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

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.