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 $\Gamma$--semigroups.One of the main results that are proved here is the analogue of the Green's Theorem for $\Gamma$--semigroups, which we call the Green's Theorem for the relation $\mathcal B$ in $\Gamma$--semigroups.Applying our Green's Theorem for relation $\mathcal B$ in $\Gamma$--semigroups, we prove that any bi-ideal of a $\Gamma$--semigroup without zero is minimal if and only if it is a $\Gamma$--subgroup.Further, we prove that, if a $\Gamma$--semigroup $M$ without zero has a cancellable element contained in a minimal bi-ideal $B$ of $M$, then $M$ is a $\Gamma$--group. Finally, we prove that, if for elements $a$, $c$ of a $\Gamma$--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 $(c)_b = (c)_q$.