The Source of $\Gamma$-semiprimeness on $\Gamma$-semigroups

Abstract

Let $S$ be a $\Gamma$-semigroup with zero. We define the $S_{S}^{\Gamma}$ subset of $S$ as $S_{S}^{\Gamma}=\{a\in S \mid a\Gamma (S\Gamma a)=(0)\}.$ This set is called the source of $\Gamma$-semiprimeness of $S$. In this study, we examined some properties of $S_{S}^{\Gamma}$ set and defined $\lvert S_{S}^{\Gamma}\rvert$-idempotent , $\lvert S_{S}^{\Gamma}\rvert$-regular and $\lvert S_{S}^{\Gamma}\rvert$-reduced $\Gamma$-semigroups. We then obtained some results for these newly defined semigroups.

Keywords

• Source of $\Gamma$-semiprimeness
• $\lvert S_{S}^{\Gamma}\rvert$-idempotent $\Gamma$-semigroup
• $\lvert S_{S}^{\Gamma}\rvert$-regular $\Gamma$-semigroup
• $\lvert S_{S}^{\Gamma}\rvert$-reduced $\Gamma$-semigroup

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