Interpolation and amalgamation in modal cylindric algebras

Abstract

Let α be an ordinal and L be a unimodal logic (like S4 or S5). A modal cylindric algebra of dimension α, an LCA α, is a cylindric algebra of dimension α, expanded with α-many L modalities. For a frame (U, R) of L, each k < α, one defines a diamond box operator on . This defines the semantics of the L modalities in set algebras, with the rest of the operations defined like in cylindric set algebras of dimension α. We study interpolation properties for the corresponding predicate logic having α-many variables. Our results are valid for any reflexive L whose frames contain the universal frames (U, U × U ). In particular, they hold for K5CA α, S4CA α (which is an algebraizable extension of topological predicate logic with semantics induced by Alexandrov topologies).