Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for S-posets, posets with an action of a pomonoid S on them. They gave a criterion for dc-injectivity and studied such injectivity for S itself, and for its poideals. In this paper, we give results about dc-injectivity of S-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of T-injectivity for S-posets, where S = T ∪ { ˙ 1} and 1 is externally adjoined to the posemigroup T, we find some classes of pomonoids such that for S-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of S-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of S-posets for some classes of pomonoids S.
