On intra-regular and some left regular $\Gamma$-semigroups

Abstract

We characterize the intra-regular $\Gamma$-semigroups and the left regular $\Gamma$-semigroups $M$ in which $x\Gamma M\subseteq M\Gamma x$ for every $x\in M$ in terms of filters and we prove, among others, that every intra-regular $\Gamma$-semigroup is decomposable into simple components, and every $\Gamma$-semigroup $M$ for which $x\Gamma M\subseteq M\Gamma x$ is left regular, is decomposable into left simple components.