Source of $\Gamma$-Semigroups' Primeness

This study aims to define the source of a $\Gamma$-semigroup $S$'s primeness and to research some of its basic properties and the $P_{S_{\Gamma}}$ subset of S as $P_{S_{\Gamma}}=\{b\in S : S\Gamma S \Gamma b=(0)\}$, for the $\Gamma$-semigroup $S$ with zero. Afterward, it investigates the relationships between $\lvert P_{S_{\Gamma}}\rvert$-reduced, $\lvert P_{S_{\Gamma}}\rvert$-idempotent, $\lvert P_{S_{\Gamma}}\rvert$-strongly idempotent, and $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup structures as follows: (\emph{i}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup, (\emph{ii}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-reduced $\Gamma$-semigroup, (\emph{iii}) If $S$ is an idempotent (regular, reduced) $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-idempotent (regular, reduced) $\Gamma$-semigroup, (\emph{iv}) If $S$ is a $\lvert P_{S_{\Gamma}}\rvert$- strongly idempotent (regular) $\Gamma$-semigroup, then $A$ is a $\lvert P_{A_{\Gamma}}\rvert$- strongly idempotent (regular) $\Gamma$-semigroup, and (\emph{v}) If $S$ is a commutative $\lvert P_{S_{\Gamma}}\rvert$-regular $\Gamma$-semigroup, then $S$ is a $\lvert P_{S_{\Gamma}}\rvert$-reduced $\Gamma$-semigroup. Moreover, this study explores the connections between the source of $\Gamma$-primeness of a $\Gamma$-semigroup and the aforementioned $\Gamma$-semigroup structures and clarifies some theoretical parts of the study with several examples.