Kleene posets and pseudo-Kleene posets

The concept of a Kleene algebra (sometimes also called lattice) was already generalized by the first author for non-distributive lattices under name pseudo-Kleene algebra. We extend these concepts to posets and show how (pseudo-)Kleene can be characterized identities implications assigned commutative meet-directoids. Moreover, we prove that Dedekind-MacNeille completion poset is finite Further, introduce strict an additional assumption organized into residuated structure. Finally, using so-called twist construction every embedded in some natural way.