Choice-free Topological Duality for Implicative Lattices and Heyting Algebras

TL;DR

This paper introduces a choice-free duality framework for implicative lattices and Heyting algebras, unifying semantics for intuitionistic and weaker logics through a novel topological duality approach.

Contribution

It provides a new choice-free representation and duality theorem for implicative lattices, extending to Heyting algebras and related topological structures.

Findings

Unified semantic framework for IPC and weaker logics

Choice-free duality for implicative lattices and Heyting algebras

Topological sorted frames with ternary relations

Abstract

We develop a common semantic framework for the interpretation both of $\mathbf{IPC}$, the intuitionistic propositional calculus, and of logics weaker than $\mathbf{IPC}$ (substructural and subintuitionistic logics). This is done by proving a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the category of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.