Tense distributive lattices: algebra, logic and topology

TL;DR

This paper explores the algebraic, logical, and topological aspects of tense distributive lattices, establishing dualities, characterizing congruences, and connecting to known tense logics.

Contribution

It introduces categorical dualities for tense distributive lattices and links algebraic structures with Kripke frames and Priestley spaces.

Findings

Categorical dualities for tdlat established

Characterization of congruence lattices in tense distributive lattices

Connection between tdlat sub-classes and existing tense logics

Abstract

Tense logic was introduced by Arthur Prior in the late 1950s as a result of his interest in the relationship between tense and modality. Prior's idea was to add four primitive modal-like unary connectives to the base language today widely known as Prior's tense operators. Since then, Prior's operators have been considered in many contexts by different authors, in particular, in the context of algebraic logic. Here, we consider the category tdlat of bounded distributive lattices equipped with Prior's tense operators. We establish categorical dualities for tdlat in terms of certain categories of Kripke frames and Priestley spaces, respectively. As an application, we characterize the congruence lattice of any tense distributive lattice as well as the subdirectly irreducible members of this category. Finally, we define the logic that preserves degrees of truth with respect to tdlat-algebras and precise the relation between particular sub-classes of tdlat and know tense logics found in the literature.