Duality for κ -additive complete atomic modal algebras

In this paper, we show that the category of κ -additive complete atomic modal algebras is dually equivalent to 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 logics, but frames for monomodal logics in which the modal operator does not distribute over (possibly infinite) disjunction, in general. We first define homomorphisms of multi-relational Kripke frames, and discuss the relationship between the category of multi-relational Kripke frames and the category of neighborhood frames. Then we give two kinds of proofs for the duality theorem between the category of κ -additive complete atomic modal algebras and the category of κ -downward directed multi-relational Kripke frames. The first proof is given by making use of Došen duality theorem between the category of modal algebras and the category of neighborhood frames, and the second one is based on the idea given by Minari.