Order dense injectivity of $S$-posets

Abstract

‎‎‎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 of‎‎$S$-posets‎.