Source of ?-Semigroups’ Primeness

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