Constructions of Kleene lattices

TL;DR

This paper introduces a straightforward construction method to generate Kleene lattices from distributive lattices, demonstrating embeddings, cardinalities, and representation limitations.

Contribution

It provides a new construction technique for Kleene lattices from distributive lattices and explores their representability and structural properties.

Findings

Every finite chain as a Kleene lattice can be constructed this way.

The construction preserves direct products.

Not all Kleene lattices are representable.

Abstract

We present an easy construction producing a Kleene lattice K from an arbitrary distributive lattice L and a non-empty subset of L. We show that L can be embedded into K and compute the cardinality of K under certain additional assumptions. We prove that every finite chain considered as a Kleene lattice can be represented in this way and that this construction preserves direct products.Moreover, we demonstrate that certain Kleene lattices that are ordinal sums of distributive lattices are representable. Finally, we prove that not every Kleene lattice is representable.