The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M :(1) Every ordered semilattice congruences is a regular congruence.(2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congruence ρ on M .(3) The regular congruences are not ordered semilattice congruences in general.
PreliminariesRecall that a semilattice congruence N on an ordered semigroup S is an ordered semilattice congruence on S. Thus the S/N endowed with the multiplication (x) N · (y) N = (xy) N is still a semigroup. If we define (x) N (y) N if and only if (x) N = (xy) N , then it can be easily seen that (S/N ; ·, ) is an ordered semigroup. We now call this semigroup (S/N ; ·, ) the natural ordered semigroup induced by the ordered semilattice congruence N on S. In 1997, Xie and Wu [1] proved that the semilattice congruence N on an ordered semigroup S is the least regular semilattice congruence. In 2004, Dutta and Adhikari [2] introduced the concepts of ordered Γ-semigroups. In 2004, Xu and Ma [3] showed that there exists an orderpreserving bijection between the set of all prime ideals of the ordered semigroup (S; ·, ≤) and the set of all prime ideals of (S/N ; ·, ). Moreover, they gave some necessary and sufficient conditions for the natural ordered semigroup (S/N ; ·, ) to be a chain. In 2006, Siripitukdet and Iampan [4] characterized the relationship among the (ordered) filters, (ordered) s-prime ideals and (ordered) semilattice congruences in ordered Γ-semigroups and gave some characterizations of semilattice congruence and ordered semilattice congruence on ordered Γ-semigroups. They proved that
Abstract. The motivation mainly comes from the conditions of the (ordered) ideals to be prime or semiprime that are of importance and interest in (ordered) semigroups and in (ordered) Γ-semigroups. In 1981, Sen [8] has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. The concept of ordered ideal extensions in ordered Γ-semigroups was introduced in 2007 by Siripitukdet and Iampan [12]. Our purpose in this paper is to introduce the concepts of the ordered n-prime ideals and the ordered n-semiprime ideals in ordered Γ-semigroups and to characterize the relationship between the ordered n-prime ideals and the ordered ideal extensions in ordered Γ-semigroups.
PreliminariesIn 1981, the concept and notion of the Γ-semigroups was introduced by Sen [8]. In 1997, Kwon and Lee [5] introduced the concepts of the weakly prime ideals and the weakly semiprime ideals in ordered Γ-semigroups and gave some characterizations of the weakly prime ideals and the weakly semiprime ideals in ordered Γ-semigroups analogous to the characterizations of the weakly prime ideals and the weakly semiprime ideals in ordered semigroups considered by Kehayopulu [3]. In 1998, Kwon and Lee [4] introduced the ideals and the weakly prime ideals in ordered Γ-semigroups and gave some characterizations of the ideals and the weakly prime ideals in ordered Γ-semigroups analogous to the characterizations of the ideals and the weakly prime ideals in ordered semigroups considered by Kehayopulu [3]. In 1999, Lee and Kwon [6] gave two new characterizations of the weakly prime ideals in ordered semigroups. They proved two theorems as follow: Let a be a quasi-completely regular element of an ordered semigroup S. If there exists an ideal not containing a, then there exists a weakly prime ideal not containing a. Let P * be the intersection of weakly prime ideals of an ordered semigroup S, a ∈ P * and I be any proper
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