Weak Factorization System for Actions of Po-monoids on Posets

Original · EN

Let S be a pomonoid. In this paper, 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 Pos-S. We show that if the identity element of S is the bottom element, then (CD, Eₛ) is a weak factorization system in Pos-S, where CD and Eₛ 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 Pos-S/B under a particular case where B has trivial action. We get a necessary condition for regular injective objects in 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 Pos-S.