A Source of Semiprimeness on Inverse and Completely Regular Semigroups

We define |S_S|-inverse semigroup and |S_S|-completely regular semigroup structures that are not encountered in the literature. They are studied in our work for the first time. The properties of these new semigroup types and their relations with the other semigroup types, such as |S_S|-idempotent, |S_S|-regular, |S_S|-nonzero, and |S_S|-reduced semigroups, are discussed. Furthermore, the relationship between the source of semiprimeness and the semiprime ideal is analyzed. It is also shown that the intersection of all semiprime ideals is equal to the source of semiprimeness. Moreover, if a function is surjective, then it can be shown that the analyzed two structures are preserved under homomorphism.