Abstract

We introduce the notion of ordered quasi--ideals of regular ordered -semigroups and study the basic properties of ordered quasi--ideals of ordered -semigroups. We also characterize regular ordered -semigroups in terms of their ordered quasi--ideals, ordered right -ideals, and left -ideals. Finally, we have shown that (i) a partially ordered -semigroup is regular if and only if for every ordered bi--ideal , every ordered -ideal , and every ordered quasi--ideal , we have and (ii) a partially ordered -semigroup is regular if and only if for every ordered quasi--ideal , every ordered left -ideal , and every ordered right--ideal , we have that .

1. Introduction

Steinfeld [1–3] introduced the notion of a quasi-ideal for semigroups and rings. Since then, this notion has been the subject of great attention of many researchers and consequently a series of interesting results have been published by extending the notion of quasi-ideals to -semigroups, ordered semigroups, ternary semigroups, semirings, -semirings, regular rings, near-rings, and many other different algebraic structures [4–15].

It is a widely known fact that the notion of a one-sided ideal of rings and semigroups is a generalization of the notion of an ideal of rings and semigroups and the notion of a quasi-ideal of semigroups and rings is a generalization of a one-sided ideal of semigroups and rings. In fact the concept of ordered semigroups and -semigroups is a generalization of semigroups. Also the ordered -semigroup is a generalization of -semigroups. So the concept of ordered quasi-ideals of ordered semigroups is a generalization of the concept of quasi-ideals of semigroups. In the same way, the notion of an ordered quasi-ideal of ordered semigroups is a generalization of a one-sided ordered ideal of ordered semigroups. Due to these motivating facts, it is naturally significant to generalize the results of semigroups to -semigroups and of -semigroups to ordered -semigroups.

In 1998, the concept of an ordered quasi-ideal in ordered semigroups was introduced by Kehayopulu [16]. He studied theory of ordered semigroups based on ordered ideals analogous to the theory of semigroups based on ideals. The concept of po--semigroup was introduced by Kwon and Lee in 1996 [17] and since then it has been studied by several authors [18–22]. Our purpose in this paper is to examine many important classical results of ordered quasi--ideals in ordered -semigroups and then to characterize the regular ordered -semigroups through ordered quasi--ideals, ordered bi--ideals and ordered one-sided -ideals.

2. Preliminaries

We note here some basic definitions and results that are relevant for our subsequent results.

Let and be two nonempty sets. Then is called a -semigroup if satisfies for all , , and , . A nonempty subset of a -semigroup is called a sub--semigroup of if for all , and . For any nonempty subsets , of , ,   and  . We also denote , , and , respectively, by , , and . Many classical results of semigroups have been generalized and extended to -semigroups [23–25]. By an ordered -semigroup (also called po--semigroups), we mean an ordered set , at the same time a -semigroup satisfying the following conditions:

Throughout this paper, will stand for an ordered -semigroup unless otherwise stated. An ordered -semigroup is called regular if for each and for each , there exists such that . Equivalent definitions of regular ordered -semigroup are as follows: (i) for each and (ii) for each . Let be an ordered -semigroup and a sub--semigroup of ; then is an ordered -semigroup. Let be a nonempty subset of . Then similarly to [26], we write for some and . We also write by simply if (see [27]). A nonempty subset of an ordered -semigroup is called an ordered right--ideal (left--ideal) of if (), and for any , . is called an ordered -ideal of if it is both a left and a right -ideals of . Also for any , we have that is an ordered left -ideal of and is an ordered right -ideal of [18]. A nonempty subset of is called an ordered quasi--ideal of if (i) and (ii) . A sub--semigroup of an ordered -semigroup is called an ordered bi--ideal of if and for any , .

Let be a nonempty subset of . Then the least right (left) ordered -ideal of containing is given by . If , , we write and , respectively, by and , and , and the ideal generated by is given by . Also, the least quasi--ideal of containing is denoted by . Moreover, we willl need some notations as follows: (i) , where and , (ii) is a set of ordered right -ideals of , (iii) is a set of ordered left -ideals of , and (iv) is a two-sided -ideal of .

Now for any two elements , , we define an operation in as follows:

Further, let be a sub--semigroup of . Then we can easily observe here the following (see [16, 18, 21, 28–30]):(i) for ,(ii)for and , we have ,(iii)for and , we have ,(iv)for and with , we have and ,(v),(vi)for every left (right, two-sided) ideal of , ,(vii)if and are ordered -ideals of , then and are also ideals of ,(viii)for any , is an ideal of .

3. Ordered -Semigroups and Ordered Quasi--Ideals

In this section, we study some classical properties of the ordered -semigroup . We start with the following lemma.

Lemma 1. Let be an ordered -semigroup. Then,(i) is an ordered -semigroup;(ii), , and are sub--semigroups of .

Proof. (i) Suppose , , . Since , we obtain . Next, we have by using . In a similar way, we can show that and therefore . Hence is a -semigroup. Suppose . Then and . Hence is an ordered -semigroup.
(ii) We have that , , and are nonempty subsets of . Suppose , . Then, obviously, we have . Moreover, using we infer that is a left -ideal of ; that is, . Thus is a sub--semigroup of .
Dually, we can prove that is a sub--semigroup of . Since , it follows that is a sub--semigroup of .

Let is an ordered quasi--ideal of . Then, obviously we have . This implies that every one-sided -ideal of an ordered -semigroup is a quasi--ideal of . Thus the class of ordered quasi--ideals of is a generalization of the class of one-sided ordered -ideals of .

Lemma 2. Each ordered quasi--ideal of an ordered -semigroup is a sub--semigroup of .

Proof. Proof is straightforward. In fact, we have .

Lemma 3. For every ordered right -ideal and an ordered left -ideal of an ordered -semigroup , is an ordered quasi--ideal of .

Proof. As and , we obtain , so . Now the fact that is an ordered quasi--ideal of follows from the following:(i),(ii).

Lemma 4. Let be an ordered quasi--ideal of , then one obtains .

Proof. The following relation
Conversely, suppose . Then or and for some , , , , , and , . As is an ordered quasi--ideal of , the former case implies that and the latter case implies that . Therefore .

We recall here that if is a nonempty subset of an ordered -semigroup , then we write the least quasi-ideal of containing by . If , we write by .

Theorem 5. Suppose is an ordered -semigroup. Then one has the following:(i)for every , ,(ii)let , .

Proof of (i). Suppose . Using Lemma 3, is a quasi--ideal of containing ; therefore , and by Lemma 4, we obtain
Hence .

Proof of (ii). Its proof can be given as (i).

The notion of a bi--ideal of -semigroups is a generalization of the notion of a quasi--ideal of -semigroups. Similarly, the class of ordered quasi--ideals of ordered -semigroups is a particular case of the class of ordered bi--ideals of ordered -semigroups. This is what we have shown in the following result.

Theorem 6. Suppose is a two-sided ordered -ideal of an ordered -semigroup and is a quasi--ideal of ; then is an ordered bi--ideal of .

Proof. Since is an ordered quasi--ideal of and , we obtain and There exists such that and .
Therefore, Hence applying these facts together with Lemma 2, we have shown that is an ordered bi--ideal of .

4. Regular Ordered -Semigroups and Ordered Quasi--Ideals

In this section, we use the concept of ordered quasi--ideals to characterize regular ordered -semigroups.

Lemma 7. Let be an ordered -semigroup. Then the ordered sub--semigroup of generated by and is in the following form:

Proof. One can easily see that
Suppose . Then the conditions that arise are as follows: (i) : in this condition by Lemma 1, we obtain ; (ii) , : in this condition, by also Lemma 1; (iii) , : in this condition, is an ordered -ideal of , so ; (iv) , : in this condition, in . Therefore for any , where , using (i)–(iv), there arise three conditions as follows.(i)′If , then .(ii)′If , then .(iii)′If and , where , then . Hence the lemma holds.

Theorem 8. Let be an ordered -semigroup. Then the following assertions on are equivalent.(i) is a regular ordered -semigroup.(ii)For every ordered left -ideal and every ordered right -ideal , one has (iii)For every ordered right -ideal and ordered left -ideal of ,(1) ,(2),(3) is an ordered quasi--ideal of .(iv) and are ordered idempotent -semigroups and is the sub--semigroup of generated by and .(v) is a regular ordered sub--semigroup of the -semigroup .(vi)Every ordered quasi--ideal of is given by .(vii) is a regular sub--semigroup of the ordered -semigroup of .

Proof. Suppose and are ordered right and left -ideals of , respectively; then we have Let be regular; we need to prove only that . Suppose . Since is regular, we obtain for some and , , and so and ; therefore . Therefore , and thus .
is an ordered quasi--ideal of that follows directly from Lemma 3 and the condition (ii). As the ordered two-sided -ideal of is generated by , the condition (ii) implies that Conversely, suppose . Then for and . From , we have , where or for some and . Therefore or for ; thus . Thus , so that . Similarly we can prove that dually.
The conditions , in (iii) and Lemma 7 show that and are idempotent -semigroups, respectively. Applying (iii) , we obtain ; therefore in .
Conversely, suppose . Then is the ordered left -ideal of generated by . The condition (iii) implies that
We can dually prove that . Therefore using these facts and Lemma 4, it follows that Therefore for , we have , and the condition (iii) together with (a) implies that
Moreover, by the assertion (iii) (2), we have and
Therefore . Dually, we can prove that
From these facts, (a) and (b), we obtain by Lemma 7. Therefore . Hence in .
It is a consequence of Lemma 7.
By , we have (b) and (c). Suppose , are two ordered quasi--ideals of . Then is the least ordered left -ideal of containing . Then the condition (iii) implies that
Dually one can prove that . These facts together with (b) show that
By Theorem 5 (ii), is an ordered quasi--ideal of ; therefore . Hence is a sub--semigroup of . For every , by (c), we obtain , and so , where . Thus is a regular sub -semigroup of .
Suppose is an ordered quasi--ideal of . Applying the condition (iv), there is an ordered quasi--ideal of so that, by Lemma 4,
and therefore .
It is straightforward.
For every , using Theorem 5, is an ordered quasi--ideal of containing . By (vii), there exists so that
Hence is a regular ordered -semigroup.