Equivariant completeness and regular injectivity of S-posets

Abstract

In this paper, considering the actions of a pomonoid S on posets, namely S-posets, we study some relations between equivariant completeness and regular injectivity of S-posets which lead to some homological classification results for pomonoids. In particular, we show that regular injectivity implies equivariant completeness, but the converse is true only if S is left simple. Finally, it is proved that regularly injective S-posets are exactly the complete and cofree-retract ones. Among other results, we also see that the Skornjakov and Baer criteria fail for regular injectivity of S-posets.