The relation B and Minimal Bi-ideals in Gamma-semigroups

Abstract

In this paper we introduce the relation $\mathcal{B}$ ``to generate the same principal bi-ideal'' in $\mathrm{\Gamma }$--semigroups.One of the main results that are proved here is the analogue of the Green's Theorem for $\mathrm{\Gamma }$--semigroups, which we call the Green's Theorem for the relation $\mathcal{B}$ in $\mathrm{\Gamma }$--semigroups.Applying our Green's Theorem for relation $\mathcal{B}$ in $\mathrm{\Gamma }$--semigroups, we prove that any bi-ideal of a $\mathrm{\Gamma }$--semigroup without zero is minimal if and only if it is a $\mathrm{\Gamma }$--subgroup.Further, we prove that, if a $\mathrm{\Gamma }$--semigroup $M$ without zero has a cancellable element contained in a minimal bi-ideal $B$ of $M$, then $M$ is a $\mathrm{\Gamma }$--group. Finally, we prove that, if for elements $a$, $c$ of a $\mathrm{\Gamma }$--semigroup without zero we have $a\mathcal{D}c$ and the principal bi-ideal $(a{)}_b$ and principal quasi-ideal $(a{)}_q$ are minimal, then $(a{)}_b=(a{)}_q$ and the principal bi-ideal $(c{)}_b$ and the principal quasi-ideal $(c{)}_q$ are minimal too, and $(c{)}_b=(c{)}_q$