Choice-free Topological Duality for Implicative Lattices and Heyting Algebras

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.