No need to say that the study of injectivity with respect to different classes of monomorphisms is crucial in any category. In this paper, the notion of injectivity with respect to down closed embeddings in the category of S-posets, posets with a monotone action of a pomonoid S on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity of S-sets, for down closed injectivity. Also, we consider such injectivity for S itself, and its (po)ideals. Further, we investigate if this kind of injectivity is preserved or reflected by products, coproducts, direct sums, and order sums of S-posets. Finally, we define and study some kinds of weak injectivity with respect to down closed embeddings.
