José L. Castiglioni

A frontal operator in a Heyting algebra is an expansive operator preserving finite meets which also satisfies the equation (x) ≤ y ∨ (y → x). A frontal Heyting algebra is a pair (H, ), where H is a Heyting algebra and  a frontal operator on H. Frontal operators are always compatible, but not necessarily new or implicit in the sense of Caicedo and Cignoli (An algebraic approach to intuitionistic connectives. Journal of Symbolic Logic, 66, No4 (2001), 1620-1636). Classical examples of new implicit frontal operators are the functions, (op. cit., Example 3.1), the successor (op. cit., Example 5.2), and Gabbay’s operation (op. cit., Example 5.3).
We study a Priestley duality for the category of frontal Heyting algebras and in particular for the varieties of Heyting algebras with each one of the implicit operations given as examples. The topological approach of the compatibility of operators seems to be important in the research of affin completeness of Heyting algebras with additional compatible operations. This problem have also a logical point of view. In fact, we look for some complete propositional intuitionistic calculus enriched with implicit connectives.

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_ω, of the latter. There is a categorical duality between Heyting algebras with suc- cessor 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 κ and a filtration on X such that X =LJλ≤κ X_λ, the height of X is the minimun ordinal ξ ≤ κ such that   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_ω.