Free Heyting Algebras: Revisited

Abstract

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.

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

“…The proof is an application of the coalgebraic Jónsson-Tarksi theorem. Recall from Theorem 1 and Diagram (1), that the canonical extension of an object A in any of our reasoning kernels C is given by GF A. This justifies the following:…”

“…The reason for proceeding in this step-bystep way will become clear in the sequel and is similar in spirit to the approach of [1]. The main difference is that in [1], the axioms of Heyting algebras are separated into rank 1 and non-rank 1 axioms, leading to the notion of weak Heyting algebras which obey the axioms HDL1-2 and also a → a = . In this work, we want to build a minimal 'pre-Heyting' logic with a strongly complete semantics and well-behaved (viz.…”

Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

“…Nelson algebras were introduced by Rasiowa [86], under the name of Nlattices, as the algebraic counterparts of the constructive logic with strong negation considered by Nelson and Markov [84]. The centered Nelson algebras with the interpolation property are represented by Heyting algebras [10]. Nelson lattices are involutive residuated lattices, and are equationally equivalent to centered Nelson algebras [21].…”

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

“…1998 ACM Subject Classification: F.4.1. CC Creative Commons 2 N. BEZHANISHVILI AND M. GEHRKE construction of free algebras in a straightforward way.This paper is an extended version of [7]. However, unlike [7], here we give a complete solution to the problem of describing finitely generated free Heyting algebras in a systematic way using methods similar to those used for rank 1 logics.…”

“…This paper is an extended version of [7]. However, unlike [7], here we give a complete solution to the problem of describing finitely generated free Heyting algebras in a systematic way using methods similar to those used for rank 1 logics.…”