🤖 AI Summary
This study addresses the complexity in point-based constructions arising from varying morphism definitions in modal logic by establishing a Stone-type duality between an algebraic category equipped with paired modal operators and a category of topological spaces endowed with binary relations. By introducing a semi-continuity condition on relations, the work reveals a direct correspondence between modal axioms and relational properties of the underlying spaces, thereby significantly simplifying point-set manipulations in traditional dualities. This approach not only unifies several existing dualities between modal frameworks and topological semantics but also provides a precise bridge between algebraic and relational semantics, offering new categorical tools for the systematic study of modal logics.
📝 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.