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 ğ‘Žâˆˆğ‘†. 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 ğ‘Ž if ğ‘¥ğ‘Žğ‘¥=𝑥. We denote by 𝑊(ğ‘Ž) the set of all weak inverses of ğ‘Ž 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 {ğ‘Žâˆˆğ‘†âˆ£ğ‘ŽğœŒâˆˆğ¸(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={ğ‘¥âˆˆğ‘†âˆ£âˆƒâ„Žâˆˆğ»,â„Žğ‘¥âˆˆğ»}. 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 ğ‘Žâˆˆğ‘†, ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), there exist ğ‘Žî…žğ¾ğ‘ŽâŠ†ğ¾, ğ‘Žğ¾ğ‘Žî…žâŠ†ğ¾. For a subset 𝐻 of 𝑆, we define a binary relation named ğœŽğ» on 𝐻 as ğœŽğ»=(ğ‘Ž,𝑏)âˆˆğ‘†Ã—ğ‘†âˆ¶âˆƒğ‘î…žâˆˆğ‘Š(𝑏),ğ‘Žğ‘î…žî€¾âˆˆğ».(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 ğ‘ŽğœŒ is an idempotent of 𝑆/𝜌, then an idempotent 𝑒 can be found in 𝑆 such that ğ‘ŽğœŒğ‘’.

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 [5, 6]). Let 𝑆 be an eventually regular semigroup. Then 𝑊(ğ‘Ž)≠∅ and ğ‘Žğ‘Žî…ž,ğ‘Žî…žğ‘Žâˆˆğ¸ğ‘† for all ğ‘Žâˆˆğ‘†, ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž).

Lemma 2.5. Let 𝐻 be a subsemigroup of an eventually regular semigroup 𝑆 and ğ‘Žğ‘âˆˆğ» for ğ‘Ž,𝑏∈𝑆. If 𝐻 is weak self-conjugate, closed, and full, then ğ‘Žğ‘¥ğ‘âˆˆğ» for 𝑥∈⟨𝐸𝑆⟩.

Proof. Suppose that there exist ğ‘Ž,𝑏∈𝑆 such that ğ‘Žğ‘âˆˆğ» and 𝑥∈⟨𝐸𝑆⟩. Since 𝐻 is full and weak self-conjugate, we obtain ğ‘â€²ğ‘Žâ€²ğ‘Žğ‘¥ğ‘âˆˆğ», ğ‘Žğ‘ğ‘â€²ğ‘Žâ€²âˆˆğ» for ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), 𝑏′∈𝑊(𝑏). It follows from ğ‘Žğ‘âˆˆğ» that (ğ‘Žğ‘)ğ‘â€²ğ‘Žâ€²ğ‘Žğ‘¥ğ‘âˆˆğ». Since 𝐻 is closed, we claim ğ‘Žğ‘¥ğ‘âˆˆğ».

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. (1)If 𝐻 is a weak self-conjugate, closed subsemigroup, then ğœŽğ» is the minimum group congruence on 𝑆 and kerğœŽğ»=𝐻.(2)If the relation ğœŽ is a group congruence on 𝑆 and kerğœŽ=𝐻, then ğœŽ is the minimum group congruence on 𝑆 and 𝐻 is weak self-conjugate, closed, and full subsemigroup with ğœŽ=ğœŽkerğœŽ.

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 ğ‘Ž,𝑏∈𝑆. If the subsemigroup 𝐻 of 𝑆 is weak self-conjugate, closed, and full, then the following statements are equivalent: (1)ğ‘ŽğœŽğ»ğ‘; (2)ğ‘Žğ‘î…žâˆˆğ», ğ‘ğ‘Žî…žâˆˆğ» for ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), ğ‘î…žâˆˆğ‘Š(𝑏); (3)ğ‘î…žğ‘Žâˆˆğ» for ğ‘î…žâˆˆğ‘Š(𝑏).

Proof. (1)⇒(2) Suppose ğ‘ŽğœŽğ»ğ‘ for ğ‘Ž,𝑏∈𝑆, then there exists ğ‘Žî…žî…žâˆˆğ‘Š(ğ‘Ž) such that ğ‘Žğ‘î…žî…žâˆˆğ», and so ğ‘Žğ‘î…žî…žğ‘ğ‘î…žâˆˆğ» for ğ‘î…žâˆˆğ‘Š(𝑏). For any ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), ğ‘Žî…žğ‘Žâˆˆğ¸ğ‘†, it follows from Lemma 2.4 that ğ‘Žğ‘î…žî…žğ‘î€·ğ‘Žî…žğ‘Žî€¸ğ‘î…ž=ğ‘Žğ‘î…žî…žğ‘ğ‘Žî…žî€·ğ‘Žğ‘î…žî€¸âˆˆğ».(3.1) Since 𝐻 is weak self-conjugate, closed, and full, we deduce ğ‘Žğ‘î…žî…žğ‘ğ‘Žî…žâˆˆğ», so that ğ‘Žğ‘î…žâˆˆğ». In a similar way, we prove ğ‘ğ‘Žî…žâˆˆğ» for ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž).
(2)⇒(3) Using the statement (2), we conclude that there exists ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘Žğ‘î…žâˆˆğ». Since 𝐻 is weak self-conjugate, we obtain ğ‘Žî…žğ‘Žğ‘î…žğ‘Žâˆˆğ» and ğ‘Žî…žğ‘Žâˆˆğ¸ğ‘†âŠ†ğ», so that ğ‘î…žğ‘Žâˆˆğ».
(3)⇒(1) For ğ‘Ž,𝑏∈𝑆, there exists ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘î…žğ‘Žâˆˆğ». From the weak self-conjugate of 𝐻, we deduce ğ‘ğ‘î…žğ‘Žğ‘î…žâˆˆğ» and ğ‘ğ‘î…žâˆˆğ». And since 𝐻 is closed, we have ğ‘Žğ‘î…žâˆˆğ», which leads to ğ‘ŽğœŽğ»ğ‘.

We now give the proof of Theorem 3.1.

Proof of Theorem 3.1. (1) To show that ğœŽğ» 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 ğ‘Žâˆˆğ‘†, there exists ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž) such that ğ‘Žğ‘Žî…žâˆˆğ¸ğ‘†âŠ†ğ», so that ğ‘ŽğœŽğ»ğ‘Ž, and so ğœŽğ» is reflexive. To prove the symmetry, suppose ğ‘ŽğœŽğ»ğ‘ for ğ‘Ž,𝑏∈𝑆, then there exists ğ‘î…žâˆˆğ‘Š(𝑏),ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž) such that ğ‘Žğ‘â€²âˆˆğ». And since 𝐻 is weak self-conjugate, full, we obtain ğ‘Žğ‘î…žğ‘ğ‘Žî…žâˆˆğ», so that ğ‘ğœŽğ»ğ‘Ž, and so ğœŽğ» is symmetry. To prove the transitivity, let ğ‘ŽğœŽğ»ğ‘, ğ‘ğœŽğ»ğ‘ for ğ‘Ž,𝑏,𝑐∈𝑆. Then there exist ğ‘î…žâˆˆğ‘Š(𝑏), ğ‘î…žâˆˆğ‘Š(𝑐) such that ğ‘Žğ‘î…žâˆˆğ», ğ‘ğ‘î…žâˆˆğ», hence ğ‘Žğ‘î…žğ‘ğ‘î…žâˆˆğ». And there exists ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž) such that ğ‘Žî…žğ‘Žâˆˆğ¸ğ‘†, and it follows from Lemma 2.4 that ğ‘Žğ‘î…žğ‘(ğ‘Žî…žğ‘Ž)ğ‘î…ž=(ğ‘Žğ‘î…žğ‘ğ‘Žî…ž)ğ‘Žğ‘î…žâˆˆğ». Since 𝐻 is weak self-conjugate and full, we deduce ğ‘Žğ‘î…žğ‘ğ‘Žî…žâˆˆğ», ğ‘Žğ‘î…žâˆˆğ», and so ğ‘ŽğœŽğ»ğ‘, which says that ğœŽğ» is transitivity. Therefore ğœŽğ» is an equivalence, as required.
We now prove that ğœŽğ» is a congruence. Suppose ğ‘ŽğœŽğ»ğ‘ for ğ‘Ž,𝑏,𝑐∈𝑆. Then there exists (𝑐𝑏)î…žâˆˆğ‘Š(𝑐𝑏), and so 𝑏(𝑐𝑏)î…žâˆˆğ‘Š(𝑐), (𝑐𝑏)î…žğ‘âˆˆğ‘Š(𝑏). Put ğ‘î…ž=𝑏(𝑐𝑏), ğ‘î…ž=(𝑐𝑏)î…žğ‘. Then ğ‘î…žğ‘î…žâˆˆğ‘Š(𝑐𝑏), (𝑐𝑏)=ğ‘î…žğ‘î…ž. It follows from Lemma 3.2 that ğ‘Žğ‘î…žî…žâˆˆğ» for ğ‘î…žî…žâˆˆğ‘Š(𝑏), and so ğ‘î…ž=(𝑐𝑏)î…žğ‘âˆˆğ‘Š(𝑏), so that ğ‘Žğ‘î…žâˆˆğ». Since 𝐻 is weak self-conjugate and (ğ‘ğ‘Ž)(𝑐𝑏)=ğ‘ğ‘Žğ‘î…žğ‘î…ž, we conclude ğ‘ğ‘Ž(𝑐𝑏)=ğ‘ğ‘Žğ‘î…žğ‘î…žâˆˆğ», so that ğ‘ğ‘ŽğœŽğ»ğ‘ğ‘. Therefore ğœŽğ» is left compatible. On the other hand, a similar argument will show that ğœŽğ» satisfies right compatible. Thus ğœŽğ» is a congruence on 𝑆.
We now turn to show ğœŽğ» is a group congruence on 𝑆. For any 𝑒,𝑓∈𝐸𝑆, there exists 𝑓∈𝑊(𝑓)⊆⟨𝐸𝑆⟩ such that 𝑒𝑓∈⟨𝐸𝑆⟩⊆𝐻, so that ğ‘’ğœŽğ»ğ‘“. It follows from Lemma 2.1 that 𝑆/ğœŽğ» has a uniue idempotent. For any ğ‘Žâˆˆğ‘†, there exists 𝑚∈𝑁 such that ğ‘Žğ‘š is regular element. Furthermore, there exists (ğ‘Žğ‘š)î…žâˆˆğ‘Š(ğ‘Žğ‘š) such that ğ‘Žğ‘š(ğ‘Žğ‘š)î…žâˆˆğ¸ğ‘†,ğ‘Žğ‘š(ğ‘Žğ‘š)î…žğ‘Ž(ğ‘Žğ‘š)=ğ‘Žğ‘š(ğ‘Žğ‘š)î…žâˆˆğ¸ğ‘†,(3.2) and so ğ‘Ž(ğ‘Žğ‘š)î…žâˆˆâŸ¨ğ¸ğ‘†âŸ©ğœ”=𝐻, which leads to ğ‘ŽğœŽğ»ğ‘Žğ‘š. Therefore, we conclude that 𝑆/ğœŽğ» is a regular semigroup. It follows from Lemma 2.3 that 𝑆/ğœŽğ» is a group, so that ğœŽğ» is a group congruence on 𝑆.
We then show that ğœŽğ» is the minimum group congruence on 𝑆. Let ğ‘ŽğœŽğ»ğ‘ for ğ‘Ž,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 ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘Žğ‘î…žâˆˆğ», and so there exists 𝑡∈⟨𝐸𝑆⟩ such that ğ‘¡ğ‘Žğ‘î…žâˆˆâŸ¨ğ¸ğ‘†âŸ©. Notice that î€·ğ‘¡ğ‘Žğ‘î…žî€¸ğœŒ=𝑒𝜌=(ğ‘ŽğœŒ)ğ‘î…žî€·ğœŒ,ğ‘Žğ‘Žî…žî€¸ğœŒ=𝑒𝜌=(ğ‘ŽğœŒ)ğ‘Žî…žğœŒ,(3.3) for ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), so that ğ‘î…žğœŒ and ğ‘Žî…žğœŒ are the group inverse of ğ‘ŽğœŒ. In view of the uniqueness of group inverses, we have ğ‘Žî…žğœŒ=ğ‘î…žğœŒ. Since ğ‘Žî…žğœŒ is the group inverse of ğ‘ŽğœŒ and ğ‘î…žğœŒ is the group inverse of 𝑏𝜌, we claim ğ‘ŽğœŒ=𝑏𝜌, which leads to ğœŽğ»âŠ†ğœŒ. Thus ğœŽğ» is the minimum group congruence on 𝑆.
We finally prove kerğœŽğ»=𝐻. For any ğ‘ŽâˆˆkerğœŽğ», it follows from Lemma 2.1 that there exists 𝑒∈𝐸𝑆 such that ğ‘ŽğœŽğ»ğ‘’. We, by Lemma 3.2, deduce that there exists ğ‘’î…žâˆˆğ‘Š(𝑒) such that ğ‘’î…žğ‘Žâˆˆğ», ğ‘’ğ‘’î…žâˆˆğ¸ğ‘†âŠ†ğ». Since 𝐻 is closed, we have ğ‘’î…žâˆˆğ», ğ‘Žâˆˆğ», and so kerğœŽğ»âŠ†ğ». To show kerğœŽğ»âŠ‡ğ», let ğ‘Žâˆˆğ». Since there exists 𝑡∈⟨𝐸𝑆⟩, (ğ‘¡ğ‘Ž)î…žâˆˆğ‘Š(ğ‘¡ğ‘Ž) such that ğ‘¡ğ‘ŽâˆˆâŸ¨ğ¸ğ‘†âŸ©,(ğ‘¡ğ‘Ž)ğœŽğ»î‚µğ‘†âŠ†ğ¸ğœŽğ»î‚¶,(ğ‘¡ğ‘Ž)(ğ‘¡ğ‘Ž)î…žâˆˆğ¸ğ‘†,(3.4) and so ğ‘Ž(ğ‘¡ğ‘Ž)î…žâˆˆğ», so that ğ‘ŽğœŽğ»(ğ‘¡ğ‘Ž). Therefore ğ‘ŽğœŽğ»âŠ†ğ¸(𝑆/ğœŽğ»), and so ğ‘ŽâˆˆkerğœŽğ». Thus kerğœŽğ»=𝐻, as required.
(2) Let ğœŽ be a group congruence on 𝑆 and ğ‘’ğœŽ the identity of 𝑆/ğœŽ. Suppose ğ‘ŽğœŽğ‘ for ğ‘Ž,𝑏∈𝑆, then there exist ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž), ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘Žî…žğœŽ is the group inverse of ğ‘ŽğœŽ and ğ‘î…žğœŽ is the group inverse of ğ‘ğœŽ. By the uniqueness of group inverses, we have ğ‘Žî…žğœŽğ‘î…ž and ğ‘Žğ‘Žî…žğœŽğ‘ğ‘î…žğœŽğ‘ğ‘Žî…žğœŽğ‘’, so that ğ‘ğ‘Žî…žâˆˆkerğœŽ=𝐻, and so there exists 𝑡∈⟨𝐸𝑆⟩ such that ğ‘¡ğ‘ğ‘Žî…žâˆˆâŸ¨ğ¸ğ‘†âŸ©. Suppose that 𝜌 is any group congruence on 𝑆, then î€·ğ‘¡ğ‘ğ‘Žî…žî€¸î€·ğœŒ=ğ‘ğ‘Žî…žî€¸ğœŒ=(𝑏𝜌)ğ‘Žî…žğœŒ=𝑒𝜌,(3.5) and so 𝑏𝜌 is the group inverse of ğ‘Žî…žğœŒ. On the other hand, ğ‘ŽğœŒ is the group inverse of ğ‘Žî…žğœŒ. By the uniqueness of group inverses, we have ğ‘ŽğœŒğ‘, so that ğœŽâŠ†ğœŒ. Therefore ğœŽ is the minimum group congruence on 𝑆.
We now prove 𝐻 is weak self-conjugate, closed, and full. It is obvious that kerğœŽ=𝐻 is a full subsemigroup. For any ğ‘Žâˆˆğ‘†,ğ‘Žî…žâˆˆğ‘Š(ğ‘Ž),𝑥∈kerğœŽ, then î€·ğ‘Žğ‘¥ğ‘Žî…žî€¸î€·ğ‘ŽğœŽ=(ğ‘ŽğœŽ)ğ‘’ğœŽî…žğœŽî€¸=î€·ğ‘Žğ‘Žî…žî€¸ğœŽ=ğ‘’ğœŽ,(3.6) which leads to ğ‘Žğ‘¥ğ‘Žî…žâˆˆkerğœŽ. A similar argument shows that ğ‘Žî…žğ‘¥ğ‘ŽâˆˆkerğœŽ. Therefore 𝐻 is weak self-conjugate. For 𝑥∈𝐻𝜔=(kerğœŽ)𝜔, then there exists 𝑡∈kerğœŽ such that 𝑡𝑥∈kerğœŽ. Hence ğ‘’ğœŽ=(𝑡𝑥)ğœŽ=ğ‘¡ğœŽ(ğ‘¥ğœŽ)=ğ‘’ğœŽ(ğ‘¥ğœŽ)=ğ‘¥ğœŽ,(3.7) and so 𝑥∈kerğœŽ, so that (kerğœŽ)𝜔⊆kerğœŽ. On the other hand, it is obvious that (kerğœŽ)𝜔⊇kerğœŽ. Thus (kerğœŽ)𝜔=kerğœŽ, and so 𝐻 is weak self-conjugate, closed, and full subsemigroup of 𝑆, as required.
We finally prove ğœŽ=ğœŽkerğœŽ. To show ğœŽâŠ†ğœŽkerğœŽ, let ğ‘ŽğœŽğ‘ for ğ‘Ž,𝑏∈𝑆. Then there exists ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘Žğ‘î…žğœŽğ‘ğ‘î…žğœŽğ‘’, and so ğ‘Žğ‘î…žâˆˆkerğœŽ, ğ‘ŽğœŽkerğœŽğ‘, which yields to ğœŽâŠ†ğœŽkerğœŽ. We now turn to proving that the converse holds. Let ğ‘ŽğœŽkerğœŽğ‘ for ğ‘Ž,𝑏∈𝑆. Then there exists ğ‘î…žâˆˆğ‘Š(𝑏) such that ğ‘Žğ‘î…žâˆˆkerğœŽ=𝐻, and so there exists 𝑡∈⟨𝐸𝑆⟩ such that ğ‘¡ğ‘Žğ‘î…žâˆˆâŸ¨ğ¸ğ‘†âŸ©. Put 𝜌 is any group congruence on 𝑆. Notice î€·ğ‘¡ğ‘Žğ‘î…žî€¸î€·ğ‘ğœŒ=(𝑡𝜌)ğ‘ŽğœŒî…žğœŒî€¸=𝑒𝜌,(3.8) so that 𝑏𝜌 and ğ‘ŽğœŒ are the group inverse of ğ‘î…žğœŒ. By the uniqueness of group inverses, we claim that ğ‘ŽğœŒğ‘. Since ğœŽ is the minimum group congruence on 𝑆, ğœŽ is the intersection of all group congruence on 𝑆. Hence ğ‘ŽğœŽğ‘, so that ğœŽâŠ‡ğœŽkerğœŽ, and so ğœŽ=ğœŽkerğœŽ. 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 ğœŽğ» is a group congruence on 𝑆 and kerğœŽğ»=𝐻.(2)If the relation ğœŽ is a group congruence on 𝑆, then kerğœŽ is a weak self-conjugate, closed, and full subsemigroup with ğœŽ=ğœŽkerğœŽ.

Acknowledgment

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