Absolute retractness of automata on directed complete posets

The notion of retractness, which is about having left inverses (reflection) for monomorphisms, is crucial in most branches of mathematics. One very important notion related to it is injectivity, which is about extending morphisms to larger domains and plays a fundamental role in many areas of mathematics as well as in computer science, under the name of complete or partial objects. Absolute retractness is tightly related to injectivity and is in fact equivalent to it in many categories. In this paper, combining the two important notions of actions of semigroups and directed complete posets, which are both crucial abstraction and useful in mathematics as well as in computer science, we consider the category Dcpo-[Formula: see text] of actions of a directed complete semigroup on directed complete posets, and study absolute retractness with respect to both monomorphisms and embeddings in this category. Among other things, we show that absolute retract ([Formula: see text]-)dcpo’s are complete but the converse is not necessarily true. Investigating the converse, we find that if we add the property of being a countable chain to completeness, over some kinds of dcpo-monoids such as dcpo-groups and commutative monoids, we get absolute retractness. Furthermore, we show that there are absolute retract [Formula: see text]-dcpo’s, which are not chains.