Semigroup Forum Vol. 65 (2002) 386–404c© 2001 Springer-Verlag New York Inc.

DOI: 10.1007/s002330010110

RESEARCH ARTICLE

Semigroups which Have a Minimum PrimitiveInverse Congruence

Anthony Hayes∗

Communicated by John B. Fountain

Abstract

We investigate primitive inverse semigroups and Brandt semigroups as ana-logues of groups for semigroups with zero. We give a description of the minimumprimitive inverse congruence on a categorical E∗ -dense E -semigroup. We showthat a categorical semigroup S with a primitive inverse congruence has a min-imum such congruence if D̃(S) is ∗ -dense in S . Here D̃(S) is the least full,weakly self-conjugate, ∗ -unitary and ∗ -reflexive subsemigroup of S . Our resultsare analogous to those of Fountain, Pin and Weil for general semigroups.

Introduction

This paper is a contribution to the growing belief that, for semigroups with zero,the classes of Brandt semigroups and primitive inverse semigroups are naturalanalogues of the class of groups. Any semigroup in either class without zero-divisors is a group with an adjoined zero. We obtain a quite general condition toensure that a semigroup with zero has a minimum primitive inverse congruence,a description of which is also given.

Our results are analogous to and extend those of Fountain, Pin and Weil[5] and Fountain and Gomes [3]. In [5] conditions for a monoid to have aminimum group congruence are investigated and a description of the congruenceis provided. The particular case of an E -dense monoid is also considered. In[3] primitive inverse congruences are studied and it is shown that the kernelsof 0-restricted primitive inverse congruences on a categorical semigroup S areprecisely the ∗ -reflexive, ∗ -unitary and ∗ -dense subsemigroups of S .

Any semigroup has a group congruence, namely the universal congruence.However, it is shown in Preston [17] that for a semigroup S with zero to havea 0-restricted primitive inverse congruence, S must be categorical, and weknow from Munn [15] that S must be strongly categorical to have a Brandtcongruence. Following the definitions and preliminaries, in Section 2 we outlinethe characterisation of the kernels of 0-restricted primitive inverse congruencesdescribed in [3]. In Section 3 we investigate a class of semigroups with zero

∗The author would like to thank Professor J. B. Fountain and Dr. V. A. R. Gould formuch helpful discussion during the preparation of this paper.

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which have a minimum primitive inverse congruence, proving results analogousto those in [5]. We study E∗-dense semigroups in Section 4, and give a newcharacterisation of the minimum primitive inverse congruence on categorical E∗-dense E -semigroups. We also consider 0-restricted surjective morphisms ontoprimitive inverse semigroups. These results are analogues of those for E -densemonoids in [5].

1. Definitions and preliminaries

For standard semigroup terminology and notation we refer the reader to [10]. Inparticular, the set of idempotents of a semigroup S is denoted by E(S) or justE , and we say that a subsemigroup T of S is full if E(S) ⊆ T . We adopt theusual convention that a semigroup with zero must contain at least two elements.A 0-restricted congruence on a semigroup with zero is a congruence ρ such that0ρ = {0} . On occasion we shall use the term 0-subsemigroup to emphasise thatthe subsemigroup being considered contains 0.

In [3], the notions of dense, reflexive and unitary subsets of a semigroupare modified to give more useful ideas in the context of semigroups with zero.We now describe these modifications. Let S be a semigroup with zero and T asubset of S such that 0 ∈ T . The set of non-zero elements of T is denoted byT ∗ ; in particular, the set of non-zero idempotents of S is E∗(S) or simply E∗ .

The subset T is dense if for all a ∈ S there exist x, y ∈ S withax , ya ∈ T . Because T is trivially dense in S , we define T to be ∗-densein S if for every a in S∗ there exist x, y ∈ S such that ax , ya ∈ T ∗ .

We say that T is ∗-reflexive if ab ∈ T ∗ implies that ba ∈ T ∗ for alla, b ∈ S .

It is easy to see that if T is unitary in S , then T = S . Therefore T isdefined to be ∗-unitary if for all a ∈ S and t ∈ T , we have a ∈ T if at or tais in T ∗ .

Now, for a ∈ S , the sets XT (a) and YT (a) are defined by:

XT (a) = {b ∈ S | atb ∈ T ∗ for all t ∈ T 1 such that at �= 0 or tb �= 0},YT (a) = {b ∈ S | bta ∈ T ∗ for all t ∈ T 1 such that ta �= 0 or bt �= 0}.

Put W ∗T (a) = XT (a) ∩ YT (a).

An element s of a semigroup S with zero is nilpotent if sn = 0 for somepositive integer n . If T has no non-zero nilpotent elements and W ∗

T (a) �= ∅ forall a ∈ S∗ , then T is said to be strongly ∗-dense. Note that a strongly ∗ -densesubset T satisfies the condition that, for all a in S∗ , there is an element b suchthat ab, ba ∈ T ∗ ; in particular, T is ∗ -dense. However, ∗ -dense does not implystrongly ∗ -dense. For example, in the semigroup Z8 under multiplication, thesubsemigroup {0̄, 2̄, 4̄} is ∗ -dense but not strongly ∗ -dense as (4̄)2 = 0̄, whence4̄ is a non-zero nilpotent.

Next we note some connections between ∗ -reflexivity and reflexivity.

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Lemma 1.1. Let T be a 0-subsemigroup of a semigroup S with zero. If Tis reflexive and has no non-zero nilpotents, then T is ∗-reflexive.

Proof. If a, b ∈ S with ab ∈ T ∗ then certainly ab ∈ T and so ba ∈ Tsince T is reflexive. But abab ∈ T ∗ as T is a subsemigroup with no non-zeronilpotents, so that ba �= 0 and hence ba ∈ T ∗ , as required.

Lemma 1.2. Let T be a 0-subsemigroup of a semigroup S with zero. If Shas no non-zero nilpotents and T is ∗-reflexive, then T is reflexive.

Proof. Suppose that a, b ∈ S with ab ∈ T . If ab �= 0 then ab ∈ T ∗ andso ba ∈ T ∗ as T is ∗ -reflexive. On the other hand, if ab = 0, then ba �= 0would imply that baba �= 0 since S has no non-zero nilpotents. But this is acontradiction because ab = 0, whence ba = 0 ∈ T , and the lemma is proved.

Combining the previous two lemmas gives the following proposition.

Proposition 1.3. Let T be a 0-subsemigroup of a semigroup S with zeroand suppose that S has no non-zero nilpotents. Then T is ∗-reflexive if andonly if T is reflexive.

A semigroup S is categorical at zero or just categorical if it has a zero andfor all a, b, c ∈ S , if abc = 0 then ab = 0 or bc = 0. It is strongly categorical ifit has the additional property that the intersection of any two non-zero idealsis non-zero. We now quote the following result from [3].

Lemma 1.4. If T is a ∗-reflexive, ∗-dense subsemigroup of a categoricalsemigroup S , then T is strongly ∗-dense.

We say that an inverse semigroup S with zero is primitive if every non-zero idempotent e in S is primitive, that is, for all f ∈ E∗(S), if e ≤ f ,then e = f . Every Brandt semigroup, that is, a completely 0-simple inversesemigroup, is primitive and in fact, from [18, Corollary 2] or [16, Theorem II.4.3]we know that every primitive inverse semigroup is a 0-direct union of Brandtsemigroups. The following result is [1, Lemma 7.61].

Lemma 1.5. A primitive inverse semigroup is categorical.

We now quote [6, Lemma 1.7].

Lemma 1.6. Let S be a primitive inverse semigroup.

(i) If e, f ∈ E∗(S) , then ef �= 0 implies e = f .

(ii) If e ∈ E∗ and s ∈ S∗ then

es �= 0 implies es = s, se �= 0 implies se = s.

(iii) If a, s ∈ S∗ and as = a , then s = a−1a . Dually, if sa = a , thens = aa−1 .

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We define a 0-restricted congruence ρ on a semigroup S with zero to bea primitive inverse (Brandt) congruence if S/ρ is a primitive inverse (Brandt)semigroup. The kernel of a congruence ρ on a semigroup S is the subset Kerρgiven by

Kerρ = {a ∈ S | aρ ∈ E(S/ρ)} = {a ∈ S | a ρ a2}.Note that if ρ is a 0-restricted congruence and a ∈ (Kerρ)∗ , then a ρ a2 anda2 �= 0. It follows that Kerρ has no non-zero nilpotents.

2. Primitive inverse congruences

In this section we state some results from [3] which will be needed in the sequel.We say that a morphism θ: S → T of semigroups with zero is 0-restricted if thecongruence induced by θ is 0-restricted, that is, if and only if 0θ−1 = {0} . Thekernel of θ , denoted Kerθ , is defined to be the inverse image of the idempotentsof T , that is, Kerθ = (E(T ))θ−1 .

Throughout the section S will be a categorical semigroup. We now statesome results from [3, Section 3], which will be needed in the sequel. We beginby defining a congruence ρT on S , where T is a strongly ∗ -dense subsemigroupof S . For all a, b ∈ S ,

(a, b) ∈ ρT if and only if a = b = 0 or xa = bt �= 0 for some x, t ∈ T.

We now quote [3, Proposition 3.1].

Proposition 2.1. Let T be a strongly ∗-dense subsemigroup of S . Thenthe relation ρT is a primitive inverse congruence on S and T ⊆ KerρT .

Next we state [3, Proposition 3.2].

Proposition 2.2. Let ρ be a primitive inverse congruence on S . Then Kerρis a ∗-unitary, ∗-reflexive, ∗-dense subsemigroup of S and ρ = ρKerρ .

It is easy to show that a non-zero intersection of ∗ -reflexive, ∗ -unitarysubsemigroups of S is again ∗ -reflexive and ∗ -unitary.

We now have the main result of this section, [3, Theorem 3.4].

Theorem 2.3. The mappings T → ρT and ρ → Kerρ are mutually inverseorder isomorphisms between the set of ∗-unitary, ∗-reflexive, ∗-dense subsemi-groups of S and the set of all primitive inverse congruences on S .

We have indicated above how 0-restricted primitive inverse congruencescan be characterised by their kernels. The following result ([3, Theorem 3.8])gives equivalent conditions for such congruences to exist. In analogy with ringtheory, we define a semigroup S with zero to be semiprime if aSa �= {0} for alla ∈ S∗ .

Theorem 2.4. Let S be a semigroup with zero and let

U = {a ∈ S|a2 �= 0} ∪ {0}.

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Then the following statements are equivalent:

(1) S has a 0-restricted primitive inverse homomorphic image,

(2) S is categorical, semiprime, and for all a, b in S ,

a2 �= 0, b2 �= 0, ab �= 0 together imply ba �= 0,

(3) S is categorical and U is a ∗-dense subsemigroup,

(4) S is categorical and U is a ∗-unitary, ∗-reflexive, ∗-dense subsemigroup.

We conclude this section by investigating alternative characterisations ofρT . If T is a subset of S , define

τT = {(a, b) ∈ S × S|a = b = 0 or ac, bc ∈ T ∗ for some c ∈ S}.

We know from Lemma 1.4 that a ∗ -reflexive, ∗ -dense subsemigroup of acategorical semigroup is strongly ∗ -dense. Therefore the relation ρT is definedfor ∗ -dense and ∗ -reflexive subsemigroups, and we have the result below.

Lemma 2.5. If T is a ∗-dense and ∗-reflexive subsemigroup of S , thenτT ⊆ ρT .

Proof. Let a, b ∈ S∗ with a τT b . Then ac, bc ∈ T ∗ for some c ∈ S .Since T is ∗ -reflexive, ca ∈ T ∗ . Put x = bc and t = ca . It follows thatxa = bca = bt �= 0 by categoricity as bc, ca �= 0. Because x, t ∈ T , we havethat a ρT b , as required.

Now we have a converse of Lemma 2.5.

Lemma 2.6. If T is a strongly ∗-dense subsemigroup of S , then ρT ⊆ τT .

Proof. Suppose that a, b ∈ S∗ are such that a ρT b . Then xa = bt �= 0for some x, t ∈ T . By Proposition 2.1, ρT is a congruence, whence at ρT bt .Since bt �= 0 and ρT is 0-restricted, at �= 0. Let a′ ∈ W ∗

T (a). It follows thatata ′ ∈ T ∗ . Further, bta ′ = xaa ′ ∈ T as x, aa ′ ∈ T . Also by categoricity,bta ′ �= 0 because xa, aa ′ �= 0. Thus a(ta′), b(ta ′) ∈ T ∗ , and a τT b .

Finally, combining Lemmas 2.5 and 2.6 gives the following proposition(see Lemma 1.4).

Proposition 2.7. If T is a ∗-dense and ∗-reflexive subsemigroup of S , thenρT = τT .

3. Semigroups which have a minimum primitive inverse congruence

We give analogues for semigroups with zero of results in [5, Section 6]. The“least weakly ∗ -self-conjugate, ∗ -unitary and ∗ -reflexive” subsemigroup D̃∗(S)

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of a semigroup S is considered and it is shown that if S is categorical with aprimitive inverse congruence and D̃∗(S) is ∗ -dense in S , then S has a minimumprimitive inverse congruence.

We remark that categoricity alone is not a sufficient condition for theexistence of a primitive inverse congruence on a semigroup with zero. For anynull semigroup is trivially categorical at zero but not semiprime and so does notadmit a 0-restricted primitive inverse homomorphic image, by Theorem 2.4.

First, for a categorical semigroup S with a primitive inverse congruence,we define a subsemigroup K(S) of S by

K(S) = {k ∈ S | kθ ∈ E(B) for all 0-restricted surjective morphisms θ

from S onto a primitive inverse semigroup B}.

We begin by giving an analogue of [5, Lemma 6.1].

Lemma 3.1. Let S be a categorical semigroup with a primitive inverse con-gruence. Then K(S) is a ∗-unitary and ∗-reflexive subsemigroup of S . If Shas a minimum primitive inverse congruence σ , then K(S) = Kerσ .

Proof. By Proposition 2.2, K(S) is the non-zero intersection of a familyof ∗ -unitary and ∗ -reflexive subsemigroups. By the remark following the sameresult, K(S) is also ∗ -unitary and ∗ -reflexive. The second part of the statementis immediate.

Lemma 3.2. A categorical semigroup S with a primitive inverse congruencehas a minimum such congruence σ if and only if K(S) is ∗-dense in S .Moreover, σ = ρK(S) whenever σ exists.

Proof. Suppose that K(S) is ∗ -dense in S . Since K(S) is ∗ -reflexiveby Lemma 3.1 and S is categorical, it follows from Lemma 1.4 that K(S)is strongly ∗ -dense. Hence, by Proposition 2.1, there is a primitive inversecongruence ρK(S) on S determined by K(S). Furthermore, since K(S) is∗ -unitary, ∗ -reflexive and ∗ -dense, it follows from Theorem 2.3 that K(S) =KerρK(S) . If ρ is any primitive inverse congruence on S , then it follows fromthe definition of K(S) that K(S) ⊆ Kerρ , that is, KerρK(S) ⊆ Kerρ . ByProposition 2.2, ρ = ρKerρ . Consequently, K(S) = KerρK(S) ⊆ Kerρ givesρK(S) ⊆ ρKerρ = ρ . Thus ρK(S) is the minimum primitive inverse congruenceon S .

Conversely, suppose that S has a minimum primitive inverse congruence.Then K(S) is ∗ -dense in S by Lemma 3.1 and Proposition 2.2.

If a and b are elements of a semigroup with bab = b , then b is said tobe a weak inverse of a . We denote by W (a) the set of all weak inverses of a ,and W ∗(a) is the set of all non-zero weak inverses of a when a is non-zero. IfS is a semigroup with zero and T is a subsemigroup of S , then T is weaklyself-conjugate or closed under weak conjugation if for all t ∈ T , a ∈ S and

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a′ ∈ W (a) we have ata ′, a′ta ∈ T . It is clear that, in a semigroup with zero,the intersection of a family of weakly self-conjugate subsemigroups is weaklyself-conjugate and contains zero. We can therefore define D(S) to be the least(under inclusion) full, weakly self-conjugate subsemigroup of S . If S has afull, weakly self-conjugate, ∗ -unitary and ∗ -reflexive subsemigroup, then byD̃(S) we mean the least full subsemigroup of S which is weakly self-conjugate,∗ -unitary and ∗ -reflexive.

Following [3], we say that a subsemigroup T of a semigroup with zeroS is weakly ∗-self-conjugate if for all a ∈ S∗ and a′ ∈ W ∗(a) and t ∈ T 1 , ifat �= 0 or ta ′ �= 0, then ata ′ ∈ T ∗ , and if ta �= 0 or a′t �= 0, then a′ta ∈ T ∗ .Note that a weakly ∗ -self-conjugate subsemigroup T is necessarily full. For,0 ∈ T by definition, and if e ∈ E∗(S), then e ∈ W ∗(e) and e1 = e �= 0,whence e1e = ee = e ∈ T ∗ . In analogy with the above D∗(S) is taken tobe the least full, weakly ∗ -self-conjugate subsemigroup of S when S containsa weakly ∗ -self-conjugate subsemigroup. Also, if there exists a weakly ∗ -self-conjugate, ∗ -unitary and ∗ -reflexive subsemigroup of S , then D̃∗(S) is theleast weakly ∗ -self-conjugate, ∗ -unitary and ∗ -reflexive subsemigroup of S ,necessarily containing E(S).

Since it is clear that if T is weakly ∗ -self-conjugate, then it is weaklyself-conjugate, we have D(S) ⊆ D∗(S) and D̃(S) ⊆ D̃∗(S) whenever thesesubsemigroups exist.

We now give an analogue of [5, Lemma 6.3].

Lemma 3.3. Let φ: S1 → S2 be a 0-restricted semigroup morphism. ThenD(S1)φ is a subsemigroup of D(S2) . If D∗(S2) (respectively D̃(S2) , D̃∗(S2))exists, then D∗(S1) (respectively D̃(S1) , D̃

∗(S1)) exists and D∗(S1)φ (respec-tively D̃(S1)φ , D̃∗(S1)φ) is a subsemigroup of D∗(S2) (respectively D̃(S2) ,D̃∗(S2)).

Proof. We prove the result for D̃∗ ; the proofs for the other cases are similar.Let T = (D̃∗(S2))φ

−1 . Then T is immediately seen to be a weakly ∗ -self-conjugate, ∗ -unitary and ∗ -reflexive subsemigroup of S1 . Hence, by definition,D̃∗(S1) exists and is contained in T . Thus D̃∗(S1)φ ⊆ D̃∗(S2).

We have the following corollary which strengthens [3, Lemma 4.3].

Corollary 3.4. Let S be a semigroup with zero, P be a primitive inversesemigroup and let φ: S → P be a 0-restricted surjective morphism. ThenD̃∗(S) exists and D̃∗(S)φ ⊆ E(P ) . Moreover, D̃∗(S) ⊆ K(S) .

Proof. The first statement will follow immediately from Lemma 3.3 if weshow that D̃∗(P ) = E(P ). Let a ∈ P ∗ , a′ ∈ W ∗(a) and e ∈ (E(P ))1 besuch that ae �= 0. Then ae = a by Lemma 1.6(ii), and so aea ′ = aa ′ ∈ E∗(P ).Similar arguments give the other three conditions needed for E(P ) to be weakly∗ -self-conjugate. From the proof of Proposition 2.2 in [3] we know that E(P ) is∗ -unitary and ∗ -reflexive, and hence by definition D̃∗(P ) = E(P ), as required.Thus D̃∗(S) is contained in (E(P ))φ−1 for each 0-restricted morphism φ from

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S onto a primitive inverse semigroup. By definition of K(S), this implies thatD̃∗(S) ⊆ K(S).

Observe that, in particular, if S is a semigroup with a primitive inversecongruence, then D∗(S), D̃(S) and D̃∗(S) all exist.

The next proposition is an analogue of part of [5, Proposition 6.9].

Proposition 3.5. Let S be a categorical semigroup with a primitive inversecongruence in which D̃∗(S) is ∗-dense. Then ρD̃∗(S) is the minimum primitive

inverse congruence on S and D̃∗(S) = K(S) . If D̃(S) is ∗-dense, then ρD̃(S)

is the minimum primitive inverse congruence on S and D̃(S) = D̃∗(S) = K(S) .

Proof. By Corollary 3.4, D̃∗(S) ⊆ K(S). It follows that K(S) is ∗ -denseand hence, by Lemma 3.2, ρK(S) is the minimal primitive inverse congruence of

S . Now Lemma 1.4 shows that D̃∗(S) is strongly ∗ -dense, and hence ρD̃∗(S)

is a primitive inverse congruence by Proposition 2.1. But D̃∗(S) ⊆ K(S)implies ρD̃∗(S) ⊆ ρK(S) , so that ρD̃∗(S) = ρK(S) . Then Theorem 2.3 shows

that D̃∗(S) = K(S).

If D̃(S) is ∗ -dense, then as D̃(S) ⊆ D̃∗(S) ⊆ K(S), exactly the aboveargument gives the required result.

We now have the following theorem, an analogue of part of [5, Theo-rem 6.10].

Theorem 3.6. For a categorical semigroup S with a primitive inverse con-gruence, the following conditions are equivalent:

(1) S has a minimum primitive inverse congruence and D̃∗(S) = K(S) ,

(2) there is a primitive inverse semigroup B and a 0-restricted surjectivemorphism φ: S → B with (E(B))φ−1 = D̃∗(S) ,

(3) D̃∗(S) is ∗-dense in S .

Proof. Conditions (1) and (3) are equivalent by Proposition 3.5 andLemma 3.2. If (2) holds, then D̃∗(S) is ∗ -dense in S by Proposition 2.2,and so (3) holds. Finally, if (1) holds, then ρK(S) is a 0-restricted primitive in-verse congruence and so θ: S → S/ρK(S) is a 0-restricted surjective morphism,where θ is the natural morphism. We have that

KerρK(S) = (E(S/ρK(S)))θ−1 = K(S) = D̃∗(S),

establishing (2).

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Corollary 3.7. For a categorical semigroup S with a primitive inverse con-gruence, the following conditions are equivalent:

(1) S has a minimum primitive inverse congruence and D̃(S) = D̃∗(S) =K(S) ,

(2) there is a primitive inverse semigroup B and a 0-restricted surjectivemorphism φ: S → B with (E(B))φ−1 = D̃(S) ,

(3) D̃(S) is ∗-dense in S .

4. E∗ -dense semigroups

A semigroup S with zero is E∗ -dense if the set E(S) of idempotents of Sis ∗ -dense in S . Trivially an E∗ -dense semigroup is E -dense and E∗ -densesemigroups are the analogue for semigroups with zero of E -dense semigroups.In [1] E -dense semigroups are called E -inversive, while E∗ -dense semigroupsare referred to as 0-inversive in [11] and weakly regular in [13].

It is shown in [8] that every E -dense semigroup has a minimum groupcongruence, an explicit description of which is given in [14, Proposition 9].However, as remarked earlier, the work of Munn and Preston shows that for asemigroup to have a (Brandt) primitive inverse congruence it must be (strongly)categorical. Preston (Munn) shows that (strong) categoricity is sufficient for aninverse semigroup to admit such a congruence and a similar result for an E∗ -dense semigroup in which the idempotents commute is proved below. However,we showed at the start of Section 3 that categoricity alone is not a sufficient con-dition for the existence of a primitive inverse congruence on a general semigroupwith zero. As we shall see, this is also the case for E∗ -dense semigroups.

Let T be a semigroup with zero in which the idempotents commute.From [11, Theorem 3] we know that T is primitive inverse if and only if it isE∗ -dense and satisfies the weak cancellation law:

if a, b, x, y ∈ T, then ax = bx �= 0 and ya = yb �= 0 together imply a = b.

Suppose that I is the set of all 0-restricted primitive inverse congruences onan E∗ -dense semigroup S , and put σ =

⋂{τ |τ ∈ I} . It follows from the above

result that σ is the minimum 0-restricted primitive inverse congruence on Swhenever I is non-empty.

By an E -semigroup we mean a semigroup in which the idempotentsform a subsemigroup. Following [3], we define an E∗ -dense semigroup S tobe strongly E∗ -dense if D(S) is a strongly ∗ -dense subsemigroup of S . It isshown in [3] that if S is an E -dense E -semigroup, then E(S) = D(S). Hence,for an E∗ -dense E -semigroup S , if S is strongly E∗ -dense, then E(S) = D(S)is strongly ∗ -dense. Conversely, if E(S) is strongly ∗ -dense, then D(S) = E(S)is strongly ∗ -dense, and S is strongly E∗ -dense.

In this section we describe the minimum primitive inverse congruence ona categorical E∗ -dense E -semigroup. We also consider 0-restricted surjective

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morphisms onto a primitive inverse semigroup and give, among other things, anequivalent condition for a categorical E∗ -dense semigroup S with a primitiveinverse congruence and where D(S) has no non-zero nilpotents to admit a 0-restricted surjective morphism onto a primitive inverse semigroup B with D(S)the inverse image of E(B). We begin by quoting [3, Theorem 4.9], which is ananalogue of Proposition 9 of [14].

Theorem 4.1. Let S be a categorical E∗ -dense semigroup. Then S hasa 0-restricted primitive inverse congruence if and only if D(S) is strongly ∗-dense. Moreover, if D(S) is strongly ∗-dense, then the congruence ρD(S) is theminimum 0-restricted primitive inverse congruence on S .

We can now recover the following result of Gomes and Howie [6] with anadditional statement from our work in Section 3. A semigroup S with zero isE∗ -reflexive if E(S) is ∗ -reflexive in S .

Theorem 4.2. Let S be an E∗ -dense, E∗ -reflexive categorical E -semi-group. Then the relation ρE(S) is the minimum 0-restricted primitive inversecongruence on S . Further, ρE(S) = ρD̃(S) = ρD(S) .

Proof. We have that E(S) is strongly ∗ -dense, by Lemma 1.4. Then byProposition 2.1, ρE(S) is a primitive inverse congruence. Since E(S) ⊆ D̃(S) it

follows that D̃(S) is ∗ -dense, whence ρD̃(S) is the minimum primitive inverse

congruence on S , by Proposition 3.5. But clearly E(S) ⊆ D̃(S) implies thatρE(S) ⊆ ρD̃(S) . Thus ρE(S) = ρD̃(S) . That ρE(S) = ρD(S) follows from

Theorem 4.1, but we can also note that E(S) = D(S), by [3, Proposition 4.7].

Next we quote the following result from [4], summarising some elementaryproperties of E∗ -dense semigroups.

Proposition 4.3. Let S be a semigroup with zero and let E = E(S) . Thenthe following conditions are equivalent:

(1) S is E∗ -dense,

(2) for all s in S∗, there exists s′ in S∗ such that ss ′, s′s ∈ E∗,

(3) for all s in S∗, there exists b in S∗ such that bs ∈ E∗,

(4) for all s in S∗, there exists b in S∗ such that sb ∈ E∗,

(5) every element of S∗ has a non-zero weak inverse.

We now have the lemma below.

Lemma 4.4. A weakly ∗-self-conjugate subsemigroup of an E∗ -dense semi-group has no non-zero nilpotents.

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Proof. Let S be an E∗ -dense semigroup and T a weakly ∗ -self-conjugatesubsemigroup of S . Suppose that a ∈ T ∗ . Since S is E∗ -dense there existsa′ ∈ W ∗(a). Then a′a �= 0 so that a′aa ∈ T ∗ by weak ∗ -self-conjugacy. Hencea2 �= 0, as required.

It is immediate from the definitions and Lemma 4.4 that a weakly ∗ -self-conjugate subsemigroup of an E∗ -dense semigroup is strongly ∗ -dense. Inparticular, we have the following corollary.

Corollary 4.5. For any E∗ -dense semigroup S , if the subsemigroup D∗(S)exists, then D∗(S) is strongly ∗-dense.

Specialising Theorem 4.1 to the case where S has a primitive inversecongruence, we obtain the following result.

Proposition 4.6. The minimum primitive inverse congruence on a categor-ical E∗ -dense semigroup S with a primitive inverse congruence is the relationρD∗(S) . Furthermore, ρD∗(S) = ρD(S) .

Proof. By Corollary 4.5 we have that D∗(S) is strongly ∗ -dense, and henceρD∗(S) is a primitive inverse congruence by Proposition 2.1. Now D̃∗(S) is ∗ -dense in S (as E ⊆ D̃∗(S)) so that ρD̃∗(S) is the minimum primitive inverse

congruence, by Proposition 3.5. But D∗(S) ⊆ D̃∗(S) so that ρD∗(S) ⊆ ρD̃∗(S) ,whence ρD∗(S) is the minimum primitive inverse congruence. The second partof the statement follows immediately from Theorem 4.1.

It is also possible to recover the result below of Fountain and Gomes [3]with an additional statement, which strengthens Theorem 4.2.

Theorem 4.7. Let S be a categorical E∗ -dense E -semigroup. Then theminimum 0-restricted primitive inverse congruence on S is given by ρE(S) .Further, ρE(S) = ρD(S) = ρD∗(S) .

Proof. We show first that E(S) is weakly ∗ -self-conjugate. Let a ∈ S∗ ,a′ ∈ W ∗(a) and e ∈ E(S)1 be such that ae �= 0. Then

(aea ′)2 = aea ′aea ′ = aea ′aea ′aa ′ = a(ea ′a)2a′ = aea ′aa ′= aea ′,

as S is an E -semigroup. Hence aea ′ ∈ E(S). Suppose that aea ′ = 0. Ife = 1, then aa ′ = 0, a contradiction. Therefore e �= 1, and by categoricityae = 0 or ea ′ = 0. But ae �= 0 by hypothesis, whence ea ′ = 0. It follows thatea ′a = 0, and so a′ae = 0, for otherwise, a′aea ′ae = a′ae �= 0, a contradictionas ea ′a = 0. Since a′a, ae �= 0, we have that a′ae �= 0 by categoricity. Henceaea ′ �= 0, and aea ′ ∈ E∗(S), as required. Similar arguments give the otherthree conditions necessary for weak ∗ -self-conjugacy.

Now, by the remark following Lemma 4.4, E(S) is strongly ∗ -dense.Hence ρE(S) is a primitive inverse congruence on S , by Proposition 2.1. Suppose

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that S/τ is a primitive inverse semigroup, where τ is a 0-restricted congruenceon S . Let a, b ∈ S∗ be such that a ρE(S) b . Then ea = bf �= 0 for somee, f ∈ E(S). Hence (ea)τ = (bf )τ , that is, (eτ)(aτ) = (bτ)(fτ) �= 0. Thereforeaτ = bτ , by Lemma 1.6(ii). Thus ρE(S) ⊆ τ , and ρE(S) is the minimumprimitive inverse congruence on S .

For the second part of the statement, since E(S) is weakly ∗ -self-conjugate it follows by definition that E(S) = D(S) = D∗(S). We can alsoappeal to Proposition 4.6.

We will often use the characterisation of E -semigroups below, which wasproved by Fountain and Hayes [4] and independently by Weipoltshammer [19].

Lemma 4.8. Let S be an E -dense semigroup. Then the following conditionsare equivalent:

(1) S is an E -semigroup,

(2) W (b)W (a) = W (ab) for all a, b ∈ S .

Now we note the following properties of E -semigroups. For (1) see [19],(2) follows from [2, Lemma 1] and was proved independently by Fountain andHayes [4].

Lemma 4.9. Let S be an E -semigroup. Then:

(1) for all a ∈ S , a′ ∈ W (a) , e, f ∈ E(S) , we have ea ′, a′f and ea ′f ∈ W (a) ,

(2) W (e) ⊆ E(S) for all e ∈ E(S) .

We also need the following result from [9]. A band with zero E is a∗-rectangular band if efe = e whenever ef �= 0, for all e, f ∈ E .

Lemma 4.10. For any categorical E∗ -dense semigroup, the following areequivalent:

(1) E(S) is a ∗-rectangular band,

(2) for all a, b ∈ S ,

W (a) ∩W (b) �= {0} implies W (a) = W (b).

The theorem below provides a description of the minimum primitiveinverse congruence on categorical E∗ -dense E -semigroups.

Theorem 4.11. Let S be a categorical E∗ -dense E -semigroup. Then theminimum 0-restricted primitive inverse congruence on S is given by

a σ b if and only if a = b = 0 or W (a) ∩W (b) �= {0}.

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Proof. Let a ∈ S∗ . By Proposition 4.3(5) there exists a′ ∈ W ∗(a), and σis reflexive. It is clear that σ is symmetric.

Suppose that a σ b and b σ c , where a, b, c ∈ S∗ . Then there existx ∈ W ∗(a) ∩ W ∗(b) and y ∈ W ∗(b) ∩ W ∗(c). Put z = xby . Then z �= 0as xb, by �= 0, and since by , xb ∈ E∗(S) it follows by Lemma 4.9(1) thatz ∈ W ∗(a) ∩W ∗(c). Therefore σ is transitive.

Now let a, b, c ∈ S , with a σ b . We require to show that ca σ cband ac σ bc . This is clearly the case if a = b = 0. Otherwise, thereexists x ∈ W ∗(a) ∩ W ∗(b). Suppose that ac = 0. Then, by Lemma 4.8,W (ac) = W (c)W (a) = {0} . If bc �= 0, there exists (bc)′ ∈ W ∗(bc). But(bc)′ = c′b′ for some c′ ∈ W ∗(c) and b′ ∈ W ∗(b) by Lemma 4.8(2). Observethat b′b, bx �= 0, whence by categoricity b′bx �= 0. As b′b ∈ E∗(S) we haveb′bx ∈ W ∗(a) ∩ W ∗(b) by Lemma 4.9(1). Further by categoricity, c′b′bx �= 0,and so c′b′bx ∈ (W (c)W (a))∗ ∩ (W (c)W (b))∗ = W ∗(ac) ∩ W ∗(bc). Becauseac = 0 this is a contradiction. Hence bc = 0 and ac σ bc .

Assume that ac �= 0. Then bc �= 0 by the above, and there exists(ac)′ ∈ W ∗(ac). Now (ac)′ = c′a′ for some c′ ∈ W ∗(c) and a′ ∈ W ∗(a). Wecan follow the above to show that c′a′ax ∈ W ∗(ac)∩W ∗(bc), whence ac σ bc .Thus σ is right compatible. Similarly ca σ cb , and σ is a congruence on S .

To see that S/σ is regular, let a ∈ S∗ and a′ ∈ W ∗(a). Then

a′(aa ′a)a′ = (a′aa ′)aa ′ = a′aa ′ = a′ �= 0,

so that a′ ∈ W ∗(a) ∩W ∗(aa ′a). Therefore a σ aa ′a .To show that S/σ is inverse, it suffices to prove that the idempotents of

S/σ commute. Let a ∈ S∗ be such that a σ a2 . Then there exists x ∈ W ∗(a)∩W ∗(a2). Hence xaax = x , and (xa)a(xa) = xa �= 0, so that xa ∈ W ∗(a). Sincexa ∈ E∗(S) it follows that xa ∈ W ∗(xa). Thus W (a) ∩ W (xa) �= {0} , anda σ xa .

Now, if aσ, bσ are idempotents of S/σ , we have just seen that there areidempotents e, f of S such that a σ e and b σ f . Suppose that ef = 0. Iffe �= 0, then as S is an E -semigroup we have fefe = fe �= 0, a contradictionas ef = 0. Hence fe = 0, and ab σ ef = 0 = fe σ ba . Assume that ef �= 0.Then ef ∈ W ∗(e) ∩W ∗(f), whence e σ f . Therefore a σ b , and E(S/σ) isa commutative subsemigroup of S/σ . Thus S/σ is inverse.

In order to complete the proof that S/σ is a primitive inverse semi-group, we have to show that every non-zero idempotent is primitive. Letaσ, bσ ∈ E∗(S/σ) be such that (aσ)(bσ) = aσ . There are idempotents e, f inS∗ such that aσ = eσ and bσ = fσ , and so (ef )σ = (eσ)(fσ) = eσ �= 0. Hencethere exists x ∈ W ∗(ef ) ∩ W ∗(e). Thus xefx = x , and (xe)f(xe) = xe �= 0,whence xe ∈ W ∗(f). Also xe ∈ W ∗(e), so that e σ f and bσ = fσ is aprimitive idempotent. Therefore S/σ is primitive. Also σ is 0-restricted sinceW (0) = {0} .

Finally, let ρ be any 0-restricted congruence on S such that S/ρ isa primitive inverse semigroup. Let a, b ∈ S∗ be such that a σ b . Then

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there exists x ∈ W ∗(a) ∩ W ∗(b), and clearly xρ ∈ W ∗(aρ) ∩ W ∗(bρ). Hence(aρ)(xρ) ∈ E∗(S/ρ), so that by Lemma 1.6(ii), (aρ)(xρ)(aρ) = (aρ). Thus aρis an inverse of xρ , and dually bρ is an inverse of xρ . By uniqueness of inversesit follows that aρ = bρ . Thus σ ⊆ ρ , and the proof is complete.

Remark. On an E∗ -dense semigroup which is not an E -semigroup, the rela-tion given by

a ρ b if and only if a = b = 0 or W (a) ∩W (b) �= {0}

is not in general even an equivalence. For example, the semigroup of the firstremark following [19, Theorem 3.1] with an adjoined zero is E∗ -dense, but ρ isnot transitive.

In the case where the idempotents commute we can obtain the followingresult which strengthens [6, Theorem 2.3].

Corollary 4.12. Let S be a categorical E∗ -dense semigroup with E(S)commutative. Then the minimum 0-restricted primitive inverse congruence onS is given by

a βE(S) b if and only if a = b = 0 or ea = eb �= 0 for some e ∈ E∗(S).

Proof. By Theorem 4.11, it suffices to show that σ = βE(S) . Let a, b ∈ S∗ besuch that a σ b . Then W (a)∩W (b) �= {0} , and there exists x ∈ W ∗(a)∩W ∗(b).Now xa, xb ∈ E∗(S), whence by assumption, (xa)(xb) = (xb)(xa), that is,(xax )b = (xbx )a , and xb = xa �= 0. If x′ ∈ W ∗(x), then by categoricity(x′x)a = (x′x)b �= 0. Hence aβE(S)b as x′x ∈ E∗(S).

Conversely, suppose that aβE(S)b . Then there exists e ∈ E∗(S) suchthat ea = eb �= 0. Let (ea)′ ∈ W ∗(ea). Now by Lemma 4.8(2), (ea)′ = a′e′

for some a′ ∈ W ∗(a) and e′ ∈ W ∗(e). By Lemma 4.9(2), e′ ∈ E∗(S), whenceby Lemma 4.9(1), a′e′ ∈ W ∗(a). Therefore we also have (ea)′ ∈ W ∗(b). ThusW (a) ∩W (b) �= {0} , and a σ b .

The one-sided nature of βE(S) is only apparent. The above results makethis clear, since σ and ρE(S) are left/right symmetric. Therefore, if S is acategorical E∗ -dense semigroup with E(S) commutative and a, b ∈ S∗ , then

a βE(S) b if and only if there is a non-zero idempotent f such that af =bf �=0.

Hall [7] shows that the minimum inverse congruence γ on an orthodoxsemigroup S is given by

γ = {(a, b) ∈ S × S|V (a) = V (b)}.

It follows from [7, Theorem 2] that

γ = {(a, b) ∈ S × S|V (a) ∩ V (b) �= ∅}.

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If we take S to be a categorical E∗ -dense E -semigroup, and in the aboveequation replace V (a) and V (b) by the corresponding sets W (a) and W (b)and ∅ by {0} , also defining {0} to be a congruence class, then by Theorem 4.11we obtain the minimum 0-restricted primitive inverse congruence σ on S .However, the relations

σ = {(a, b) ∈ S × S|a = b = 0 or W (a) ∩W (b) �= {0}}

and

{(a, b) ∈ S × S|W (a) = W (b)}

only coincide in the following case.

Corollary 4.13. Let S be a categorical E∗ -dense semigroup with a primitiveinverse congruence. Then the minimum 0-restricted primitive inverse congru-ence on S is given by

σ = {(a, b) ∈ S × S|W (a) = W (b)}

if and only if E(S) is a ∗-rectangular band.

Proof. Necessity. Let e, f ∈ E(S) be such that ef �= 0. Then (eσ)(fσ) =(ef )σ �= {0} , whence by Lemma 1.6(i), eσ = fσ . Therefore, by assumption,W (e) = W (f). Since e ∈ W (e) = W (f) it follows that efe = e , and E(S) is a∗ -rectangular band.Sufficiency. Let a, b ∈ S∗ and suppose that a σ b . Then W (a) ∩W (b) �= {0}by Theorem 4.11, so that, by Lemma 4.10, W (a) = W (b). Conversely, ifW (a) = W (b), then W (a) ∩ W (b) = W (a) �= {0} , and by Theorem 4.11,a σ b .

Next we make the following observation.

Lemma 4.14. A full, ∗-unitary subset of an E∗ -dense semigroup containsall the weak inverses of its elements.

Proof. Let S be an E∗ -dense semigroup and T a full, ∗ -unitary subset ofS . Suppose that a ∈ T and a′ ∈ W (a). If a′ = 0 then a′ ∈ T . Suppose thata′ �= 0. Then a �= 0 and a′a �= 0. Since T is full and a′a ∈ E∗(S) we havea, a′a ∈ T ∗ so that a′ ∈ T ∗ as T is ∗ -unitary.

Corollary 4.15. A weakly ∗-self-conjugate and ∗-unitary subsemigroup ofan E∗ -dense semigroup contains all the weak inverses of its elements.

Proof. We have seen that a weakly ∗ -self-conjugate subsemigroup is full.The result now follows from Lemma 4.14.

We now quote [3, Proposition 4.12].

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Proposition 4.16. Let S be a categorical E∗ -dense semigroup and let Tbe a weakly ∗-self-conjugate subsemigroup of S . If T is ∗-unitary, then T is∗-reflexive.

When the semigroup S is categorical and E∗ -dense, the propositionapplies to D∗(S) when it exists and we define S to be D∗ -unitary if D∗(S)is ∗ -unitary in S . The corollary below is an immediate consequence of theproposition and the definitions of D∗(S) and D̃∗(S).

Corollary 4.17. If S is a categorical E∗ -dense, D∗ -unitary semigroup,then D̃∗(S) exists and D∗(S) = D̃∗(S) .

Specialising Theorem 3.6 to the case where S is E∗ -dense, we obtain thefollowing theorem.

Theorem 4.18. For a categorical E∗ -dense semigroup S with a primitiveinverse congruence, the following conditions are equivalent:

(1) there is a primitive inverse semigroup B and a 0-restricted surjectivemorphism φ: S → B with (E(B))φ−1 = D∗(S) ,

(2) S is D∗ -unitary.

Proof. If (1) holds, then D∗(S) is ∗ -unitary by Proposition 2.2, and so (2)holds. Conversely, if (2) holds, then by Corollary 4.17 D∗(S) = D̃∗(S), and so(1) holds by Theorem 3.6.

Next we note the following lemma.

Lemma 4.19. Let S be a categorical E∗ -dense semigroup and T a subsemi-group of S . Then T is weakly ∗-self-conjugate if and only if T is full and weaklyself-conjugate with no non-zero nilpotents.

Proof. We have already seen from Lemma 4.4 and the remarks precedingLemma 3.3 that a weakly ∗ -self-conjugate subsemigroup is full and weakly self-conjugate with no non-zero nilpotents.

Conversely, suppose that T is full and weakly self-conjugate with no non-zero nilpotents. Let a ∈ S∗, a′ ∈ W ∗(a) and t ∈ T 1 be such that at �= 0. Ift = 1 then ata ′ = aa ′ ∈ T ∗ as T is full. If t �= 1, then certainly ata ′ ∈ T as Tis weakly self-conjugate. Since T is full we have that a′a ∈ T , whence a′at ∈ Tas T is a subsemigroup. Further, a′a �= 0 and at �= 0, so that a′at �= 0 bycategoricity. Since T has no non-zero nilpotents we have a′ata ′at �= 0, whenceata ′ �= 0. Thus ata ′ ∈ T ∗ as required. Similar arguments give the other threeconditions for T to be weakly ∗ -self-conjugate.

The proposition below is obtained by combining Proposition 4.16 andLemma 4.19.

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Proposition 4.20. Let S be a categorical E∗ -dense semigroup and let T bea full, weakly self-conjugate subsemigroup with no non-zero nilpotents. If T is∗-unitary, then T is ∗-reflexive.

By analogy with the above we say that S is D -unitary if D(S) is ∗ -unitary in S . We again have an immediate corollary.

Corollary 4.21. If S is a categorical E∗ -dense, D -unitary semigroup andD(S) has no non-zero nilpotents, then D̃(S) exists and D(S) = D̃(S) .

Specialising Corollary 3.7 to the case where D(S) has no non-zero nilpo-tents we obtain the following theorem which is an analogue of part of [5, Theo-rem 7.13].

Theorem 4.22. For a categorical E∗ -dense semigroup S with a primitiveinverse congruence and D(S) having no non-zero nilpotents, the following con-ditions are equivalent:

(1) there is a primitive inverse semigroup B and a 0-restricted surjectivemorphism φ: S → B with (E(B))φ−1 = D(S) ,

(2) S is D -unitary.

Proof. If (1) holds, then D(S) is ∗ -unitary by Proposition 2.2, and so (2)holds. Conversely, if (2) holds, then by Corollary 4.21 D(S) = D̃(S), and so(1) holds by Corollary 3.7.

We next note the following. A semigroup S with zero is E∗ -unitary ifE(S) is ∗ -unitary in S .

Lemma 4.23. If S is an E∗ -dense, E∗ -unitary semigroup, then E(S) is aweakly self-conjugate subsemigroup of S , and E(S) = D(S) = D̃(S) .

Proof. We show first that E(S) is a subsemigroup. Let e, f ∈ E(S). Ifef = 0 then ef ∈ E(S). Suppose that ef �= 0. By ∗ -denseness there existsan element x ∈ S such that (ef )x ∈ E∗(S). Since E(S) is ∗ -unitary, we havefx ∈ E∗(S), so x ∈ E∗(S), whence ef ∈ E∗(S).

That E(S) = D(S) now follows from [3, Proposition 4.7], but for com-pleteness we give the short proof. Let a ∈ S , a′ ∈ W (a) and e ∈ E(S). Thena′aa ′ = a′ and a′a ∈ E(S) so that

(aea ′)2 = aea ′aea ′ = aea ′aea ′aa ′ = a(ea ′a)2a′ = a(ea ′a)a′ = aea ′.

Hence aea ′ ∈ E(S). Similarly, a′ea ∈ E(S) and so E(S) is weakly self-conjugate. Therefore E(S) = D(S) by definition of D(S). Now, as S isE∗ -unitary, it follows by [6, Lemma 1.3] that S is E∗ -reflexive. Thus E(S) =D(S) = D̃(S) as required.

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Specialising to E -semigroups we obtain the following analogue of part of[5, Corollary 7.15]. Note that each condition forces E(S) to be a subsemigroupof S .

Corollary 4.24. For a categorical semigroup S with a primitive inversecongruence, the following conditions are equivalent:

(1) there is a primitive inverse semigroup B and a 0-restricted surjectivemorphism φ: S → B with (E(B))φ−1 = E(S) ,

(2) S is E∗ -dense and E∗ -unitary.

Proof. If (1) holds, then by Proposition 2.2, E(S) is ∗ -dense and ∗ -unitary.Conversely, if (2) holds, then D(S) = E(S) by Lemma 4.23, so clearly D(S)has no non-zero nilpotents, and hence (1) holds by Theorem 4.22.

Remarks. (1) In the finite case, we can obtain two structure theorems bysimple modifications to the proofs of Theorems 4.18 and 4.22. For E -semi-groups we have the finite version of Corollary 4.24.

(2) If, at the start of Section 3, we replaced “primitive inverse” by“Brandt” in the definition of K(S), then of course there are analogues of allsubsequent results, with S strongly categorical and “primitive inverse” replacedby “Brandt.”

References

[1] Clifford, A. H. and G. B. Preston, “The Algebraic Theory of Semigroups”,Vols. I and II, American Math. Soc., Providence, RI, 1961 and 1967.

[2] Fitzgerald, J. G., On inverses of products of idempotents in regular semi-groups, J. Austral. Math. Soc. 13 (1972), 335–337.

[3] Fountain, J. B. and G. M. S. Gomes, Primitive inverse congruences oncategorical semigroups, Proc. Edinburgh Math. Soc. 43 (2000), 395–413.

[4] Fountain, J. B. and A. D. Hayes, E∗ -dense semigroups whose idempotentsform a subsemigroup, preprint.

[5] Fountain, J. B., J.-E. Pin, and P. Weil, Covers for monoids, submitted.

[6] Gomes, G. M. S. and J. M. Howie, Semigroups with zero whose idempotentsform a subsemigroup, Proc. Royal Soc. Edinburgh A 128 (1998), 265–281.

[7] Hall, T. E., On regular semigroups whose idempotents form a subsemigroup,Bull. Austral. Math. Soc. 1 (1969), 195–208.

[8] Hall, T. E. and W. D. Munn, The hypercore of a semigroup, Proc. Edin-burgh Math. Soc. 28 (1985), 107–112.

[9] Hayes, A. D., E∗ -dense E -semigroups, in preparation.

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[10] Howie, J. M., “Fundamentals of Semigroup Theory”, Oxford UniversityPress, 1995.

[11] Lallement, G. and M. Petrich, Some results concerning completely 0-simplesemigroups, Bull. Amer. Math. Soc. 70 (1964), 777–778.

[12] LaTorre, D. R., Group congruences on regular semigroups, Semigroup Fo-rum 24 (1982), 327–340.

[13] McAlister, D. B., Matrix representations of semigroups, Glasgow Math. J.8 (1967), 1–13.

[14] Mitsch, H., Subdirect products of E -inversive semigroups, J. Austral. Math.Soc. 48 (1990), 66–78.

[15] Munn, W. D., Brandt congruences on inverse semigroups, Proc. LondonMath. Soc. 14(3) (1964), 154–164.

[16] Petrich, M.,“Inverse Semigroups”, Wiley, New York, 1984.

[17] Preston, G. B., Congruences on Brandt semigroups, Math. Ann. 139(1959), 91–94.

[18] Preston, G. B., Matrix representations of inverse semigroups, J. Austral.Math. Soc. 9 (1969), 29–61.

[19] Weipoltshammer, B., E -inversive E -semigroups, submitted.

Department of MathematicsUniversity of YorkHeslingtonYork YO10 5DD, U.K.tonydhayes@hotmail.com

Received July 18, 2001and in final form October 9, 2001Online publication December 28, 2001