AlgebraVolume 2013 (2013), Article ID 565848, 7 pageshttp://dx.doi.org/10.1155/2013/565848

Research Article

On Ordered Quasi-Gamma-Ideals of Regular Ordered Gamma-Semigroups

M. Y. Abbasi and Abul Basar

Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India

Received 31 March 2013; Accepted 7 October 2013

Academic Editor: Sorin Dascalescu

Copyright © 2013 M. Y. Abbasi and Abul Basar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Lemma 9. Every two-sided ordered -ideal of a regular ordered -semigroup is a regular sub--semigroup of .

Proof. Suppose . As is regular, there exists so that, for , we have As , we observe that .

Theorem 10. Suppose is a regular ordered -semigroup. Then the following statements are true. (i)Every ordered quasi--ideal of can be expressed as follows: where and are, respectively, the ordered right and left -ideals of generated by .(ii)Let be an ordered quasi--ideal of ; then .(iii)Every ordered bi--ideal of is an ordered quasi--ideal of .(iv)Every ordered bi--ideal of any ordered two sided--ideal of is a quasi--ideal of .(v)For every , and , , one obtains

Proof. Because is a regular ordered -semigroup, then by Lemma 4 and Theorem 8, the statement (i) is done. Since is always true, we need to show that . We have that is also an ordered quasi--ideal of by Theorem 8. Moreover we have the following equation:Suppose is an ordered bi--ideal of . Then is an ordered left -ideal and is an ordered right -ideal of . Applying Theorem 8, we obtain Therefore is an ordered quasi--ideal of .Suppose is a two-sided ordered -ideal of and is an ordered bi--ideal of . By the relation (iii) and Lemma 9, is an ordered quasi--ideal of ; therefore using Theorem 6, is an ordered bi--ideal of . Also from the relation (iii) again, we obtain as an ordered quasi--ideal of .Lastly, suppose , . Because is regular and is an ordered quasi--ideal of , using Theorem 8, we obtain Dually, we can prove that for all , .

Theorem 11. A partially ordered -semigroup is regular if and only if for every ordered bi--ideal , every ordered -ideal , and every ordered quasi--ideal , one has

Proof. Let be regular. Then for any there exists such that Hence , where .Conversely, let for every ordered bi--ideal , every ordered -ideal , and every ordered quasi--ideal of . Suppose . Let and be the ordered bi--ideal and ordered quasi--ideal of generated by , respectively. So we have the following: Hence is regular.

Next consider in place of in Theorem 11 to obtain the following.

Corollary 12. An ordered -semigroup is regular if and only if for every ordered bi--ideal , every ordered -ideal , and every right -ideal of ,

Theorem 13. A partially ordered -semigroup is regular if and only if for every ordered quasi--ideal , every ordered left -ideal , and every ordered right--ideal , one has

Proof. Let be regular; then for any , there exists such that , for . Hence .Conversely, let for every ordered right -ideal , every ordered quasi--ideal , and every ordered left -ideal of . Suppose . So we have So for , or for some . If , then . If for some , then . So, finally we obtain . Hence is regular.

Corollary 14. If one considers an ordered left -ideal (or an ordered right -ideal ) in place of the ordered quasi--ideal in Theorem 13, one obtains

Acknowledgment

The authors are grateful to the referee for the useful comments and valuable suggestions.

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