On Ordered Ideal Extensions in po-Γ-Semigroups.

In this paper, we introduce the concept of the extension ≪ A; I ≫ of an ordered ideal I by a set A in a po-Γ-semigroup M and many properties of ≪ A; I ≫ are provided. Moreover, the equivalence relation ΦI on M defined by (x; y) ∈ ΦI ⇔≪ x; I ≫=≪ y; I ≫ is considered. It is shown that if M is commutative and I is an ordered s-semiprime ideal, then ΦI is an ordered semilattice congruence. Moreover, if I is an ordered prime ideal of M, then (x; y) ∈ ΦI ⇔ x; y ∈ I or x; y ∉ I. [ABSTRACT FROM AUTHOR]