Free Heyting Algebras: Revisited

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Free Heyting Algebras: Revisited
Abstract. 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 perspective does 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.
Nick Bezhanishvili, Mai Gehrke
Added 26 May 2010
Updated 26 May 2010
Type Conference
Year 2009
Authors Nick Bezhanishvili, Mai Gehrke
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