Representability and amalgamation for Heyting polyadic algebras

Abstract

We prove that every (not necessarily locally finite) polyadic Heyting algebra of infinite dimension is representable in some concrete sense. We also show that this class has the super amalgamation property. As a byproduct we infer that a certain infinitary extension of predicate intuitionistic logic, or equivalently, the intuitionistic fragment of Keisler’s infinitary logics, is complete and enjoys the Craig interpolation property.