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