Kleene posets and pseudo-Kleene posets
Abstract
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.
Cite
@article{arxiv.2006.04417,
title = {Kleene posets and pseudo-Kleene posets},
author = {Ivan Chajda and Helmut Länger},
journal= {arXiv preprint arXiv:2006.04417},
year = {2020}
}