## Idempotent residuated structures: Some category equivalences and their applications

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- by N. Galatos and J. G. Raftery PDF
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**367**(2015), 3189-3223 Request permission

## 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
- 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) - © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3314806