In this paper, we show that the category of \(\kappa \)-additive complete atomic modal algebras is dually equivalent to 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 logics, 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 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 \(\kappa \)-additive complete atomic modal algebras and the category of \(\kappa \)-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.

在本文中，我们表明，对于任何基数\（\ kappa \）的加法完全原子模态代数的类别与\（\ kappa \）向下定向的多关系Kripke框架的类别双重等效。 \ kappa \）。多关系Kripke框架不是用于多模态逻辑的Kripke框架，而是用于单模态逻辑的框架，其中模态运算符\（\ Diamond \）通常不分布（可能是无限）析取。我们首先定义多关系Kripke框架的同构性，并讨论多关系Kripke框架的类别与邻域框架的类别之间的关系。然后我们给出了范畴之间的对偶定理的两种证明。\（\ kappa \）-可加的完整原子模态代数和\（\ kappa \）的类别-向下定向的多关系Kripke框架。第一个证明是利用模态代数的类别与邻域框架的类别之间的Došen对偶定理，第二个证明是基于Minari给出的思想。