An eventually regular semigroup is a semigroup in which some power of any element is regular. The minimum group congruence on an eventually regular semigroup is investigated by means of weak inverse. Furthermore, some properties of the minimum group congruence on an eventually regular semigroup are characterized.

1. Introduction

Throughout this paper, we follow the notation and conventions of Howie [1].

Recall that a semigroup is said to be eventually regular if each of its elements which has some power is regular. From the definition we conclude that eventually regular semigroups generalize both regular and finite semigroups. Edwards [2] was successful in showing that many results for regular semigroups can be obtained for eventually regular semigroups. The strategy to study eventually regular semigroups is to generalize known results for regular semigroups to eventually regular semigroups. Group congruences on regular semigroups have been investigated by many algebraists. Latorre [3] explored group congruences on regular semigroups extensively and gave the representation of group congruences on regular semigroups. Hanumantha [4] generalized the results in [3] for regular semigroups to eventually regular semigroups. Moreover, group congruences on E-inversive semigroups were studied in [5, 6].

In this paper, the author explores the minimum group congruences on eventually regular semigroups by means of weak inverses. A new representation of the minimum group congruence on an eventually regular semigroup is given. Furthermore, group congruences on eventually regular semigroups are described in the same technique.

2. Preliminaries

Let S be a semigroup and a∈S. As usual, ES is the set of all idempotents of S, 〈ES〉 is the subsemigroup of S generated by ES and N the positive integers. An element x of S is called a weak inverse of a if xax=x. We denote by W(a) the set of all weak inverses of a in S.

Let ρ be a congruence on a semigroup S. Then ρ is called group congruence if the quotient S/ρ is a group. In particular, a congruence ρ is said to be the minimum group congruence if S/ρ is the maximum group morphic image of S. For a congruence ρ of S, the subset {a∈S∣aρ∈E(S/ρ)} of S is called the kernel of ρ denoted by kerρ.

Let S be a semigroup and H a subset of S. Then the subset Hω is called closure of H if Hω={x∈S∣∃h∈H,hx∈H}. In this case, H is said to be closed if Hω=H. Moreover, a subset H of S is called full if ES⊆H. A subsemigroup K of an eventually regular semigroup S is called weak self-conjugate if for any a∈S, a′∈W(a), there exist a′Ka⊆K, aKa′⊆K. For a subset H of S, we define a binary relation named σH on H as
σH={(a,b)∈S×S:∃b′∈W(b),ab′∈H}.

We give some lemmas which will be used in the sequel.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B2">2</xref>, <xref ref-type="bibr" rid="B3">7</xref>]).

Let S be an eventually regular semigroup and ρ a congruence on S. If aρ is an idempotent of S/ρ, then an idempotent e can be found in S such that aρe.

Remark 2.2.

Since S is an eventually regular semigroup and ρ is a group congruence on S, xρ is an idempotent of S/ρ for all x∈〈ES〉.

Lemma 2.3.

Let S be a regular semigroup with a unique idempotent, then S is a group.

Lemma 2.4 (see [<xref ref-type="bibr" rid="B7">5</xref>, <xref ref-type="bibr" rid="B8">6</xref>]).

Let S be an eventually regular semigroup. Then W(a)≠∅ and aa′,a′a∈ES for all a∈S, a′∈W(a).

Lemma 2.5.

Let H be a subsemigroup of an eventually regular semigroup S and ab∈H for a,b∈S. If H is weak self-conjugate, closed, and full, then axb∈H for x∈〈ES〉.

Proof.

Suppose that there exist a,b∈S such that ab∈H and x∈〈ES〉. Since H is full and weak self-conjugate, we obtain b′a′axb∈H, abb′a′∈H for a′∈W(a), b′∈W(b). It follows from ab∈H that (ab)b′a′axb∈H. Since H is closed, we claim axb∈H.

3. Main Results

We begin the section with the main result of this paper.

Theorem 3.1.

Let S be an eventually regular semigroup and H=〈ES〉ω. Then the following statements are true.

If H is a weak self-conjugate, closed subsemigroup, then σH is the minimum group congruence on S and kerσH=H.

If the relation σ is a group congruence on S and kerσ=H, then σ is the minimum group congruence on S and H is weak self-conjugate, closed, and full subsemigroup with σ=σkerσ.

The following lemma plays an important role in the proof of Theorem 3.1.

Lemma 3.2.

Let S be an eventually regular semigroup and a,b∈S. If the subsemigroup H of S is weak self-conjugate, closed, and full, then the following statements are equivalent:

aσHb;

ab′∈H, ba′∈H for a′∈W(a), b′∈W(b);

b′a∈H for b′∈W(b).

Proof.

(1)⇒(2) Suppose aσHb for a,b∈S, then there exists a′′∈W(a) such that ab′′∈H, and so ab′′bb′∈H for b′∈W(b). For any a′∈W(a), a′a∈ES, it follows from Lemma 2.4 that
ab′′b(a′a)b′=ab′′ba′(ab′)∈H.
Since H is weak self-conjugate, closed, and full, we deduce ab′′ba′∈H, so that ab′∈H. In a similar way, we prove ba′∈H for a′∈W(a).

(2)⇒(3) Using the statement (2), we conclude that there exists b′∈W(b) such that ab′∈H. Since H is weak self-conjugate, we obtain a′ab′a∈H and a′a∈ES⊆H, so that b′a∈H.

(3)⇒(1) For a,b∈S, there exists b′∈W(b) such that b′a∈H. From the weak self-conjugate of H, we deduce bb′ab′∈H and bb′∈H. And since H is closed, we have ab′∈H, which leads to aσHb.

We now give the proof of Theorem 3.1.

Proof of Theorem <xref ref-type="statement" rid="thm3.1">3.1</xref>.

(1) To show that σH is an equivalence, let H=〈ES〉ω be a weak self-conjugate, closed subsemigroup. It is obvious that H is full and 〈ES〉⊆H. For a∈S, there exists a′∈W(a) such that aa′∈ES⊆H, so that aσHa, and so σH is reflexive. To prove the symmetry, suppose aσHb for a,b∈S, then there exists b′∈W(b),a′∈W(a) such that ab′∈H. And since H is weak self-conjugate, full, we obtain ab′ba′∈H, so that bσHa, and so σH is symmetry. To prove the transitivity, let aσHb, bσHc for a,b,c∈S. Then there exist b′∈W(b), c′∈W(c) such that ab′∈H, bc′∈H, hence ab′bc′∈H. And there exists a′∈W(a) such that a′a∈ES, and it follows from Lemma 2.4 that ab′b(a′a)c′=(ab′ba′)ac′∈H. Since H is weak self-conjugate and full, we deduce ab′ba′∈H, ac′∈H, and so aσHc, which says that σH is transitivity. Therefore σH is an equivalence, as required.

We now prove that σH is a congruence. Suppose aσHb for a,b,c∈S. Then there exists (cb)′∈W(cb), and so b(cb)′∈W(c), (cb)′c∈W(b). Put c′=b(cb)′, b′=(cb)′c. Then b′c′∈W(cb), (cb)′=b′c′. It follows from Lemma 3.2 that ab′′∈H for b′′∈W(b), and so b′=(cb)′c∈W(b), so that ab′∈H. Since H is weak self-conjugate and (ca)(cb)′=cab′c′, we conclude ca(cb)′=cab′c′∈H, so that caσHcb. Therefore σH is left compatible. On the other hand, a similar argument will show that σH satisfies right compatible. Thus σH is a congruence on S.

We now turn to show σH is a group congruence on S. For any e,f∈ES, there exists f∈W(f)⊆〈ES〉 such that ef∈〈ES〉⊆H, so that eσHf. It follows from Lemma 2.1 that S/σH has a uniue idempotent. For any a∈S, there exists m∈N such that am is regular element. Furthermore, there exists (am)′∈W(am) such that
am(am)′∈ES,am(am)′a(am)′=am(am)′∈ES,
and so a(am)′∈〈ES〉ω=H, which leads to aσHam. Therefore, we conclude that S/σH is a regular semigroup. It follows from Lemma 2.3 that S/σH is a group, so that σH is a group congruence on S.

We then show that σH is the minimum group congruence on S. Let aσHb for a,b∈S, and let ρ be any group congruence on S with eρ as the unique idempotent of S/ρ. It follows from Lemma 3.2 that there exists b′∈W(b) such that ab′∈H, and so there exists t∈〈ES〉 such that tab′∈〈ES〉. Notice that
(tab′)ρ=eρ=(aρ)b′ρ,(aa′)ρ=eρ=(aρ)a′ρ,
for a′∈W(a), so that b′ρ and a′ρ are the group inverse of aρ. In view of the uniqueness of group inverses, we have a′ρ=b′ρ. Since a′ρ is the group inverse of aρ and b′ρ is the group inverse of bρ, we claim aρ=bρ, which leads to σH⊆ρ. Thus σH is the minimum group congruence on S.

We finally prove kerσH=H. For any a∈kerσH, it follows from Lemma 2.1 that there exists e∈ES such that aσHe. We, by Lemma 3.2, deduce that there exists e′∈W(e) such that e′a∈H, ee′∈ES⊆H. Since H is closed, we have e′∈H, a∈H, and so kerσH⊆H. To show kerσH⊇H, let a∈H. Since there exists t∈〈ES〉, (ta)′∈W(ta) such that
ta∈〈ES〉,(ta)σH⊆E(SσH),(ta)(ta)′∈ES,
and so a(ta)′∈H, so that aσH(ta). Therefore aσH⊆E(S/σH), and so a∈kerσH. Thus kerσH=H, as required.

(2) Let σ be a group congruence on S and eσ the identity of S/σ. Suppose aσb for a,b∈S, then there exist a′∈W(a), b′∈W(b) such that a′σ is the group inverse of aσ and b′σ is the group inverse of bσ. By the uniqueness of group inverses, we have a′σb′ and aa′σbb′σba′σe, so that ba′∈kerσ=H, and so there exists t∈〈ES〉 such that tba′∈〈ES〉. Suppose that ρ is any group congruence on S, then
(tba′)ρ=(ba′)ρ=(bρ)a′ρ=eρ,
and so bρ is the group inverse of a′ρ. On the other hand, aρ is the group inverse of a′ρ. By the uniqueness of group inverses, we have aρb, so that σ⊆ρ. Therefore σ is the minimum group congruence on S.

We now prove H is weak self-conjugate, closed, and full. It is obvious that kerσ=H is a full subsemigroup. For any a∈S,a′∈W(a),x∈kerσ, then
(axa′)σ=(aσ)eσ(a′σ)=(aa′)σ=eσ,
which leads to axa′∈kerσ. A similar argument shows that a′xa∈kerσ. Therefore H is weak self-conjugate. For x∈Hω=(kerσ)ω, then there exists t∈kerσ such that tx∈kerσ. Hence
eσ=(tx)σ=tσ(xσ)=eσ(xσ)=xσ,
and so x∈kerσ, so that (kerσ)ω⊆kerσ. On the other hand, it is obvious that (kerσ)ω⊇kerσ. Thus (kerσ)ω=kerσ, and so H is weak self-conjugate, closed, and full subsemigroup of S, as required.

We finally prove σ=σkerσ. To show σ⊆σkerσ, let aσb for a,b∈S. Then there exists b′∈W(b) such that ab′σbb′σe, and so ab′∈kerσ, aσkerσb, which yields to σ⊆σkerσ. We now turn to proving that the converse holds. Let aσkerσb for a,b∈S. Then there exists b′∈W(b) such that ab′∈kerσ=H, and so there exists t∈〈ES〉 such that tab′∈〈ES〉. Put ρ is any group congruence on S. Notice
(tab′)ρ=(tρ)aρ(b′ρ)=eρ,
so that bρ and aρ are the group inverse of b′ρ. By the uniqueness of group inverses, we claim that aρb. Since σ is the minimum group congruence on S, σ is the intersection of all group congruence on S. Hence aσb, so that σ⊇σkerσ, and so σ=σkerσ. The proof is then completed.

As a specialization of Theorem 3.1, the following corollary is immediate.

Corollary 3.3.

Let S be an eventually regular semigroup. Then the following statements are true.

If H is a weak self-conjugate, closed, and full subsemigroup, then σH is a group congruence on S and kerσH=H.

If the relation σ is a group congruence on S, then kerσ is a weak self-conjugate, closed, and full subsemigroup with σ=σkerσ.

Acknowledgment

This paper was partially supported by the National Natural Science Foundation of China (nos. 60873144 and 10971086).

HowieJ. M.EdwardsP. M.Eventually regular semigroupsLatorreD. R.Group congruences on regular semigroupsHanumanthaR. S.Group congruences on eventually regular semigroupsWeipoltshammerB.Certain Congruences on E-inversive E-semigroupsZhengH. W.Group congruences on an E-inverse semigroupEdwardsP. M.Congruences and Green's relations on eventually regular semigroups