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Distributive residuated frames and generalized bunched implication algebras

Algebra universalis volume 78, pages 303–336 (2017)Cite this article

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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References

  1. Blok W. J., van Alten C. J.: On the finite embeddability property for residuated ordered groupoids. Trans. Amer. Math. Soc. 357, 4141–4157 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cardona, R., Galatos, N.: The FEP for some varieties of fully-distributive knotted residuated lattices. Algebra Universalis (to appear)

  3. Ciabattoni, A., Galatos, N., Terui, K.: From axioms to analytic rules in nonclassical logics. Proceedings of LICS’08, pp. 229–240 (2008)

  4. Ciabattoni A., Galatos N., Terui K.: MacNeille completions of FL-algebras. Algebra Universalis 66, 405–420 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ciabattoni A., Galatos N., Terui K.: Algebraic proof theory for substructural logics: cut-elimination and completions. Ann. Pure Appl. Logic 163, 266–290 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Galatos N., Horčík R.: Cayley’s and Holland’s theorems for idempotent semirings and their applications to residuated lattices. Semigroup Forum 87, 569–589 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galatos N., Jipsen P.: Residuated frames with applications to decidability. Trans. Amer. Math. Soc. 365, 1219–1249 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Studies in Logic and the Foundations of Mathematics 151 (2007)

  9. Galatos, N., Terui, K.: An introduction to substructural logics. (in progress)

  10. Galmiche D., Méry D., Pym D. J.: The semantics of BI and resource tableaux. Mathematical Structures in Computer Science 15, 1033–1088 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gehrke, M., Generalized Kripke frames. Studia Logica 84, 241–275 (2006)

  12. Kozak M.: Distributive full Lambek calculus has the finite model property. Studia Logica 91, 201–216 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pym, D.J.: The Semantics and Proof Theory of the Logic of Bunched Implications. Kluwer Academic Publishers, Applied Logic Series 26 (2002)

  14. Reynolds, J. C.: Separation Logic: A logic for shared mutable data structures. 17th IEEE Symposium on Logic in Computer Science (LICS 2002), 22-25 July, Copenhagen, Proceedings, pp. 55–74 (2002)

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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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Corresponding author

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

2010 Mathematics Subject Classification

  • Primary: 06F05
  • Secondary: 08B15
  • 03B47
  • 03G10

Key words and phrases

  • substructural logic
  • Gentzen system
  • residuated lattice
  • residuated frame
  • cut elimination
  • decidability
  • finite model property
  • finite embeddability property