Tense operators on distributive lattices with implication

TL;DR

This paper introduces and studies tense operators on distributive lattices with implication, establishing categorical equivalences and describing algebraic congruences using tense filters and deductive systems.

Contribution

It extends the theory of tense operators to distributive lattices with implication and establishes categorical equivalences with tense centered Kleene algebras.

Findings

Categorical equivalence between tense distributive lattices with implication and tense centered Kleene algebras.

Description of congruences via tense 1-filters and deductive systems.

Extension of tense operator theory to a broader algebraic setting.

Abstract

Inspired by the definition of tense operators on distributive lattices presented by Chajda and Paseka in 2015, in this paper, we introduce and study the variety of tense distributive lattices with implication and we prove that these are categorically equivalent to a full subcategory of the category of tense centered Kleene algebras with implication. Moreover, we apply such an equivalence to describe the congruences of the algebras of each variety by means of tense 1-filters and tense centered deductive systems, respectively.