Algebraisable versions of predicate topological logic

Abstract

Motivated by questions like: which spatial structures may be characterized by means of modal logic, what is the logic of space, how to encode in modal logic different geometric relations, topological logic provides a framework for studying the confluence of the topological semantics for $\sf S4$ modalities, based on topological spaces rather than Kripke frames. Following research initiated by Sgro, and further pursued algebraically by Georgescu, we prove an interpolation theorem and an omitting types theorem for various extensions of predicate topological logic and Chang's modal logic. Our proof is algebraic addressing expansions of cylindric algebras using interior operators and boxes, respectively. Then we proceed like is done in abstract algebraic logic by studing algebraisable extensions of both logics; obtaining a plethora of results on the amalgamation property for various subclasses of their algebraic counterparts, which are varieties. Notions like atom-canonicity and complete representations are approached for finite dimensional topological cylindric algebras. The logical consequences of our algebraic results are carefully worked out for infinitary extensions of Chang's predicate modal logic and finite versions thereof, by restricting to $n$ variables, $n$ finite, viewed as a propositional multi-dimensional modal logic, and $n$ products of bimodal whose frames are of the form $(U, U\times U, R)$ where $R$ is a pre-order, endowed with diagonal constants.