ON THE ORDERED n-PRIME IDEALS IN ORDERED Γ-SEMIGROUPS

Abstract

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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