Minimum Group Congruences on Eventually Regular Semigroups

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 𝑎∈𝑆. As usual, 𝐸_𝑆 is the set of all idempotents of 𝑆, ⟨𝐸_𝑆⟩ is the subsemigroup of 𝑆 generated by 𝐸_𝑆 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 𝐸_𝑆⊆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 𝜎_𝐻 on 𝐻 as 𝜎_𝐻=(𝑎,𝑏)∈𝑆×𝑆∶∃𝑏^∈𝑊(𝑏),𝑎𝑏^∈𝐻.(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 𝑥∈⟨𝐸_𝑆⟩.

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^,𝑎^𝑎∈𝐸_𝑆 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∈⟨𝐸_𝑆⟩.

Proof. Suppose that there exist 𝑎,𝑏∈𝑆 such that 𝑎𝑏∈𝐻 and 𝑥∈⟨𝐸_𝑆⟩. 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=⟨𝐸_𝑆⟩𝜔. Then the following statements are true. (1)If 𝐻 is a weak self-conjugate, closed subsemigroup, then 𝜎_𝐻 is the minimum group congruence on 𝑆 and ker𝜎_𝐻=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=𝜎_ker𝜎.

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: (1)𝑎𝜎_𝐻𝑏; (2)𝑎𝑏^∈𝐻, 𝑏𝑎^∈𝐻 for 𝑎^∈𝑊(𝑎), 𝑏^∈𝑊(𝑏); (3)𝑏^𝑎∈𝐻 for 𝑏^∈𝑊(𝑏).

Proof. (1)⇒(2) Suppose 𝑎𝜎_𝐻𝑏 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∈𝐸_𝑆, it follows from Lemma 2.4 that 𝑎𝑏^𝑏𝑎^𝑎𝑏^=𝑎𝑏^𝑏𝑎^𝑎𝑏^∈𝐻.(3.1) 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∈𝐸_𝑆⊆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 𝑎𝜎_𝐻𝑏.

We now give the proof of Theorem 3.1.

Proof of Theorem 3.1. (1) To show that 𝜎_𝐻 is an equivalence, let 𝐻=⟨𝐸_𝑆⟩𝜔 be a weak self-conjugate, closed subsemigroup. It is obvious that 𝐻 is full and ⟨𝐸_𝑆⟩⊆𝐻. For 𝑎∈𝑆, there exists 𝑎^∈𝑊(𝑎) such that 𝑎𝑎^∈𝐸_𝑆⊆𝐻, so that 𝑎𝜎_𝐻𝑎, and so 𝜎_𝐻 is reflexive. To prove the symmetry, suppose 𝑎𝜎_𝐻𝑏 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 𝑏𝜎_𝐻𝑎, and so 𝜎_𝐻 is symmetry. To prove the transitivity, let 𝑎𝜎_𝐻𝑏, 𝑏𝜎_𝐻𝑐 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∈𝐸_𝑆, 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 𝑎𝜎_𝐻𝑐, which says that 𝜎_𝐻 is transitivity. Therefore 𝜎_𝐻 is an equivalence, as required.
We now prove that 𝜎_𝐻 is a congruence. Suppose 𝑎𝜎_𝐻𝑏 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 𝑐𝑎𝜎_𝐻𝑐𝑏. Therefore 𝜎_𝐻 is left compatible. On the other hand, a similar argument will show that 𝜎_𝐻 satisfies right compatible. Thus 𝜎_𝐻 is a congruence on 𝑆.
We now turn to show 𝜎_𝐻 is a group congruence on 𝑆. For any 𝑒,𝑓∈𝐸_𝑆, there exists 𝑓∈𝑊(𝑓)⊆⟨𝐸_𝑆⟩ such that 𝑒𝑓∈⟨𝐸_𝑆⟩⊆𝐻, so that 𝑒𝜎_𝐻𝑓. It follows from Lemma 2.1 that 𝑆/𝜎_𝐻 has a uniue idempotent. For any 𝑎∈𝑆, there exists 𝑚∈𝑁 such that 𝑎^𝑚 is regular element. Furthermore, there exists (𝑎^𝑚)^∈𝑊(𝑎^𝑚) such that 𝑎^𝑚(𝑎^𝑚)^∈𝐸_𝑆,𝑎^𝑚(𝑎^𝑚)^𝑎(𝑎^𝑚)^=𝑎^𝑚(𝑎^𝑚)^∈𝐸_𝑆,(3.2) and so 𝑎(𝑎^𝑚)^∈⟨𝐸_𝑆⟩𝜔=𝐻, which leads to 𝑎𝜎_𝐻𝑎^𝑚. Therefore, we conclude that 𝑆/𝜎_𝐻 is a regular semigroup. It follows from Lemma 2.3 that 𝑆/𝜎_𝐻 is a group, so that 𝜎_𝐻 is a group congruence on 𝑆.
We then show that 𝜎_𝐻 is the minimum group congruence on 𝑆. Let 𝑎𝜎_𝐻𝑏 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∈⟨𝐸_𝑆⟩ such that 𝑡𝑎𝑏^∈⟨𝐸_𝑆⟩. Notice that 𝑡𝑎𝑏^𝜌=𝑒𝜌=(𝑎𝜌)𝑏^𝜌,𝑎𝑎^𝜌=𝑒𝜌=(𝑎𝜌)𝑎^𝜌,(3.3) 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 𝜎_𝐻⊆𝜌. Thus 𝜎_𝐻 is the minimum group congruence on 𝑆.
We finally prove ker𝜎_𝐻=𝐻. For any 𝑎∈ker𝜎_𝐻, it follows from Lemma 2.1 that there exists 𝑒∈𝐸_𝑆 such that 𝑎𝜎_𝐻𝑒. We, by Lemma 3.2, deduce that there exists 𝑒^∈𝑊(𝑒) such that 𝑒^𝑎∈𝐻, 𝑒𝑒^∈𝐸_𝑆⊆𝐻. Since 𝐻 is closed, we have 𝑒^∈𝐻, 𝑎∈𝐻, and so ker𝜎_𝐻⊆𝐻. To show ker𝜎_𝐻⊇𝐻, let 𝑎∈𝐻. Since there exists 𝑡∈⟨𝐸_𝑆⟩, (𝑡𝑎)^∈𝑊(𝑡𝑎) such that 𝑡𝑎∈⟨𝐸_𝑆⟩,(𝑡𝑎)𝜎_𝐻⊆𝐸𝑆𝜎_𝐻,(𝑡𝑎)(𝑡𝑎)^∈𝐸_𝑆,(3.4) and so 𝑎(𝑡𝑎)^∈𝐻, so that 𝑎𝜎_𝐻(𝑡𝑎). Therefore 𝑎𝜎_𝐻⊆𝐸(𝑆/𝜎_𝐻), and so 𝑎∈ker𝜎_𝐻. Thus ker𝜎_𝐻=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∈⟨𝐸_𝑆⟩ such that 𝑡𝑏𝑎^∈⟨𝐸_𝑆⟩. Suppose that 𝜌 is any group congruence on 𝑆, then 𝑡𝑏𝑎^𝜌=𝑏𝑎^𝜌=(𝑏𝜌)𝑎^𝜌=𝑒𝜌,(3.5) 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,(3.6) 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,(3.7) 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=𝜎_ker𝜎. To show 𝜎⊆𝜎_ker𝜎, let 𝑎𝜎𝑏 for 𝑎,𝑏∈𝑆. Then there exists 𝑏^∈𝑊(𝑏) such that 𝑎𝑏^𝜎𝑏𝑏^𝜎𝑒, and so 𝑎𝑏^∈ker𝜎, 𝑎𝜎_ker𝜎𝑏, which yields to 𝜎⊆𝜎_ker𝜎. We now turn to proving that the converse holds. Let 𝑎𝜎_ker𝜎𝑏 for 𝑎,𝑏∈𝑆. Then there exists 𝑏^∈𝑊(𝑏) such that 𝑎𝑏^∈ker𝜎=𝐻, and so there exists 𝑡∈⟨𝐸_𝑆⟩ such that 𝑡𝑎𝑏^∈⟨𝐸_𝑆⟩. Put 𝜌 is any group congruence on 𝑆. Notice 𝑡𝑎𝑏^𝜌=(𝑡𝜌)𝑎𝜌𝑏^𝜌=𝑒𝜌,(3.8) 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⊇𝜎_ker𝜎, and so 𝜎=𝜎_ker𝜎. 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 𝜎_𝐻 is a group congruence on 𝑆 and ker𝜎_𝐻=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=𝜎_ker𝜎.

Acknowledgment

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