Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

TL;DR

This paper explores the construction of finitely generated free Heyting algebras using Birkhoff duality and coalgebraic methods, extending rank 1 axiomatization techniques to more complex cases.

Contribution

It introduces a modified coalgebraic approach to handle Heyting algebras with mixed rank axiomatizations, expanding the applicability of existing methods.

Findings

Finitely generated free Heyting algebras can be constructed via duality and coalgebraic techniques.

Heyting algebras are nearly rank 1 axiomatized, allowing adapted coalgebraic methods.

The approach broadens the scope of coalgebraic methods for algebraic structures.

Abstract

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods.