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.

中文翻译：

---

关于交换半群的谱

本研究旨在研究交换半群 S 的素理想上的 Zariski 拓扑，用 Spec(S) 表示。首先，我们证明拓扑空间 X 同胚于某些可交换半群 S 的 Spec(S) 当且仅当 X 是可以纯粹用拓扑术语描述的 SS 空间。接下来，我们证明交换半群的范畴和 SS 空间的范畴之间存在一个附加。我们进一步证明，交换幂等半群的范畴与 SS 空间的范畴是对偶等价的。