On Ordered-Primeness and Ordered-Semiprimeness in Ordered Semigroups

In this paper, we investigate the structure of the source of ordered-semiprimeness and the source of ordered-primeness within the class of ordered semigroups. We focus on extending the sources of semiprimeness and primeness and results that have previously been established in more restrictive settings. By working in this broader framework, we establish generalizations of these results and show that many of the underlying structural properties are preserved under suitable conditions. Our approach not only unifies existing results but also provides a deeper understanding of the behavior of these properties in a more general context.