1.4). Let u E L(S) and v = u v t. Let @ = S/v. Then, since v is a semilattice congruence on S, each U E 9 is of the form U = u 8, where A is an interval of r.

and (1.4) may be replaced by

,€A

Lemma 12. I / U = n 8, and 1111 > 1 then uU = u I U is a group congruence an U .

Proof . For a E A let F , denote the a-filter of U . Also, let 0" = Q 1 Fa and urn = c I 8,. Define, for a > /? and a, /? E A , / 4 , s : P,l@ + F , / d by /,,p(xa") = xu". Since a" E d , f a , , is a homomorphism. Clearly Ulur, 3 (F,/u4; /,,,). Thus to show U/," is a group it suffices to show that for each u E A , Fala" is a group.

Let a E A. Let I = {x E F , 1 say for some y E 8,) I is clearly an ideal of F , and since each S,, y E r, is simple we have I = Fa. Hence F,l@ S,/@. We may assume a is not maximal in A. Let /? > a and choose e2 = e E S , then by (1.3) eua n S, is an identity for $,/a,. Hence S,/aa is a completely simple seniigroup with an identity. It follows that SJu, is a group.

Lemma 13. Let U = U S, with IAl > 1 then /or each x E U and a E A xu n S, 4 0.

Proof . Let x E S , 2 U and u E A. Let y = up. Using notation as in the proof of lemma 12 we have uy is a group congruence on S,. Let e E S , with eu, the identity of Saloy. Then by (1.4) there exists a E S, with ea E eS, n (ex) u,. Thus aueauexm by lemma 12 and lemma 9.

4 E A

aEA

Lemma 14. u is perfect.

Proof . Let b, c E S with b 6 S , and c 6 S,. We may assume that /3 2 y. Let U = u S, = bv and V = u S, = cv. We consider cases:

# € A oEA

Case 1. U = V.

Hamilton, Pcrfect Completely Regular Semigroups 173

If lAl = 1 then U = S g = S, = V and ba ca = bag * cap = (bc) us = bca as uBg is perfect (lemma 6). Thus we assume IAl > 1. Then by lemma 12 au = ay is a group congruencc, and by lemma 13 each au-class meets each S,,, a 6 A. By lemma 8 (a n T) - U is perfect. Hence bas cu = U (ba n S,) - U (ca n S,) = U (ba n S,) (co n S,) = u [(bc) u n S,] = (bc) a. ,€A ,€A o € A

,€A

Cwe 2. If U =+ V then for all a E A and a' E A we have a > a'. In this case (bc) u 2 V . Let d E (bc) a. Say d E Sd. Then there exists c' E S, with c'ac by lemma 13 if Id1 > 1 and if Id1 = 1 we can choose c' = c. Using lemma 8 we have b = a I Sg u $6

is perfect. Hence bb - c'b = (bc') 6. Thus, since d E (bc') it 2 (bc) a, we have ba . cu 2 (bc) a. Therefore ba - cu = (bc) a and we have shown that u is perfect in all cases.

3. Structure Theorems for Completely Regular Semigroups

If S = U S, is a completely regular semigroup then S is said to satisfy D-couering

if whenever e2 = e, p = f and e E S,, f E Sp with a > ,9 then e > F (i.e. ef = fe = f ) . From lemma 7 we see that a perfect completely regular semigroup satisfies D-covering.

In [l] CLIFFORD and PETRICH gave the following structure theorem for completely regular semigroups satisfying D-covering.

First we introduce some notation. Since S. is completely simple for each a E I' we assume S, is represented as a REES matrix semigroup M(Q,; I,, A*, Pa) for each a E r with Pa = (piei) normalized at (la, 1,). We identify a E a, with (a; la, 1,) and, in particular, Sa with Q, if S, is a group. Also, e, will denote the identity of a,.

acr

Theorem 2 [l]. With each element a of a eemilattice r associate a Rees matrix semi- group s, = M(Q,,; I., A,, Pa) with Pa normalized at (la, la), and with s, n Sg = 0 i f a + B. Assume that if a + up =+ B, then S,, = On,+ For each a > in I', assume that Q, acts by permutatium on I B from the left and on A , from the right. With each pair a > B in r assmiale a mapping 0,,@: Q. --f Qp w h that the following condilions are sattkfied, for all a > /3 in (2.1)-(2.4) and all a > /3 > y in (2.5) and (2.6):

(2.1) ab) 0a,g = (a%@) P!o.bl(bfl..p) (all a, E Q a ) ;

(2.2) d,o1(a0a.@) p$.i = d,oi(a0a.s) P L , ~ (all a E an, i E I,, 1 E A,) ; (2.3) p&0,.p = ea (all i E I., 1 E A, ) ; (2.4) ppiij = j and

(2.6) (a0.,,) i = ai a d ?.(~0,,~) = l a (all a E Q,, i E I , , 1 E A,).

Define a product in S = u S, as follows. I f (a ; i, ?.) E S, and ( b ; j , p )

= p (all i E I,, j E Ig, 1 E A., p E A@); (2.5) 0 u . g 0 0 p . y = f l u . ? ;

Sg let aer

(a ; i, 2.1 (a; j , p ) = ((a@.,@) P!o,jb; aj, a) if a > B , if a < B , = (aP:.*l(b0g,,); i, q

= (c7P!,,k i, P ) i / a = p

= (a0a,.g) (b0g.ua) if a * . p * p .

174 Math. Nachr. 128 (1986)

Then S i s a completely regular sem@roup satisfying D-covering. Conversely, every S W ? ~

semigroup is isomorphic to one constructed in this way.

Lemma 15. A wmpktdy reguhir semigroup S = U 8, satisfief, D-covering if and only if it is permud&.bn group composed.

Proof . By lemma 5.1 of [l] each idempotent of S, acts a8 a two sided identity on Sp for OL > p. The proof of the lemma is thus almost a duplicate of the proof of lemma 7.

For each u E r if S, = M(Qa; I , , i l , ; Pa) then the greatest group homomorphic image of S, is QolNU where Nu is the smallest normal subgroup of containing { p i i } . Thus the smallest group congruence r ] , on S, is given by ( a ; i, A ) q,(b; j , p ) if and only if ab-1 E Nu. Also, let v, denote the congruence on (3, such that G.,fv, L% QafN..

a c r

Theorem 3. Let S = u S,, where S, = M((3,; I , , A.; Pa) for each a E r, be a completely

regular eemigroup satis/ying D-covering determined by (0,,p: (3. -+ a,, a > a s in theorem 2 then S is perfect i f and only i f r is a chain and (a&@: a E Qa} is a seclioii

a e r

of for all dl < /?. Proof. If S is perfect then for each a E r by lemma 11 q,, has the r-section pro-

perty. Let a > /? then (ea; la, la) S, isa section of vp Xow, (ep, la, 1s) S, = {(aOa,p, l a , lp) a E (3,) is a section of q p if and only if {a0,.p I a E (3,) is a section of YE.

Conversely, if r is a chain and {a0,,@: a E a,} is a section of va for each u > 6 then (ep, la, la) S, = { ( ~ 0 , , ~ ; 18, 1p) I a E a,} is a section of qa and thus (1.6) is satisfied. Also, by lemma 16 S satisfies (1.3), therefore S is perfect by theorem 1.

Let S be a semigroup of the type described in theorem 2.

Lemma 16. If m > p and a E (3, then (a0.$l = P&,,- ,~. ( ~ ~ 0 4 .

Proof. As a a - I = e,, by (2.1) and (2.3) we have ep = e,Oa,a = (a0..p) p$.o-ll x ( ~ - ~ 0 , , ~ ) therefore (a0,,$l = ~O,O-l l (a- lO. ,a) .

Lemma 17. I / a > p then (Nu) 0..a 5 N p . Proof. Let g E a, then

a (gP!,ig-') 0o.b = (gP!,i) 0a.5 * PlgpI,il-'l . (g-'0n.,9) by (2.1) B a a

a

= (slra.a) P l g . p ~ , i l ( P i i 0 a . a ) * P l g p i , + g - 1 1 * (g- ' f la ,B) by (2.1)

= ( f 7 0 u . p ) P l u p ~ , i * - ~ r ( s - ' O a . a ) by (2.3) and (2.4)

= ( s s a , s ) PlgP;#i.u-'l * P!;;-~l(@m,a) by lemma 16.

Therefore (gpi,g-I) 0,,p from (2.1) that

N p Thus 0,,a maps a generator of Nu into N , and it follows maps Nu into N P For each a > /I in rdef ine 8.,p: G J N , -+ QBfN,

by (aNa) %a,@ = (aflu,,) Ng-

Lemma 18. The maps

Proof . Let a, a' E (3, with aN, = a". then a' = an with n E N,. Thus U ' B , , ~ = (an) 0a.p = (aB.,p) ~ { ~ , ~ l ( n @ . . ~ ) and ~ f ~ , ~ ~ . ( n 0 , ~ ) E Np by lemma 17. Hence (a '0 ,~ )Np = (a@,,p) N,.

are well-defined.

Hamilton, Perfect Completely Regular Semigroups 175

Lemma 19. The maps g,,, are hmwnwrphisms.

Proof . IA aN., bN, E Q a I N , then (aN, - bN,) 0,.8 = (ab) 0,.p N , = (aOa,Bp~a,,,bO,,p)NB

Theorem 4. Let S be as in theorem 2 with r a chain. Then S is perfect if and only if 8-.,, is surjective fm all a > 8 in r.

Proof. By theorem 3 S is perfect if and only if (aB.,,: a E Q,) is a section of vB for each u > /? in T. That is, S is perfect if and only if for each a > /? in I' (a0,,,: a E a,) meets each coset of N , in a,, which is equivalent to the surjectiveness of ia,p Next we consider some special cases :

If S = U S, is an orthogroup satisfying D-covering then for each cx c r S ,

= Q. x I, x A, is a rectangular group. And the maps 0,,, for u > /? of theorem 2 are homomorphisms (theorem 6.1 [l]), and 0,,p = 0,,, for a > 8. Hence we have

Corollary 1. Let S be a n orthogroup satzkfying D-covering determined as in theorem 2 by (0.,,] then S is perfect i f and only i f T is a chain and 0,,, th surjective for a > 8 in I'.

Corollary 2. Let S = (J a, be a semilattice of g roup t?, d e t e r m i d by the transitive

= a0a.gNB * bBo.gp = (aNa) K., * (bNm) 3a.b

-

aEr system of h m w p h i s m (0,,, I bi > /?). Then S is perfed i f a d only i f I' zk a chain and 0,,8 is surjectke for all > a.

Corollary 3. A cornmuttdive semigroup G perfect if and only if it is a perfect semilattice

Corollary 4. A band ie perfect if and only if it G a n ordinal Bum of rectangular bands. Proof . Let S be a band then S = u S, where S, = {e,) x I,, x A, is a rectangular

band for each u E I'. In this setting D-covering if and only if x y = y x = y for all x E S,, y E S, with u > 8. Also, 0,,, is trivial and surject.ive because a, = lea) and a, = (e,). Thus S is perfect if and only if I' is a chain and x y = y x = y for all x E S,, y E SB with u > 8.

of groups.

aEr

4. Perfect Finite Semigroups

"heorem 6. Let S = u S, be a finite semigroup. S is perfect if and only i f a e r

(5.1) r is a finite chain with least element a,; (5.2) S,, is simple for cx > uo, and S,, -is simple or o-simple with zero divisors; (5.3) S is permutation group composed; (5.4) For each 01 E r every group congruence on S, has the T-section property. Proof. The necessity of conditions (5.1) thru (5.4) is a consequence of lemmas 3, 4,

7 and 11. To see sufficiency let S be a finite semigroup satisfying (5.1) thru (5.4). Since a

finite (o-) simple semigroup is completely (o-) simple, if S,. is simple then S is a com- pletely regular semigroup satisfying (1.1) thru (1.4) and is, therefore, perfect.

176 Math. Nachr. 128 (1985)

Assume that S, is o-simple with zero-divisors. Without going thru all of the details as in theorem 1, close inspection of t,hat proof will show that if we can show that a congruence a on S such that for u e SaO we have a(a v t) Q Sn0 has the property that

a I a(av t) = ua(aVr).

Let u E L(S) and suppose for some a E S,, that U = a(o v t) $ S,,. By an argument like that used in thc proof of leinma 12 we may show that for each x e U there exists y e S , such that xay and that Ul,, is o-simple with zero-divisors. Hence Ul,, is ‘ com- pletely o-simple with zero divisors. Let e2 = e E U - Sap. By (5.3) e acts as a two sided identity on S,, and since every element of U is a-related to some element of S,, we see that U/,, has a two sided identity element. Now, a completely o-simple semigroup has an identity if and only if it is a group with zero adjoined or trivial. But Ul,, has zero divisors or aC = ow. The only alternative is thus aw = mu,.

References

[l] A. H. CLIFFORD and M. PETRICH, Some clasees of completely regular semigroups, J. Algebra

[2] A. H. CLIFFORD and G. B. PRESTON, The Algebraic Theory of Semigroups, Vol. 1, Math.

[3] V. A. FORTUNATOV, Perfect semigroups decomposable in a semilattice of rectangular groups,

[4] -, Perfect Semigroups, Izv. Kyss. Ucebn. Zaved. Matem. 3 (1972) 80-90 (in Russian) [6] -, Varieties of perfect algebras. In “Studies in algebra”, Saratov Univ. Press 4 (1974) 110-114

[6] H. HAMILTON and T. TAMURA, Finito inverse perfect semigroups and their congruences, J.

[7] M. PETRICH, Introduction to semigroups, Merrill Publ. Co., Columbus, Ohio (1973) [8] T. TAMURA and H. B. HAMILTON, The study of commutative semigroups with greatest group-

[9] V. V. VAONER, Algebraic topics of the general theory of partial connections in fiber bundles,

48(1977)462-480

Surveys, No. 7, A.M.S., Providencc, R.1. (1961)

in “Studies in algebra”, Sarstov Univ. Press 2 (1970) 67-78 (in Russian)

(in Ruesian)

Aust. Math. SOC., Ser. A (to appear)

homomorphisms, Trans. Amer. Math. SOC. 178 (1972) 401-419

Ivz. Vyss. Ucebn. Zaved. Matem. 11 (1968) 26-32

California State University Dept. of Math. gacramento, C A 95819 U S A