Duality for $\kappa$-additive complete atomic modal algebras

TL;DR

This paper establishes a duality between $ppa$-additive complete atomic modal algebras and $ppa$-downward directed multi-relational Kripke frames, expanding the algebraic and relational understanding of certain modal logics.

Contribution

It introduces a duality theorem linking $ppa$-additive modal algebras with multi-relational Kripke frames for monomodal logics, including a new proof approach.

Findings

Duality between $ppa$-additive modal algebras and multi-relational Kripke frames

Equivalence of $ppa$-downward directed multi-relational Kripke frames and $ppa$-complete neighborhood frames

Alternative proof of the duality based on Minari's technique

Abstract

In this paper, we give a duality theorem between the category of $\kappa$-additive complete atomic modal algebras and the category of $\kappa$-downward directed multi-relational Kripke frames, for any cardinal number $\kappa$. Multi-relational Kripke frames are not Kripke frames for multi-modal logic, but frames for monomodal logics in which the modal operator $\Diamond$ does not distribute over (possibly infinite) disjunction, in general. We first define homomorphisms of multi-relational Kripke frames, and then show the equivalence between the category of $\kappa$-downward directed multi-relational Kripke frames and the category $\kappa$-complete neighborhood frames, from which the duality theorem follows. We also present another direct proof of this duality based on the technique given by Minari.