RESEARCH ARTICLE
Weak factorization systems and fibrewise regular injectivity for actions of pomonoids on posets
Farideh Farsad,
Ali Madanshekaf Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, P. O. Box 35131-19111, Semnan, Iran
Abstract:
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.
Keywords:
$S$-poset, slice category, regular injectivity, weak factorization system.
MSC: 06F05,
18A32,
18G05,
20M30,
20M50 Received: 21.04.2015
Language: English
Fulltext:
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