On locally finite varieties of Heyting algebras

TL;DR

This paper constructs specific varieties of Heyting algebras demonstrating a unique property where the finiteness of free algebras depends on the number of generators, revealing new structural insights.

Contribution

It introduces varieties of Heyting algebras with finite $n$-generated free algebras and infinite $(n+1)$-generated free algebras, highlighting a novel phenomenon.

Findings

Existence of such varieties for all natural numbers n

Finite $n$-generated free algebras

Infinite $(n+1)$-generated free algebras

Abstract

For every $n \in \mathbb{N}$, we construct a variety of Heyting algebras, whose $n$-generated free algebra is finite but whose $(n+1)$-generated free algebra is infinite.