Representability of Kleene posets and Kleene lattices

TL;DR

This paper studies the structure and representability of Kleene lattices and posets, extending construction methods, analyzing embedding properties, and exploring their completions and extensions.

Contribution

It extends the construction of Kleene lattices and posets using subsets, investigates their representability, and examines their completions and extensions, providing new insights into their structure.

Findings

Direct products of representable Kleene posets are representable.

Examples of non-representable Kleene posets are provided.

Extensions G(A) of Kleene posets are also Kleene lattices.

Abstract

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper [5], we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset A, namely its Dedekind-MacNeille completion DM(A) and a completion G(A) which coincides with DM(A) provided A is finite. In particular we prove that if A is a Kleene poset then its extension G(A) is also a Kleene lattice. If the subset X of principal order ideals of A is involutionclosed and doubly dense in G(A) then it generates G(A) and it is isomorphic to A itself.