Transformation semigroups associated to Γ-semigroups

Abstract

The concept of Γ-semigroups is a generalization of semigroups. In this paper, we associate two transformation semigroups to a Γ-semigroup and we call them the left and right transformation semigroups. We prove some relationships between the ideals of a Γ-semigroup and the ideals of its left and right transformation semigroups. Finally, we study some relationships between Green's equivalence relations of a Γ-semigroup and its left (right) transformation semigroup.

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

“…Researchers have extended this line of inquiry to characterise Green's relations within the framework of Γ-semigroups, as evidenced by the contributions in [4,5,7,16]. This body of research, coupled with the insights gleaned from [12], serves as the impetus behind our paper. Within this context, we present, among other findings, an elucidation of the relationships between the Green's equivalence relations of a Γ-semigroup and operator semigroups.…”

“…The authors delved into the intricate relationships between the set of Γ-ideals and operator semigroups, echoing the groundwork laid by Dutta and Adhikari [9]. Additionally, Heidari and Amooshashi [12] linked transformation semigroups to Γ-semigroups, exploring interconnectedness with Γ-ideals and Green's relations based on the idea that a transformation semigroup τ is a function from τ to itself (see [1]). The foundational work of Green [11] in establishing Green's equivalence relations for semigroups has been instrumental in unravelling the structural intricacies of semigroups (for further elaboration, see [6]).…”