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-posets. First we give some necessary and sucient conditions for an S-poset to be a cogenerator. Then we see that under some conditions, regular in- jectivity implies generator and cogenerator. Recalling Birkhoʹs Representation Theorem for algebras, we study subdirectly irre- ducible S-posets and prove this theorem for the category of ordered right acts over an ordered monoid. Among other things, we present the relationship between cogenerators and subdirectly irreducible S-posets.