Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets

Let $S$ be a pomonoid. In this paper, $\mathbf{Pos}$-$S$, 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 $\mathbf{Pos}$-$S$. We show that if $S$ is a pogroup, or the identity element of $S$ is the bottom (or top) element, then $(\mathcal{DU}, \mathrm{SplitEpi})$ is a weak factorization system in $\mathbf{Pos}$-$S$, where $\mathcal{DU}$ and $\mathrm{SplitEpi}$ are the class of du-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 $\mathbf{Pos}$-$S/B$ under a particular case that $B$ has trivial action. We show that every regular injective object in $\mathbf{Pos}$-$S/B$ is topological functor. Finally, we characterize them under a special case, where $S$ is a pogroup.