On Local Finiteness of Modal K4 Algebras

We study local finiteness for modal $K4$ algebras via the tunability of their dual general frames. In particular, we provide a sufficient condition for modal $K4$ algebras to be locally finite by identifying a structure which must be present in non-locally finite modal $K4$ algebras. We then show that this condition becomes both necessary and sufficient for complex modal $K4$ algebras. Next, we translate this condition into a pair of order-theoretic conditions on transitive Kripke frames, providing a classification of local finiteness on their dual modal algebras. We further show that the logic of any class of well-founded transitive relations with no infinite antichains has the finite model property, and conclude that the logic of the class of well-quasi orderings has the finite model property.