Locally Finite Reducts of Heyting Algebras and Canonical Formulas

Abstract

The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the $\to$-free reducts of Heyting algebras, while the variety of implicative semilattices is generated by the $\vee$-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short).

The $\vee$-free reducts of Heyting algebras give rise to the $(\wedge ,\to )$-canonical formulas that we studied in an earlier work. Here we introduce the $(\wedge ,\vee )$-canonical formulas, which are obtained from the study of the $\to$-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by $(\wedge ,\vee )$-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.

One of the main ingredients of these formulas is a designated subset $D$ of pairs of elements of a finite subdirectly irreducible Heyting algebra $A$. When $D=A^2$, we show that the $(\wedge ,\vee )$-canonical formula of $A$ is equivalent to the Jankov formula of $A$. On the other hand, when $D=\varnothing$, the $(\wedge ,\vee )$-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.