Orthogonal Frames and Indexed Relations

We define and study the notion of an indexed frame. This is a bi-dimensional structure consisting of a Cartesian product equipped with relations which only relate pairs if they coincide in one of their components. We show that these structures are quite ubiquitous in modal logic, showing up in the literature as products of Kripke frames, subset spaces, or temporal frames for STIT logics. We show that indexed frames are completely characterised by their 'orthogonal' relations, and we provide their sound and complete logic. Using these 'orthogonality' results, we provide necessary and sufficient conditions for an arbitrary Kripke frame to be isomorphic to certain well-known bi-dimensional structures.