Weak Factorization System for Actions of Po-monoids on Posets

Let $S$ be a pomonoid. In this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. One of the main aims of this paper is to draw attention to the notion of weak factorization systems in {\bf Pos}-$S.$ We show that if the identity element of $S$ is the bottom element, then $(\mathcal{C_D}, \mathcal{E_S})$ is a weak factorization system in {\bf Pos}-$S,$ where $\mathcal{C_D}$ and $\mathcal{E_S}$ are the class of down-closed embedding $S$-poset maps and the class of all split $S$-poset epimorphisms, respectively. Among other things, we use a fibrewise notion of complete posets in the category {\bf Pos}-$S/B$ under a particular case where $B$ has trivial action. We get a necessary condition for regular injective objects in {\bf Pos}-$S/B$. Finally, we characterize them under a spacial case, where $S$ is, a pogroup and conclude $(Emb, Top)$ is a weak factorization system in {\bf Pos}-$S$.