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

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Abstract

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.

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Authors and Affiliations

  1. ILLC, Universiteit van Amsterdam, Science Park 107, 1098XG, Amsterdam, The Netherlands

    Nick Bezhanishvili

  2. Department of Mathematics, University of Denver, 2360 S. Gaylord St., Denver, CO, 80208, USA

    Nick Galatos

  3. Dipartimento di Matematica, Università degli Studi di Salerno., Via Giovanni Paolo II 132, 84084, Fisciano (SA), Italy

    Luca Spada

Authors
  1. Nick Bezhanishvili
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  2. Nick Galatos
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  3. Luca Spada
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Corresponding author

Correspondence to Luca Spada.

Additional information

Presented by C. Tsinakis.

The project leading to this joint paper has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176. The second-named author acknowledges the support of the grants Simons Foundation 245805 and FWF project START Y544-N23. The third-named author gratefully acknowledges partial support by the Marie Curie Intra-European Fellowship for the project “ADAMS” (PIEF-GA-2011-299071) and from the Italian National Research Project (PRIN2010–11) entitled Metodi logici per il trattamento dell'informazione.

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Bezhanishvili, N., Galatos, N. & Spada, L. Canonical formulas for k-potent commutative, integral, residuated lattices. Algebra Univers. 77, 321–343 (2017). https://doi.org/10.1007/s00012-017-0430-7

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  • DOI: https://doi.org/10.1007/s00012-017-0430-7

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