Finitely generated free Heyting algebras: the well-founded initial segment

Abstract

In this paper we describe the well-founded initial segment of the free Heyting algebra ${\mathbb A}_\alpha$ on finitely many, $\alpha$, generators. We give a complete classification of initial sublattices of ${\mathbb A}_2$ isomorphic to ${\mathbb A}_1$ (called ‘low ladders'), and prove that for $2 \le \alpha < \omega$, the height of the well-founded initial segment of ${\mathbb A}_\alpha$ is $\omega^2$.