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.1⁵ 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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