On Bellissima's construction of the finitely generated free Heyting algebras, and beyond

TL;DR

This paper explores the structure of finitely generated free Heyting algebras through topological and model-theoretic methods, extending Bellissima's representation and analyzing their automorphisms and embeddings.

Contribution

It provides an embedding of Bellissima's representation into the profinite completion and offers an algebraic interpretation of the Kripke model as a principal ideal spectrum.

Findings

Embedding of Bellissima's representation into the profinite completion.

The principal ideal spectrum is first order interpretable in the Heyting algebra.

The automorphism group is the permutation group over the generators.

Abstract

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima's representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal sprectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. For example we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.