COGENERATOR AND SUBDIRECTLY IRREDUCIBLE IN THE CATEGORY OF S_POSETS

In this paper we study the notions of cogenerator and subdirectly irreducible in the category of S-poset. First we give some necessary and sufficient conditions for a cogenerator $S$-posets. Then we see that under some conditions, regular injectivity implies generator and cogenerator. Recalling Birkhoff's Representation Theorem for algebra, we study subdirectly irreducible S-posets and give this theorem for the category of ordered right acts over an ordered monoid. Among other things, we give the relations between cogenerators and subdirectly irreducible S-posets.