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

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 $\mathrm{◊}$ 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.