Completeness and interpolation for intuitionistic infinitary predicate logic, in connection to finitizing the class of representable Heyting polyadic algebras

TL;DR

This paper explores representation theorems and properties of Heyting polyadic algebras, connecting their algebraic structures with Kripke semantics, and addresses the finitizability problem in intuitionistic logic.

Contribution

It introduces new representation and neat embedding theorems for Heyting polyadic algebras, linking algebraic and Kripke semantics, and advances understanding of finitizability in intuitionistic logic.

Findings

Superamalgamation proved for several reducts.

Class of representable algebras coincides with those having the neat embedding property.

Representation theorems connect algebraic structures with Kripke semantics.

Abstract

We study different representation theorems for various reducts of Heyting polyadic algebras. Superamalgamation is proved for several (natural reducts) and our results are compared to the finitizability problem in classical algebraic logic dealing with cylindric and polyadic (Boolean algebras). We also prove several new neat embedding theorems, and obtain that the class of representable algebras based on (a generalized) Kripke semantics coincide with the class of algebras having the neat embedding property, that is those algebras that are subneat reducts of algebras having $\omega$ extra dimensions.