An algebraic look at filtrations in modal logic

Filtration constructions are among the oldest and best known methods for obtaining finite model properties for modal logics, and appear in the literature in both model-theoretic and algebraic versions. In this article we investigate definitions of algebraic filtrations by means of different types of binary relations on modal algebras, and the relationships between these. We generalize the notion of a model-theoretic filtration somewhat while simultaneously lifting it to the level of frames. We proceed to link algebraic filtrations with their model- or frame-theoretic counterparts by showing how our filtration notions interface neatly with the well-known duality theory of modal algebras and Kripke frames. We illustrate, by means of some examples, how this theory enables one to easily translate between algebraic and model-theoretic versions of some well-known filtrations. We obtain some order theoretic insights regarding the (usually model-theoretically specified) smallest and largest filtrations by considering their algebraic versions, thus demonstrating the utility of having ready access to both versions.