On Frontal Heyting Algebras

A frontal operator in a Heyting algebra is an expansive operator preserving finite meets which also satisfies the equation $\tau(\x) \le y \vee (y\rightarrow x)$. A frontal Heyting algebra is a pair $(H,\tau)$, where $H$ is a Heyting algebra and $\tau$ 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 $\gamma$, (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.