Hindawi Publishing Corporation
Algebra
Volume 2013, Article ID 565848, 7 pages
http://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
Correspondence should be addressed to Abul Basar; basar.jmi@gmail.com
Received 31 March 2013; Accepted 7 October 2013
Academic Editor: Sorin Dascalescu
Copyright © 2013 M. Y. Abbasi and A. Basar. his 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.
We introduce the notion of ordered quasi-Γ-ideals of regular ordered Γ-semigroups and study the basic properties of ordered
quasi-Γ-ideals of ordered Γ-semigroups. We also characterize regular ordered Γ-semigroups in terms of their ordered quasi-Γ-
ideals, ordered right Γ-ideals, and let Γ-ideals. Finally, we have shown that (i) a partially ordered Γ-semigroup is regular if and
only if for every ordered bi-Γ-ideal , every ordered Γ-ideal , and every ordered quasi-Γ-ideal , we have ∩∩⊆(ΓΓ]
and (ii) a partially ordered Γ-semigroup is regular if and only if for every ordered quasi-Γ-ideal , every ordered let Γ-ideal ,
and every ordered right-Γ-ideal , we have that ∩∩⊆(ΓΓ].
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 conse-
quently a series of interesting results have been published by
extending the notion of quasi-ideals to Γ-semigroups,
ordered semigroups, ternary semigroups, semirings, Γ-semi-
rings, regular rings, near-rings, and many other diferent
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 Γ-semigroups is a generalization of
semigroups. Also the ordered Γ-semigroup is a generalization
of Γ-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 general-
ization of a one-sided ordered ideal of ordered semigroups.
Due to these motivating facts, it is naturally signiicant to
generalize the results of semigroups to Γ-semigroups and of
Γ-semigroups to ordered Γ-semigroups.
In 1998, the concept of an ordered quasi-ideal in ordered
semigroups was introduced by Kehayopulu [16]. He stud-
ied theory of ordered semigroups based on ordered ideals
analogous to the theory of semigroups based on ideals.
he concept of po-Γ-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-
Γ-ideals in ordered Γ-semigroups and then to characterize
the regular ordered Γ-semigroups through ordered quasi-Γ-
ideals, ordered bi-Γ-ideals and ordered one-sided Γ-ideals.
2. Preliminaries
We note here some basic deinitions and results that are
relevant for our subsequent results.
Let and Γ be two nonempty sets. hen is called a Γ-
semigroup if satisies () = () for all , , ∈
and , ∈Γ. A nonempty subset of a Γ-semigroup is
called a sub-Γ-semigroup of if ∈ for all , ∈ and
∈Γ. For any nonempty subsets , of , Γ = { :
∈, ∈ and ∈ Γ}. We also denote {}Γ, Γ{}, and
{}Γ{}, respectively, by Γ, Γ, and Γ. Many classical
results of semigroups have been generalized and extended
to Γ-semigroups [23–25]. By an ordered Γ-semigroup