On the spectra of commutative semigroups.

This study aims to investigate the Zariski topology on the prime ideals of a commutative semigroup S, denoted by Spec(S). First, we show that a topological space X is homeomorphic to Spec(S) for some commutative semigroup S if and only if X is an SS-space that can be described purely in topological terms. Next, we show that an adjunction exists between the category of commutative semigroups and that of SS-spaces. We further show that the category of commutative idempotent semigroups is dually equivalent to that of SS-spaces. [ABSTRACT FROM AUTHOR]