Injective and projective poset acts

Abstract

In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {\bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {\bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {\bf PosAct}-$S$. Finally, we study injectivity and projectivity in {\bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {\bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.