An extension of Jónsson-Tarski representation and model existence in predicate non-normal modal logics

In this paper, we give an extension of the Jónsson-Tarski representation theorem for both normal and non-normal modal algebras so that it preserves countably many infinitary meets and joins. To extend the Jónsson-Tarski representation to non-normal modal algebras we consider neighborhood frames instead of Kripke frames just as Došen's duality theorem for modal algebras, and to deal with infinite meets and joins, we make use of Q-filters instead of prime filters. Then, we show that every predicate modal logic, whether it is normal or non-normal, has a model defined on a neighborhood frame with constant domains, and give completeness theorem for some predicate modal logics. We also show the same results for infinitary modal logics.