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Canonical formulas for $$k$$-potent commutative, integral, residuated lattices. (English) Zbl 1420.03147
Summary: Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising 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 ($$k$$-CIRL). We show that any subvariety of $$k$$-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples.

##### MSC:
 03G25 Other algebras related to logic 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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