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Smooth Finite T-norms and Their Equational Axiomatization

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Fuzzy Logic and Information Fusion

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 339))

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Abstract

In this paper, as homage to Professor Gaspar Mayor in his 70 anniversary, we present a summary of results on BL-algebras and related structures that, using the one-to-one correspondence between divisible finite t-norms and finite BL-chains, allows us to provide an equational characterization of any divisible finite t-norm.

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Notes

  1. 1.

    Take into account that there are no finite product chains different from the Boolean chain of two elements.

  2. 2.

    Note however, that Bou has shown [4] that there is at least one equation in the language \((*, \wedge , \vee , 0, 1)\) that is valid for all finite divisible t-norms but fails in some finite non-divisible t-norm. In particular the exhibited equation in [4] has 9 variables and it fails on a t-norm over a chain of 33 elements.

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Acknowledgments

The authors acknowledge support from the MINECO project EdeTRI (TIN2012-39348-C02-01), the LODISCO network TIN2014-56381-REDTLODISCO, and the grants 2014SGR-788 and 2014SGR-118. García-Cerdaña also acknowledges the MICINN project MTM 2011-25747.

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Authors and Affiliations

  1. IIIA-CSIC, Campus de la UAB, 08013, Bellaterra, Spain

    Francesc Esteva, Àngel García-Cerdaña & Lluís Godo

  2. DTIC, Universitat Pompeu Fabra, 08002, Barcelona, Spain

    Àngel García-Cerdaña

Authors
  1. Francesc Esteva
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  2. Àngel García-Cerdaña
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  3. Lluís Godo
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Corresponding author

Correspondence to Lluís Godo .

Editor information

Editors and Affiliations

  1. University of Alcalá, Alcalá de Henares, Spain

    Tomasa Calvo Sánchez

  2. University of Balearic Islands, Palma, Spain

    Joan Torrens Sastre

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Esteva, F., García-Cerdaña, À., Godo, L. (2016). Smooth Finite T-norms and Their Equational Axiomatization. In: Calvo Sánchez, T., Torrens Sastre, J. (eds) Fuzzy Logic and Information Fusion. Studies in Fuzziness and Soft Computing, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-319-30421-2_2

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  • DOI: https://doi.org/10.1007/978-3-319-30421-2_2

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30419-9

  • Online ISBN: 978-3-319-30421-2

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