Free Heyting Algebras: Revisited

We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras—the rank 1 reducts of Heyting algebras—and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspectives do allow us, in the case of Heyting algebras, to derive Ghilardi’s (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices.