Retractable Compact Directed Complete Poset (Acts)

Taking domains in the one hand and actions of a semigroup (automaton) on the other, as two crucial notions in mathematics as well as in computer science, we consider the notion of compact directed complete poset (acts), and investigate the interesting notion of absolute retractness for such ordered structures. As monomorphisms and embeddings for domain acts are different notions, we study absolute retractness with respect to both the class of monomorphisms and that of embeddings for compact directed complete poset (acts). We characterize the absolutely retract compact dcpos as complete compact chains. Also, we give some examples of compact directed complete poset acts which are (ε-)absolutely retract (with respect to embeddings) and show that completeness is not a sufficient condition for (ε-)absolute retractness.