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Distributive residuated frames and generalized bunched implication algebras


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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Correspondence to Nikolaos Galatos.

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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).

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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