On the Variety of Heyting Algebras with Successor Generated by All Finite Chains

Contrary to the variety of Heyting algebras, finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor. In particular, all finite chains generate a proper subvariety, $SLH_\omega$, of the latter. There is a categorical duality between Heyting algebras with successor and certain Priestley spaces. Let $X$ be the Heyting space associated by this duality to the Heyting algebra with successor $H$. If there is an ordinal $\kappa$ and a filtration on $X$ such that $X=\cup_{\lambda\le\kappa} X_\lambda$, the height of $X$ is the minimun ordinal $\xi\le\kappa$ such that $X_xi^c = \emptyset$. In this case, we also say that $H$ has height $\xi$. This filtration allows us to write the space $X$ as a disjoint union of antichains. We may think that these antichains define levels on this space. We study the way of characterize subalgebras and homomorphic images in finite Heyting algebras with successor by means of their Priestley spaces. We also depict the spaces associated to the free algebras in various subcategories of $SLH_\omega$.