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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Residuated frames with applications to decidability


Authors: Nikolaos Galatos and Peter Jipsen
Journal: Trans. Amer. Math. Soc. 365 (2013), 1219-1249
MSC (2010): Primary 06F05; Secondary 08B15, 03B47, 03G10, 03F05
Published electronically: October 31, 2012
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Abstract: Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the finite model property and the finite embeddability property, which imply the decidability of the equational/universal theories of the associated residuated lattice-ordered groupoids. In particular these techniques allow us to prove that the variety of involutive FL-algebras and several related varieties have the finite model property.


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Additional Information

Nikolaos Galatos
Affiliation: Department of Mathematics, University of Denver, 2360 S. Gaylord Street, Denver, Colorado 80208
Email: ngalatos@du.edu

Peter Jipsen
Affiliation: Mathematics and CS, Faculty of Mathematics, School of Computer Science, Chapman University, One University Drive, Orange, California 92866
Email: jipsen@chapman.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05573-5
PII: S 0002-9947(2012)05573-5
Keywords: Substructural logic, Gentzen system, residuated lattice, residuated frame, cut elimination, decidability, finite model property, finite embeddability property
Received by editor(s): August 29, 2008
Received by editor(s) in revised form: February 22, 2011
Published electronically: October 31, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.