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Idempotent residuated structures: Some category equivalences and their applications


Authors: N. Galatos and J. G. Raftery
Journal: Trans. Amer. Math. Soc. 367 (2015), 3189-3223
MSC (2010): Primary 03B47, 03G25, 06F05; Secondary 03G27, 08C05, 08C15
DOI: https://doi.org/10.1090/S0002-9947-2014-06072-8
Published electronically: December 4, 2014
MathSciNet review: 3314806
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Abstract: This paper concerns residuated lattice-ordered idempotent commutative monoids that are subdirect products of chains. An algebra of this kind is a generalized Sugihara monoid (GSM) if it is generated by the lower bounds of the monoid identity; it is a Sugihara monoid if it has a compatible involution $ \neg $. Our main theorem establishes a category equivalence between GSMs and relative Stone algebras with a nucleus (i.e., a closure operator preserving the lattice operations). An analogous result is obtained for Sugihara monoids. Among other applications, it is shown that Sugihara monoids are strongly amalgamable, and that the relevance logic $ \mathbf {RM}^\mathbf {t}$ has the projective Beth definability property for deduction.


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Additional Information

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

J. G. Raftery
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Private Bag X20, Hatfield, Pretoria 0028, South Africa
Email: james.raftery@up.ac.za

DOI: https://doi.org/10.1090/S0002-9947-2014-06072-8
Keywords: Residuation, idempotent, semilinear, representable, nucleus, Sugihara monoid, relative Stone algebra, category equivalence, epimorphism, amalgamation, Beth definability, interpolation, R-mingle
Received by editor(s): February 23, 2012
Received by editor(s) in revised form: January 2, 2013
Published electronically: December 4, 2014
Additional Notes: The work of the first author was supported in part by Simons Foundation grant 245806
The second author was supported in part by the National Research Foundation of South Africa (UID 85407)
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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