#### Abstract

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 -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 be a semigroup and . As usual, is the set of all idempotents of , is the subsemigroup of generated by and the positive integers. An element of is called a *weak inverse* of if . We denote by the set of all weak inverses of in .

Let be a congruence on a semigroup . Then is called *group congruence* if the quotient is a group. In particular, a congruence is said to be *the minimum group congruence* if is the maximum group morphic image of . For a congruence of , the subset of is called *the kernel* of denoted by .

Let be a semigroup and a subset of . Then the subset is called *closure* of if . In this case, is said to be *closed* if . Moreover, a subset of is called full if . A subsemigroup of an eventually regular semigroup is called *weak self-conjugate* if for any , , there exist , . For a subset of , we define a binary relation named on as

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

Lemma 2.1 (see [2, 7]). *Let be an eventually regular semigroup and a congruence on . If is an idempotent of , then an idempotent can be found in such that .*

*Remark 2.2. *Since is an eventually regular semigroup and is a group congruence on , is an idempotent of for all .

Lemma 2.3. *Let be a regular semigroup with a unique idempotent, then is a group.*

Lemma 2.4 (see [5, 6]). *Let be an eventually regular semigroup. Then and for all , .*

Lemma 2.5. *Let be a subsemigroup of an eventually regular semigroup and for . If is weak self-conjugate, closed, and full, then for .*

*Proof. *Suppose that there exist such that and . Since is full and weak self-conjugate, we obtain , for , . It follows from that . Since is closed, we claim .

#### 3. Main Results

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

Theorem 3.1. *Let be an eventually regular semigroup and . Then the following statements are true. *(1)*If is a weak self-conjugate, closed subsemigroup, then is the minimum group congruence on and .*(2)*If the relation is a group congruence on and , then is the minimum group congruence on and is weak self-conjugate, closed, and full subsemigroup with . *

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

Lemma 3.2. *Let be an eventually regular semigroup and . If the subsemigroup of is weak self-conjugate, closed, and full, then the following statements are equivalent: *(1)*; *(2)*, for , ; *(3)* for .*

*Proof. * Suppose for , then there exists such that , and so for . For any , , it follows from Lemma 2.4 that
Since is weak self-conjugate, closed, and full, we deduce , so that . In a similar way, we prove for .

Using the statement (2), we conclude that there exists such that . Since is weak self-conjugate, we obtain and , so that .

For , there exists such that . From the weak self-conjugate of , we deduce and . And since is closed, we have , which leads to .

We now give the proof of Theorem 3.1.

*Proof of Theorem 3.1. *(1) To show that is an equivalence, let be a weak self-conjugate, closed subsemigroup. It is obvious that is full and . For , there exists such that , so that , and so is reflexive. To prove the symmetry, suppose for , then there exists such that . And since is weak self-conjugate, full, we obtain , so that , and so is symmetry. To prove the transitivity, let , for . Then there exist , such that , , hence . And there exists such that , and it follows from Lemma 2.4 that . Since is weak self-conjugate and full, we deduce , , and so , which says that is transitivity. Therefore is an equivalence, as required.

We now prove that is a congruence. Suppose for . Then there exists , and so , . Put , . Then , . It follows from Lemma 3.2 that for , and so , so that . Since is weak self-conjugate and , we conclude , so that . Therefore is left compatible. On the other hand, a similar argument will show that satisfies right compatible. Thus is a congruence on .

We now turn to show is a group congruence on . For any , there exists such that , so that . It follows from Lemma 2.1 that has a uniue idempotent. For any , there exists such that is regular element. Furthermore, there exists such that
and so , which leads to . Therefore, we conclude that is a regular semigroup. It follows from Lemma 2.3 that is a group, so that is a group congruence on .

We then show that is the minimum group congruence on . Let for , and let be any group congruence on with as the unique idempotent of . It follows from Lemma 3.2 that there exists such that , and so there exists such that . Notice that
for , so that and are the group inverse of . In view of the uniqueness of group inverses, we have . Since is the group inverse of and is the group inverse of , we claim , which leads to . Thus is the minimum group congruence on .

We finally prove . For any , it follows from Lemma 2.1 that there exists such that . We, by Lemma 3.2, deduce that there exists such that , . Since is closed, we have , , and so . To show , let . Since there exists , such that
and so , so that . Therefore , and so . Thus , as required.

Let be a group congruence on and the identity of . Suppose for , then there exist , such that is the group inverse of and is the group inverse of . By the uniqueness of group inverses, we have and , so that , and so there exists such that . Suppose that is any group congruence on , then
and so is the group inverse of . On the other hand, is the group inverse of . By the uniqueness of group inverses, we have , so that . Therefore is the minimum group congruence on .

We now prove is weak self-conjugate, closed, and full. It is obvious that is a full subsemigroup. For any , then
which leads to . A similar argument shows that . Therefore is weak self-conjugate. For , then there exists such that . Hence
and so , so that . On the other hand, it is obvious that . Thus , and so is weak self-conjugate, closed, and full subsemigroup of , as required.

We finally prove . To show , let for . Then there exists such that , and so , , which yields to . We now turn to proving that the converse holds. Let for . Then there exists such that , and so there exists such that . Put is any group congruence on . Notice
so that and are the group inverse of . By the uniqueness of group inverses, we claim that . Since is the minimum group congruence on , is the intersection of all group congruence on . Hence , so that , and so . The proof is then completed.

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

Corollary 3.3. *Let be an eventually regular semigroup. Then the following statements are true. *(1)*If is a weak self-conjugate, closed, and full subsemigroup, then is a group congruence on and .*(2)*If the relation is a group congruence on , then is a weak self-conjugate, closed, and full subsemigroup with . *

#### Acknowledgment

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