Dualities in modal logic from the point of view of triples

TL;DR

This paper explores how monads can unify duality theorems in modal logic, linking relational categories and algebraic structures, and examines the monoidal structure related to bimorphisms.

Contribution

It introduces a uniform monadic approach to duality theorems in modal logic and analyzes the monoidal structures involved.

Findings

Unified monadic framework for duality theorems

Identification of bimorphisms as algebraic counterparts

Analysis of monoidal structures induced by Cartesian product

Abstract

In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms.