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A representation of residuated lattices satisfying the double negation law

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Abstract

Every residuated lattice can be considered as an idempotent semiring. Conversely, if an idempotent semiring is finite, then it can be organized into a residuated lattice. Unfortunately, this does not hold in general. We show that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the double negation law. Also conversely, every residuated lattice satisfying the double negation law can be considered as an idempotent semiring with an involution satisfying the mentioned conditions.

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References

  • Belluce LP, Di Nola A, Ferraioli AR (2013) MV-semirings and their sheaf representation. Order 30:165–179

    Article  MathSciNet  MATH  Google Scholar 

  • Bonzio S, Chajda I, Ledda A (2016) Representing quantum structures as near semiring. Logic J IGPL 24(2016):719–742

    Article  MathSciNet  Google Scholar 

  • Cignoli R, D’Ottaviano I, Mundici D (1999) Algebraic foundations of many-valued reasoning. Trends in logic. Kluwer Academic Publishers, Dordrecht, Boston, Heidelberg

    MATH  Google Scholar 

  • Chajda I, Länger H (2015) Commutative basic algebras and coupled near semirings. Soft Comput 19:1129–1134

    Article  MATH  Google Scholar 

  • Chajda I, Länger H (2016) Near semirings and semirings with involution. Miskolc Math Notes 17(2016):801–810

    MathSciNet  MATH  Google Scholar 

  • Ciungu LC (2014) Non-commutative Multiple-Valued Logic Algebras. Springer Verlag, Heidelberg, New York, Dordrecht, ISBN 978-3-319-01588-0

  • Galatos N, Jipsen P, Kowalski T, Ono H (2007) Residuated lattices: An algebraic glimpse at substructural logic. Elsevier, Amsterdam, Boston, Dordrecht

    MATH  Google Scholar 

  • Gispert J (2003) Axiomatic extensions of nilpotent minimum logic. Rep Math Log 37:113–123

    MathSciNet  MATH  Google Scholar 

  • Golan JS (1992) The theory of semirings with applications in mathematics and theoretical computer science. Wiley, New York

    MATH  Google Scholar 

  • Hájek P (1998) Mathematics of fuzzy logic, trends in logic. Kluwer Academic Publishers, Dordrecht, Boston, London

    MATH  Google Scholar 

  • Kuich W, Salomaa A (1986) Semirings, automata, languages. Springer, Berlin

    Book  MATH  Google Scholar 

  • Rachůnek J, Svoboda Z (2014) Interior and closure operators on bounded residuated lattices. Cent Eur J Math 12:534–544

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of the author is supported by the project IGA PrF 2017012 Palacky University Olomouc and by the Austrian Science Fund (FWF), Project I 1923-N25, and the Czech Science Foundation (GAR): Project 15-34697L.

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  1. Department of Algebra and Geometry, Palacký University Olomouc, 17.listopadu 12, 771 46, Olomouc, Czech Republic

    Ivan Chajda

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  1. Ivan Chajda
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Correspondence to Ivan Chajda.

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This article does not contain any studies with human participants or animals performed by any of the authors.

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Communicated by A. Di Nola.

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Chajda, I. A representation of residuated lattices satisfying the double negation law. Soft Comput 22, 1773–1776 (2018). https://doi.org/10.1007/s00500-017-2673-9

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  • DOI: https://doi.org/10.1007/s00500-017-2673-9

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