Discrete Duality for Tense Symmetric Heyting Algebras

TL;DR

This paper develops a discrete duality theory for tense symmetric Heyting algebras, introduces a propositional calculus based on these algebras, and proves its completeness, extending previous duality results for Heyting algebras.

Contribution

It introduces a novel duality for TSH-algebras, a propositional calculus, and establishes its completeness, advancing the algebraic and logical understanding of tense symmetric Heyting algebras.

Findings

Established a discrete duality for TSH-algebras.

Proposed a propositional calculus with TSH-algebras as models.

Proved the completeness theorem for the calculus.

Abstract

In this article, we continue the study of tense symmetric Heyting algebras (or TSH-algebras). These algebras constitute a generalization of tense algebras. In particular, we describe a discrete duality for TSHalgebras bearing in mind the results indicated by E. Or lowska and I. Rewitzky in [E. Or lowska and I. Rewitzky, Discrete Dualities for Heyting Algebras with Operators, Fund. Inform. 81 (2007), no.1-3, 275-295.] for Heyting algebras. In addition, we introduce a propositional calculus and prove this calculus has TSH-algebras as algebraic counterpart. Finally, the duality mentioned above allowed us to show the completeness theorem for this calculus.