An Algebraic Approach to Subframe Logics. Modal Case

We prove that if a modal formula is refuted on a wK4-algebra (B,□), then it is refuted on a finite wK4-algebra which is isomorphic to a subalgebra of a relativization of (B,□). As an immediate consequence, we obtain that each subframe and cofinal subframe logic over wK4 has the finite model property. On the one hand, this provides a purely algebraic proof of the results of Fine and Zakharyaschev for K4. On the other hand, it extends the Fine-Zakharyaschev results to wK4.