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arXiv · 1804.10873

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

Yoshihito Tanaka
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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.

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Yoshihito Tanaka. 2018-04-29. Duality for $\kappa$-additive complete atomic modal algebras. https://doi.org/10.1007/s00012-021-00724-7

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