چکیده
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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.
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