Order dense injectivity of S-posets

ýýýIn this paperý, ýtheý notion of injectivity with respect to order dense embeddings in ýýthe category of S S -posetsý, ýposets with a monotone action of aý pomonoid S S on themý, ýis studiedý. ýWe give a criterioný, ýlike the Baer condition for injectivity of modulesý, ýor Skornjakov criterion for injectivity of S -setsý, ýfor the order dense injectivityý. ýAlsoý, ýwe consider such injectivity for S itselfý, and its order dense idealsý. ýFurtherý, ýwe define and study some kinds of weak injectivity with respect to order dense embeddingsý, ýconsider their relations with order dense injectivityý. ýAlso investigate if these kinds of injectivity are preserved or reflected by productsý, ýcoproductsý, ýand direct sums ofýýS S -posetsý.