Categorical properties of regular monomorphisms of $S$-posets

Abstract

Recall that monomorphisms in some categories such as the category of posets, and the category of topological spaces, are not necessarily embeddings. The notion of regular monomorphism, solves this problem in these two categories. We have the same situation in the category {\bf Pos-$S$} of $S$-posets; that is posets with an action of a pomonoid $S$ which preserves the order. In this category, regular monomorphisms exactly determine sub $S$-posets. In this paper, we study some categorical properties of regular monomorphisms in the category of $S$-posets with action-preserving monotone maps.