Journal of the
Korean Mathematical Society JKMS

For a pomonoid $S$, let us denote {\bf Pos}-$S$ the category of $S$-posets and $S$-poset maps. In this paper, we consider the slice category {\bf Pos}-$S/B$ for an $S$-poset $B,$ and study some categorical ingredients. We first show that there is no non-trivial injective object in {\bf Pos}-$S/B$. Then we investigate injective objects with respect to the class of regular monomorphisms in this category and show that {\bf Pos}-$S/B$ has enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the category {\bf Pos}-$S/B$ under a particular case where $B$ has trivial action. Among other things, we also prove that the necessary condition for a map (an object) here to be regular injective is being convex and present an example to show that the converse is not true, in general.

Keywords: $S$-poset, slice category, regular monomorphism, injectivity

MSC numbers: Primary 20M30; Secondary 06A11, 18A25, 18G05