On Natural Partial Orders of IC-Abundant Semigroups

Chunhua Li^{*}, Baogen Xu

School of Science, East China Jiaotong University, Nanchang, Jiangxi, China

Abstract

In this paper, we will investigate the natural partial orders on IC-abundant semigroups. After giving some properties and characterizations of natural partial orders on abundant semigroups, we consider IC-abundant semigroups. We prove that an IC-abundant semigroup is locally ample if and only if the natural partial order on the semigroup is compatible with the multiplication.

Keywords

Abundant Semigroup, Natural Partial Order, IC-Abundant Semigroup, Locally Ample

Received: March 4, 2015

Accepted: March 20, 2015

Published online: March 23, 2015

@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/

1. Introduction

The concepts of the natural partial orders on a regular semigroup were introduced by Nambooripad [3] in 1980. As a generalization of regular semigroups in the range of abundant semigroups, El-Qallali and Fountain [1] introduced abundant semigroups. After that, various classes of abundant semigroups are researched (see, [2, 6-14] ). In 1987, Lawson defined natural partial orders on abundant semigroups, and extend Nambooripad’s results. The relation is defined on a semigroup by the rule that if and only if the elements of are related by Green’s relation in some oversemigroup of . The relation is defined dually. In this paper, we shall study the natural partial orders on IC-abundant semigroups by using the notion of majorization of the relations and . Some properties and constructions on IC-abundant semigroups will be described in terms of its natural partial order. We shall proceed as follows: section 2 provides some known results. In section 3, we give some characterizations of the natural partial orders on abundant semigroups. The last section we consider the natural partial orders on IC-abundant semigroups.

2. Preliminaries

Throughout this paper we shall use the notions and notations of [2,4,5]. Here we provide some known results used repeatedly in the sequel. At first, we recall some basic facts about the relation and .

Lemma* II*. 1** **^{1}^{ }Let be a semigroup and . Then the following statements are equivalent:

(1) ();

(2) for all if and only if

As an easy but useful consequence of Lemma II.1, we have

*Corollary** II.1* ^{1 }Letbe a semigroup and .Then the following statements are equivalent:

(1) ();

(2) and for all implies

Evidently, is a right congruence while is a left congruence. In an arbitrary semigroup, we have and. But for regular elements we get if and only if For convenience, we denote by [] a typical idempotent related [related ] to () denotes the class (class ) containing And denotes the set of idempotents of ; denotes the set of regular elements of We denote by the set of all inverses of

A semigroup is called *abundant* if and only if eachclass and eachclass contains at least one idempotent. An abundant semigroup is called *quasi-adequate* if its set of idempotents constitutes a subsemigroup (i.e., its set of idempotents is a band). Moreover, a quasi-adequate semigroup is called *adequate* if its bands of idempotents is a semilattice (i.e., the idempotents commute). An abundant semigroup is called *ample,** *if for all , and Follwing[1], an abundant semigroup is called *idempotent-connected*, for short,* IC*, provided for each and for some , there exists a bijection : such that for all where [resp. ] is the subsemigroup of generated by the set [resp.]. In fact, an ample semigroup is an IC-adequate semigroup and vice versa. A semigroup is called a *locally P-semigroup* if for all , is a P-semigroup. An equivalence relation on is called* -unipotent *if each -class of contains exactly one idempotent. Evidently, an adequate semigroup is both - and - unipotent. It is well known that - unipotent [ resp. - unipotent ] regular semigroup is an orthdox semigroup whose band of idempotents is a right [ resp. left ] regular band ( a band is left [resp. right] regular band if for , [resp. ][ see,4]).

*Definition** II**.1*^{5}^{} Let be an abundant semigroup. We define three relations on , as follows:

(1) and there exists an idempotent such that

(2) and there exists an idempotent such that

(3) i.e. there exist idempotents such that

*Lemma **II.** 2 *^{5} Let be an abundant semigroup and . Then if and only if there exists such that and

*Lemma **II.3*** ^{ 5}** Let be an abundant semigroup. Thenis IC if and only if

*Definition** **II.2 *** ^{ 4} **Let be an equivalence relation on semigroup , be a subset of is called to satisfy - majorization if for any and implies that

3. Properties and Characterizations

The aim of this section is to introduce the natural partial on abundant semigroups, and to give some properties and characterizations of the natural partials on such semigroups.

*PropositionIII**.1 *Let be an abundant semigroup and . Then if and only if there exist such that and

*Proof. *We only prove the sufficiency part. To see this, let such that and Then Hence, by Corollary II.1, and Furthermore, we get Thus . This means that .

*LemmaIII**.1 *Let be an abundant semigroup. If satisfy - majorization, then for any is - unipotent.

*Proof. *Let Then and If , then . Since satisfy - majorization, we have So, is - unipotent.

*LemmaIII**.2 *Let be an abundant semigroup. If is - unipotent, then satisfy - majorization.

*Proof. *Let such that and By the dual argument of Lemma II.2, there exist such that Hence, But, is - unipotent , we have So, that is, satisfy - majorization.

*Proposition III**.2 *Let be an abundant semigroup. Then the following statements are equivalent:

(1) for any is - unipotent;

(2) for any satisfy - majorization;

(3) satisfy - majorization;

(4) satisfy - majorization.

*Proof. *(1)(2) It follows immediately from Lemma Ⅲ.2.

(2)(3) Suppose that (2) holds. Let and . Then and . Hence, and so satisfy - majorization.

(3)(4) Assume that satisfy - majorization. Let and Then there exist such that On the other hand, since is regular, we have such that Again, since

we have that Notice that we get that Similarly, we can prove that and Hence, By the hypothesis, satisfy - majorization, we have that Therefore, That is, satisfy - majorization.

(4)(1) Assume that satisfy - majorization. Let and . Hence, and . By (4), we have and so is - unipotent.

*Theorem **III.** 1 *Let be an abundant semigroup satisfying the regularity condition. Then the following statements are equivalent:

(1) is compatible with respect to ;

(2) is a locally adequate semigroup;

(3) for any satisfies both - and -unipotent;

(4) for any satisfies both - and - majorization;

(5) satisfies both - and - majorization;

(6) satisfies both - and - majorization.

*Proof. *(1)(2) Suppose that is compatible with respect to . Then is a locally inverse semigroup (see, [3]). Noticing that for any is an abundant semigroup, which implies that is a semilattice and We observe that is a semilattice (since is a inverse semigroup ). Thus, is an adequate semigroup, that is, is a locally adequate semigroup.

(2)(3) This is clear.

(3)(4)(5)(6) Follow from Proposition Ⅲ.2.

(6)(1) Suppose that satisfies both - and - majorization. By Proposition Ⅲ.2, we have satisfies both - and -unipotent. But, we have that is a regular semigroup satisfying both - and -unipotent. Furthermore, is an inverse semigroup. Therefore, is a locally inverse semigroup, and so is compatible with respect to (see,[3]).

4. Natural Partial Orders on IC-Abundant Semigroups

In this section, we will consider the natural partial orders on IC-abundant semigroups.

Theorem IV.1 Let be an IC-abundant semigroup. Then the following statements are equivalent:

(1) is right compatible with respect to ;

(2)for any and

(3)for any satisfies both - majorization and the regularity condition;

(4) for any satisfies both - unipotent and the regularity condition.

Proof. (1)(2) Let and By (1), we have that Hence, By Proposition 2.7 of [5], we have Therefore,

(2)(3) Let Then and By (2), we have that

Hence, which implies that satisfies the regularity condition.

Next , we show that satisfies - majorization.

Let such that and By the duality of Lemma II.2, there exist such that Obviously, and Hence, so that Thus Therefore, satisfies - majorization.

(3)(4) It follows from Proposition III.2.

(4)(1) Suppose that (4) holds. Then is an - unipotent semigroup. Hence,

is a right regular band. Let and By Lemma II.2, for ,

there exists such that and Hence, Since by LemmaII.1, we have It is easy to see that Thus since Take We have

and

that is,

Since is a right regular band, by assumpation, we have

Multiplying the prior formula on the right by , we obtain that Hence, and so, Notice that by the fact that is - unipotent, we have that If we multiply this equality on the right by we can obtain that Hence,. That is, Note that , by (4), we observe Hence, Since by Lemma II.2, we have Again, since is an IC-abundant semigroup, we get that . The proof is completed.

*Theorem **IV.2 *Let be an IC-abundant semigroup. Then the following statements are equivalent:

(1) is compatible with respect to ;

(2)for any and

(3) is a locally ample semigroup;

(4) for any satisfies both - and -majorization and the regularity condition.

*Proof. *It follows from Theorem IV.1 and its dual.

5. Conclusions

In this paper, we investigate the natural partial orders on IC-abundant semigroups and give some properties and characterizations of natural partial orders on abundant semigroups by using the notion of majorization. We generalize and strengthen the results of Fountain on abundant semigroups.

Acknowledgements

This work is supported by the National Science Foundation (China) (No. 11261018; No. 11361024), the Natural Science Foundation of Jiangxi Province ( No. 20122BAB201018 ), the Science Foundation of the Education Department of Jiangxi Province (No. GJJ14381;No. KJLD12067).

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