On the Generators in the Category of Actions of Pomonoids on Posets and its Slices

TL;DR

This paper investigates the structure of the category of S-posets for a pomonoid S, characterizing projectives and exploring homological properties like regular injectivity and their relationships with generators and cyclic projectives.

Contribution

It provides new characterizations of pomonoids where all projectives are generators or free, and explores homological classifications related to regular injectivity in the category.

Findings

Characterization of pomonoids with all projectives as generators or free

Relationships between regular injectivity and generators in slice categories

Homological classification results for pomonoids

Abstract

Let $S$ be a pomonoid, in this paper, {\bf Pos}-$S$, the category of $S$-posets and $S$-poset maps, is considered. First, we characterize some pomonoids on which all projectives in this category are generator or free. Then, we study regular injectivity and weakly regularly $d$-injectivity which lead to some homological classification results for pomonoids. Among other things, we get some relationships between regular injectivity in the slice category {\bf Pos}-$S/B_S$ and generators or cyclic projectives in {\bf Pos}-$S$.