Table of Contents
ISRN Algebra
Volume 2011, Article ID 458093, 5 pages
http://dx.doi.org/10.5402/2011/458093
Research Article

Minimum Group Congruences on Eventually Regular Semigroups

Department of Mathematics, Anhui University of Science and Technology, Anhui 232001, China

Received 29 May 2011; Accepted 3 July 2011

Academic Editors: G. Militaru and P.-H. Zieschang

Copyright © 2011 Wang Yu and Yin ZhiXiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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 𝑎𝑆. 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 {𝑎𝑆𝑎𝜌𝐸(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={𝑥𝑆𝐻,𝑥𝐻}. In this case, 𝐻 is said to be closed if 𝐻𝜔=𝐻. Moreover, a subset 𝐻 of 𝑆 is called full if 𝐸𝑆𝐻. A subsemigroup 𝐾 of an eventually regular semigroup 𝑆 is called weak self-conjugate if for any 𝑎𝑆, 𝑎𝑊(𝑎), there exist 𝑎𝐾𝑎𝐾, 𝑎𝐾𝑎𝐾. For a subset 𝐻 of 𝑆, we define a binary relation named 𝜎𝐻 on 𝐻 as 𝜎𝐻=(𝑎,𝑏)𝑆×𝑆𝑏𝑊(𝑏),𝑎𝑏𝐻.(2.1)

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

Lemma 2.1 (see [2, 7]). 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 𝑥𝐸𝑆.

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

Lemma 2.4 (see [5, 6]). Let 𝑆 be an eventually regular semigroup. Then 𝑊(𝑎) and 𝑎𝑎,𝑎𝑎𝐸𝑆 for all 𝑎𝑆, 𝑎𝑊(𝑎).

Lemma 2.5. Let 𝐻 be a subsemigroup of an eventually regular semigroup 𝑆 and 𝑎𝑏𝐻 for 𝑎,𝑏𝑆. If 𝐻 is weak self-conjugate, closed, and full, then 𝑎𝑥𝑏𝐻 for 𝑥𝐸𝑆.

Proof. Suppose that there exist 𝑎,𝑏𝑆 such that 𝑎𝑏𝐻 and 𝑥𝐸𝑆. Since 𝐻 is full and weak self-conjugate, we obtain 𝑏𝑎𝑎𝑥𝑏𝐻, 𝑎𝑏𝑏𝑎𝐻 for 𝑎𝑊(𝑎), 𝑏𝑊(𝑏). It follows from 𝑎𝑏𝐻 that (𝑎𝑏)𝑏𝑎𝑎𝑥𝑏𝐻. Since 𝐻 is closed, we claim 𝑎𝑥𝑏𝐻.

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=𝐸𝑆𝜔. Then the following statements are true. (1)If 𝐻 is a weak self-conjugate, closed subsemigroup, then 𝜎𝐻 is the minimum group congruence on 𝑆 and ker𝜎𝐻=𝐻.(2)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,𝑏𝑆. If the subsemigroup 𝐻 of 𝑆 is weak self-conjugate, closed, and full, then the following statements are equivalent: (1)𝑎𝜎𝐻𝑏; (2)𝑎𝑏𝐻, 𝑏𝑎𝐻 for 𝑎𝑊(𝑎), 𝑏𝑊(𝑏); (3)𝑏𝑎𝐻 for 𝑏𝑊(𝑏).

Proof. (1)(2) Suppose 𝑎𝜎𝐻𝑏 for 𝑎,𝑏𝑆, then there exists 𝑎𝑊(𝑎) such that 𝑎𝑏𝐻, and so 𝑎𝑏𝑏𝑏𝐻 for 𝑏𝑊(𝑏). For any 𝑎𝑊(𝑎), 𝑎𝑎𝐸𝑆, it follows from Lemma 2.4 that 𝑎𝑏𝑏𝑎𝑎𝑏=𝑎𝑏𝑏𝑎𝑎𝑏𝐻.(3.1) Since 𝐻 is weak self-conjugate, closed, and full, we deduce 𝑎𝑏𝑏𝑎𝐻, so that 𝑎𝑏𝐻. In a similar way, we prove 𝑏𝑎𝐻 for 𝑎𝑊(𝑎).
(2)(3) Using the statement (2), we conclude that there exists 𝑏𝑊(𝑏) such that 𝑎𝑏𝐻. Since 𝐻 is weak self-conjugate, we obtain 𝑎𝑎𝑏𝑎𝐻 and 𝑎𝑎𝐸𝑆𝐻, so that 𝑏𝑎𝐻.
(3)(1) For 𝑎,𝑏𝑆, there exists 𝑏𝑊(𝑏) such that 𝑏𝑎𝐻. From the weak self-conjugate of 𝐻, we deduce 𝑏𝑏𝑎𝑏𝐻 and 𝑏𝑏𝐻. And since 𝐻 is closed, we have 𝑎𝑏𝐻, which leads to 𝑎𝜎𝐻𝑏.

We now give the proof of Theorem 3.1.

Proof of Theorem 3.1. (1) To show that 𝜎𝐻 is an equivalence, let 𝐻=𝐸𝑆𝜔 be a weak self-conjugate, closed subsemigroup. It is obvious that 𝐻 is full and 𝐸𝑆𝐻. For 𝑎𝑆, there exists 𝑎𝑊(𝑎) such that 𝑎𝑎𝐸𝑆𝐻, so that 𝑎𝜎𝐻𝑎, and so 𝜎𝐻 is reflexive. To prove the symmetry, suppose 𝑎𝜎𝐻𝑏 for 𝑎,𝑏𝑆, then there exists 𝑏𝑊(𝑏),𝑎𝑊(𝑎) such that 𝑎𝑏𝐻. And since 𝐻 is weak self-conjugate, full, we obtain 𝑎𝑏𝑏𝑎𝐻, so that 𝑏𝜎𝐻𝑎, and so 𝜎𝐻 is symmetry. To prove the transitivity, let 𝑎𝜎𝐻𝑏, 𝑏𝜎𝐻𝑐 for 𝑎,𝑏,𝑐𝑆. Then there exist 𝑏𝑊(𝑏), 𝑐𝑊(𝑐) such that 𝑎𝑏𝐻, 𝑏𝑐𝐻, hence 𝑎𝑏𝑏𝑐𝐻. And there exists 𝑎𝑊(𝑎) such that 𝑎𝑎𝐸𝑆, and it follows from Lemma 2.4 that 𝑎𝑏𝑏(𝑎𝑎)𝑐=(𝑎𝑏𝑏𝑎)𝑎𝑐𝐻. Since 𝐻 is weak self-conjugate and full, we deduce 𝑎𝑏𝑏𝑎𝐻, 𝑎𝑐𝐻, 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,𝑐𝑆. Then there exists (𝑐𝑏)𝑊(𝑐𝑏), and so 𝑏(𝑐𝑏)𝑊(𝑐), (𝑐𝑏)𝑐𝑊(𝑏). Put 𝑐=𝑏(𝑐𝑏), 𝑏=(𝑐𝑏)𝑐. Then 𝑏𝑐𝑊(𝑐𝑏), (𝑐𝑏)=𝑏𝑐. It follows from Lemma 3.2 that 𝑎𝑏𝐻 for 𝑏𝑊(𝑏), and so 𝑏=(𝑐𝑏)𝑐𝑊(𝑏), so that 𝑎𝑏𝐻. Since 𝐻 is weak self-conjugate and (𝑐𝑎)(𝑐𝑏)=𝑐𝑎𝑏𝑐, we conclude 𝑐𝑎(𝑐𝑏)=𝑐𝑎𝑏𝑐𝐻, 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,𝑓𝐸𝑆, there exists 𝑓𝑊(𝑓)𝐸𝑆 such that 𝑒𝑓𝐸𝑆𝐻, so that 𝑒𝜎𝐻𝑓. It follows from Lemma 2.1 that 𝑆/𝜎𝐻 has a uniue idempotent. For any 𝑎𝑆, there exists 𝑚𝑁 such that 𝑎𝑚 is regular element. Furthermore, there exists (𝑎𝑚)𝑊(𝑎𝑚) such that 𝑎𝑚(𝑎𝑚)𝐸𝑆,𝑎𝑚(𝑎𝑚)𝑎(𝑎𝑚)=𝑎𝑚(𝑎𝑚)𝐸𝑆,(3.2) and so 𝑎(𝑎𝑚)𝐸𝑆𝜔=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,𝑏𝑆, 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 𝑏𝑊(𝑏) such that 𝑎𝑏𝐻, and so there exists 𝑡𝐸𝑆 such that 𝑡𝑎𝑏𝐸𝑆. Notice that 𝑡𝑎𝑏𝜌=𝑒𝜌=(𝑎𝜌)𝑏𝜌,𝑎𝑎𝜌=𝑒𝜌=(𝑎𝜌)𝑎𝜌,(3.3) for 𝑎𝑊(𝑎), so that 𝑏𝜌 and 𝑎𝜌 are the group inverse of 𝑎𝜌. In view of the uniqueness of group inverses, we have 𝑎𝜌=𝑏𝜌. Since 𝑎𝜌 is the group inverse of 𝑎𝜌 and 𝑏𝜌 is the group inverse of 𝑏𝜌, we claim 𝑎𝜌=𝑏𝜌, which leads to 𝜎𝐻𝜌. Thus 𝜎𝐻 is the minimum group congruence on 𝑆.
We finally prove ker𝜎𝐻=𝐻. For any 𝑎ker𝜎𝐻, it follows from Lemma 2.1 that there exists 𝑒𝐸𝑆 such that 𝑎𝜎𝐻𝑒. We, by Lemma 3.2, deduce that there exists 𝑒𝑊(𝑒) such that 𝑒𝑎𝐻, 𝑒𝑒𝐸𝑆𝐻. Since 𝐻 is closed, we have 𝑒𝐻, 𝑎𝐻, and so ker𝜎𝐻𝐻. To show ker𝜎𝐻𝐻, let 𝑎𝐻. Since there exists 𝑡𝐸𝑆, (𝑡𝑎)𝑊(𝑡𝑎) such that 𝑡𝑎𝐸𝑆,(𝑡𝑎)𝜎𝐻𝑆𝐸𝜎𝐻,(𝑡𝑎)(𝑡𝑎)𝐸𝑆,(3.4) and so 𝑎(𝑡𝑎)𝐻, so that 𝑎𝜎𝐻(𝑡𝑎). Therefore 𝑎𝜎𝐻𝐸(𝑆/𝜎𝐻), and so 𝑎ker𝜎𝐻. 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,𝑏𝑆, then there exist 𝑎𝑊(𝑎), 𝑏𝑊(𝑏) such that 𝑎𝜎 is the group inverse of 𝑎𝜎 and 𝑏𝜎 is the group inverse of 𝑏𝜎. By the uniqueness of group inverses, we have 𝑎𝜎𝑏 and 𝑎𝑎𝜎𝑏𝑏𝜎𝑏𝑎𝜎𝑒, so that 𝑏𝑎ker𝜎=𝐻, and so there exists 𝑡𝐸𝑆 such that 𝑡𝑏𝑎𝐸𝑆. Suppose that 𝜌 is any group congruence on 𝑆, then 𝑡𝑏𝑎𝜌=𝑏𝑎𝜌=(𝑏𝜌)𝑎𝜌=𝑒𝜌,(3.5) and so 𝑏𝜌 is the group inverse of 𝑎𝜌. On the other hand, 𝑎𝜌 is the group inverse of 𝑎𝜌. By the uniqueness of group inverses, we have 𝑎𝜌𝑏, so that 𝜎𝜌. 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 𝑎𝑆,𝑎𝑊(𝑎),𝑥ker𝜎, then 𝑎𝑥𝑎𝑎𝜎=(𝑎𝜎)𝑒𝜎𝜎=𝑎𝑎𝜎=𝑒𝜎,(3.6) which leads to 𝑎𝑥𝑎ker𝜎. A similar argument shows that 𝑎𝑥𝑎ker𝜎. Therefore 𝐻 is weak self-conjugate. For 𝑥𝐻𝜔=(ker𝜎)𝜔, then there exists 𝑡ker𝜎 such that 𝑡𝑥ker𝜎. Hence 𝑒𝜎=(𝑡𝑥)𝜎=𝑡𝜎(𝑥𝜎)=𝑒𝜎(𝑥𝜎)=𝑥𝜎,(3.7) and so 𝑥ker𝜎, so that (ker𝜎)𝜔ker𝜎. On the other hand, it is obvious that (ker𝜎)𝜔ker𝜎. 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 𝜎𝜎ker𝜎, let 𝑎𝜎𝑏 for 𝑎,𝑏𝑆. Then there exists 𝑏𝑊(𝑏) such that 𝑎𝑏𝜎𝑏𝑏𝜎𝑒, and so 𝑎𝑏ker𝜎, 𝑎𝜎ker𝜎𝑏, which yields to 𝜎𝜎ker𝜎. We now turn to proving that the converse holds. Let 𝑎𝜎ker𝜎𝑏 for 𝑎,𝑏𝑆. Then there exists 𝑏𝑊(𝑏) such that 𝑎𝑏ker𝜎=𝐻, and so there exists 𝑡𝐸𝑆 such that 𝑡𝑎𝑏𝐸𝑆. Put 𝜌 is any group congruence on 𝑆. Notice 𝑡𝑎𝑏𝑏𝜌=(𝑡𝜌)𝑎𝜌𝜌=𝑒𝜌,(3.8) so that 𝑏𝜌 and 𝑎𝜌 are the group inverse of 𝑏𝜌. 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 𝜎𝜎ker𝜎, 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. (1)If 𝐻 is a weak self-conjugate, closed, and full subsemigroup, then 𝜎𝐻 is a group congruence on 𝑆 and ker𝜎𝐻=𝐻.(2)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).

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