Pure Ideals in Ordered Semigroups

Abstract

In this paper the concepts of pure ideals, weakly pure ideals and purely prime ideals in ordered semigroups are introduced. We obtain some characterizations of pure ideals and prove that the set of all pure prime ideals is topologized. 1. Introduction and Preliminaries In (1), Ahsan and Takahashi introduced the notions of pure ideals and purely prime ideals in a semigroup without order. Recently, Bashir and Shabir (3) dened the concepts of pure ideals, weakly pure ideals and purely prime ideals in a ternary semigroup. The authors gave some characterizations of pure ideals and showed that the set of all purely prime ideals of a ternary semigroup is topologized. In this paper, we do in the line of Bashir and Shabir. We introduce the concepts of pure ideals, weakly pure ideals and purely prime ideals on an ordered semigroup. We characterize pure ideals and prove that the set of all purely prime ideals of an ordered semigroup is topologized. Note that the results on semigroups without order become then special cases. For the rest of this section, we recall some denitions and results used through- out the paper. A semigroup S with an order relation ≤ is called an ordered semigroup ((2), (5)) if for x;y;z ∈ S, x ≤ y implies zx ≤ zy and xz ≤ yz. An element 0 of S is called a zero element of S if 0x = x0 = 0 for all x ∈ S and 0 ≤ x for all x ∈ S. A nonempty subset A of S is called a subsemigroup of S if xy ∈ A for all x;y ∈ A. Note that every subsemigroup of S is an ordered semigroup under the order relation on S. For nonempty subsets A and B of S, let AB = {xy | x ∈ A;y ∈ B}. For x ∈ S, let Ax = A{x} and xA = {x}A. A nonempty subset A of S is a subsemigroup of S if and only if AA ⊆ A.