Topological Dualities for Modal Algebras

TL;DR

This paper explores dualities between categories of modal algebras and relational spaces, highlighting how different morphisms affect the duality and simplifying in certain cases.

Contribution

It introduces a family of Stone-type dualities linking modal algebra categories with relational spaces, clarifying the impact of morphism variations.

Findings

Different morphism notions lead to significant duality variations.

Simplification occurs with semicontinuous relations, enabling easier modal-relational correspondence.

The dualities facilitate understanding of modal axioms through relational properties.

Abstract

We display a family of Stone-type dualities linking categories of frames carrying pairs of modal operators to categories of spaces carrying a binary relation. Different notions of morphism used on the relational side lead to significant variations in the point construction. We show how the situation simplifies in the case of semicontinuous relations, allowing for straightforward correspondences between modal axioms and relational properties.