Injectivity is one of the most central notions in algebra, as well as in many other branches of mathematics and the study of injectivity with respect to di erent classes of monomorphisms is crucial in almost all categories. Down closed monomorphisms and injectivity with respect to these monomorphisms were fi rst introduced and studied by the authors for S-posets over the pomonoid S. They gave a criterion for down closed injectivity and studied such injectivity for S itself, and its (po)ideals. In this paper, we study more dc-injectivity of S-posets and some homological classi cation of pomonoids and pogroups are obtained. More precisely, we introduce and characterize some pomonoids over which dc-injectivity is equivalent to having a zero top element.
