A characterization of ordered Γ-semigroups by ordered (m,n)-Γ-ideals

Abstract

In this paper, we study (m,n)-regular ordered Γ-semigroups through ordered (m,n)-Γ-ideals. It is shown that if (S,Γ,·,≤) is an ordered Γ-semigroup; m,n are non-negative integers and A(m,n) is the set of all ordered (m,n)-Γ-ideals of S. Then, S is (m,n)-regular⇐⇒ ∀A ∈ A(m,n), A = (AmΓSΓAn]. It is also proved that if (S,Γ,·,≤) is an ordered Γ-semigroup and m,n are nonnegative integers and R(m,0) and L(0,n) is the set of all (m,0)-Γideals and (0,n)-Γ-ideals of S, respectively. Then, S is (m,n)-regular ordered Γ semigroup ⇐⇒∀R ∈R(m,0)∀L ∈L(0,n),R∩L = (RmΓL∩RΓLn].