Injectivity of directed complete poset acts

Action of a monoid on a set have always been of interest for mathematicians and computer scientists. On the other hand, Domain Theory, which studies directed complete partially ordered sets, was introduced by Scott in 1970 when he explored the possibility of using ordered topological spaces to give meaning first to typed and then untyped λ-calculi.

In this paper, combining the above two notions, we consider actions of a monoid on domains and study the algebraic notion of injectivity with respect to monomorphisms and embeddings in the category so obtained.