| Summary: | The aim of this study is to develop the theory of prime ideals in ordered semigroups. First, to ensure the existence of prime ideals, we study a class of ordered semigroups which will be denoted by SIP{{\mathbb{S}}}_{IP}. And then we introduce the hull-kernel topology for the prime ideals P(S){\mathcal{P}}\left(S) and the topological properties like separation axioms, compactness and connectedness are studied. Finally, we focus on the subspace ℳ(S,I){\mathcal{ {\mathcal M} }}\left(S,I), minimal prime ideals containing the ideal II in an ordered semigroup SS. We investigate topological properties of this subspace and connections between this subspace and the ordered semigroup SS.
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