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

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-Γideals, ordered right Γ-ideals, and left Γ-ideals. Finally, we have shown that (i) a partially ordered Γ-semigroup S is regular if and only if for every ordered bi-Γ-ideal B, every ordered Γ-ideal I, and every ordered quasi-Γ-ideal Q, we have B ∩ I ∩ Q ⊆ (BΓIΓQ] and (ii) a partially ordered Γ-semigroup S is regular if and only if for every ordered quasi-Γ-ideal Q, every ordered left Γ-ideal L, and every ordered right-Γ-ideal R, we have that R ∩ Q ∩ L ⊆ (RΓQΓL].

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 quasiideal of semigroups and rings is a generalization of a onesided 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 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 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 studied theory of ordered semigroups based on ordered ideals analogous to the theory of semigroups based on ideals.The 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][19][20][21][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-Γideals, ordered bi-Γ-ideals and ordered one-sided Γ-ideals.

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

Ordered Γ-Semigroups and Ordered Quasi-Γ-Ideals
In this section, we study some classical properties of the ordered Γ-semigroup .We start with the following lemma.
Let   = { :  is an ordered quasi-Γ-ideal of }.Then, obviously we have   ∪   ⊆   ⊆   .This implies that every one-sided Γ-ideal of an ordered Γ-semigroup is a quasi-Γ-ideal of .Thus the class of ordered quasi-Γ-ideals of  is a generalization of the class of one-sided ordered Γ-ideals of .

Proof of (ii). Its proof can be given as (i).
The notion of a bi-Γ-ideal of Γ-semigroups is a generalization of the notion of a quasi-Γ-ideal of Γ-semigroups.Similarly, the class of ordered quasi-Γ-ideals of ordered Γsemigroups is a particular case of the class of ordered bi-Γideals of ordered Γ-semigroups.This is what we have shown in the following result.Theorem 6. Suppose  is a two-sided ordered Γ-ideal of an ordered Γ-semigroup  and  is a quasi-Γ-ideal of ; then  is an ordered bi-Γ-ideal of .
Hence applying these facts together with Lemma 2, we have shown that  is an ordered bi-Γ-ideal of .

Regular Ordered Γ-Semigroups and Ordered Quasi-Γ-Ideals
In this section, we use the concept of ordered quasi-Γ-ideals to characterize regular ordered Γ-semigroups.
Proof.(i) ⇒ (ii) Suppose  and  are ordered right and left Γ-ideals of , respectively; then we have Let  be regular; we need to prove only that  ∩  ⊆ (Γ].
(v) ⇒ (vi) Suppose  is an ordered quasi-Γ-ideal of .Applying the condition (iv), there is an ordered quasi-Γ-ideal  1 of  so that, by Lemma 4, Hence  is a regular ordered Γ-semigroup.
Lemma 9. Every two-sided ordered Γ-ideal  of a regular ordered Γ-semigroup  is a regular sub-Γ-semigroup of .
Theorem 10.Suppose  is a regular ordered Γ-semigroup.Then the following statements are true.
(i) Every ordered quasi-Γ-ideal of  can be expressed as follows: where  and  are, respectively, the ordered right and left Γ-ideals of  generated by .
(iv) Every ordered bi-Γ-ideal of any ordered two sided-Γideal of  is a quasi-Γ-ideal of .
(v) For every  1 ,  2 ∈   and  1 ,  2 ∈   , one obtains Proof.Because  is a regular ordered Γ-semigroup, then by Lemma 4 and Theorem 8, the statement (i) is done.Since (ΓΓ] ⊆ (Γ] is always true, we need to show that (Γ] ⊆ (ΓΓ].We have that (Γ] is also an ordered quasi-Γ-ideal of  by Theorem 8. Moreover we have the following equation: Suppose  1 is an ordered bi-Γ-ideal of .Then (Γ 1 ] is an ordered left Γ-ideal and ( 1 Γ] is an ordered right Γ-ideal of .Applying Theorem 8, we obtain Therefore  1 is an ordered quasi-Γ-ideal of .Suppose  is a two-sided ordered Γ-ideal of  and  is an ordered bi-Γ-ideal of .By the relation (iii) and Lemma 9,  is an ordered quasi-Γ-ideal of ; therefore using Theorem 6,  is an ordered bi-Γ-ideal of .Also from the relation (iii) again, we obtain  as an ordered quasi-Γ-ideal of .