Canonical formulas for k-potent commutative, integral, residuated lattices

TL;DR

This paper extends the concept of canonical formulas to k-potent, commutative, integral, residuated lattices, providing a uniform algebraic method to axiomatize all subvarieties of these structures and applying it to substructural logics.

Contribution

It introduces canonical formulas for k-potent, commutative, integral, residuated lattices and demonstrates their effectiveness in axiomatizing all subvarieties of these lattices.

Findings

Any subvariety of k-CIRL is axiomatized by canonical formulas.

The paper provides applications and examples demonstrating the method.

Extends canonical formula techniques to substructural logics.

Abstract

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for $k$-potent, commutative, integral, residuated lattices ($k$-$\mathsf{CIRL}$). We show that any subvariety of $k$-$\mathsf{CIRL}$ is axiomatised by canonical formulas. The paper ends with some applications and examples.