Characteristic formulas over intermediate logics

TL;DR

This paper extends characteristic formulas to infinite finitely presentable algebras, revealing a continuum of intermediate logics axiomatizable by these formulas, surpassing standard Jankov formulas, with applications to interior algebras and S4 extensions.

Contribution

It introduces a generalized notion of characteristic formulas for infinite algebras and demonstrates their capacity to axiomatize a vast class of intermediate logics beyond traditional methods.

Findings

Continuum of varieties of Heyting algebras with infinite finitely presentable subdirectly irreducible algebras.

Existence of a continuum of intermediate logics axiomatizable by characteristic formulas of infinite algebras.

Examples of intermediate logics not axiomatizable by characteristic formulas of infinite algebras.

Abstract

We expand the notion of characteristic formula to infinite finitely presentable subdirectly irreducible algebras. We prove that there is a continuum of varieties of Heyting algebras containing infinite finitely presentable subdirectly irreducible algebras. Moreover, we prove that there is a continuum of intermediate logics that can be axiomatized by characteristic formulas of infinite algebras while they are not axiomatizable by standard Jankov formulas. We give the examples of intermediate logics that are not axiomatizable by characteristic formulas of infinite algebras. Also, using the Goedel-McKinsey-Tarski translation we extend these results to the varieties of interior algebras and normal extensions of S4