Order dense injectivity of S-posets

‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense 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 the order dense injectivity‎. ‎Also‎, ‎we consider such injectivity for S itself‎, and its order dense ideals‎. ‎Further‎, ‎we define and study some kinds of weak injectivity with respect to order dense embeddings‎, ‎consider their relations with order dense injectivity‎. ‎Also investigate if these kinds of injectivity are preserved or reflected by products‎, ‎coproducts‎, ‎and direct sums of‎‎S-posets‎.

Keywords:

regular monomorphism , order dense sub S-poset , order dense injective