Source of Γ-Semigroups' Primeness

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