Connections of Green's relations of a Γ-semigroup with operator semigroups

Abstract

The theory of Γ-semigroups is an extension of the semigroup theory. In this paper, we examine the left operator semigroup L and the right operator semigroup R via modified definition of Γ-semigroup and deduce some results of operator semigroups acting on a Γ-semigroup. Further, we study some relationships between Green’s equivalence relations of a Γ-semigroup and its left (right) operator semigroup. In particular, we show that if two elements of a Γ-semigroup S are L(R)-related, then the two elements of L(R) resulting from S for every α ∈ Γ are also L(R)-related. Also, we describe that if two elements of S are α and β-idempotent such that the two elements are R-related in L, then their R-relation holds in S for some α, β ∈ Γ.