In this paper, we give a duality theorem between the category of
κ-additive complete atomic modal algebras and the category of
κ-downward directed multi-relational Kripke frames, for any cardinal
number κ. Multi-relational Kripke frames are not Kripke frames for
multi-modal logic, but frames for monomodal logics in which the modal operator
â—Š 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 κ-downward directed
multi-relational Kripke frames and the category κ-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