Errata on "On the variety of Heyting algebras with successor generated by all finite chains"

In [3] we have claimed that finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor, and in particular all finite chains generate a proper subvariety of the latter. As Xavier Caicedo made us notice, this claim is not true. He proved, using techniques of Kripke models, that the intuitionistic calculus with S has finite model property and from this result he concluded that the variety of Heyting algebras with successor is generated by its finite members [2]. This fact particularly affects Section 3.2 of our article. Concretely, in Remark 3.3, our claim “Let K be a class of S-Heyting algebras of height less or equal to a fixed ordinal ξ. Using the categorical duality between SHeyting algebras and S-Heyting spaces, it can be shown that the elements of classes H(K), S(K) and P(K) have also height less or equal to ξ. Here H, S and P are the class operators of universal algebra. Hence for each ordinal ξ, the class of S-Heyting algebras of height less or equal to ξ is a variety“ is not true as stated. It remains valid only if ξ is a finite ordinal.