  In this paper, we give an extension of the J\'{o}nsson-Tarski representation
theorem for both normal and non-normal modal algebras so that it preserves
countably many infinitary meets and joins. To extend the J\'{o}nsson-Tarski
representation to non-normal modal algebras we consider neighborhood frames,
instead of Kripke frames, 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.
