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Distributive residuated frames and generalized bunched implication algebras

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Abstract

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Aspen Hall, 2280 S. Vine St., Denver, CO, 80208, USA

    Nikolaos Galatos

  2. Faculty of Mathematics, Chapman University, One University Drive, Orange, CA, 92866, USA

    Peter Jipsen

Authors
  1. Nikolaos Galatos
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  2. Peter Jipsen
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Correspondence to Nikolaos Galatos.

Additional information

Presented by J. Raftery.

The first author acknowledges the support of the grants Simons Foundation 245805 and FWF project START Y544-N23.

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Galatos, N., Jipsen, P. Distributive residuated frames and generalized bunched implication algebras. Algebra Univers. 78, 303–336 (2017). https://doi.org/10.1007/s00012-017-0456-x

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  • DOI: https://doi.org/10.1007/s00012-017-0456-x

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