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A Note on the Distributivity of Fuzzy Implications over Representable Uninorms

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Book cover Advances in Computational Intelligence (IPMU 2012)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 298))

Abstract

Recently, in [1] we have presented the solutions of the following two distributive equations I(x,U 1(y,z)) = U 2(I(x,y),I(x,z)) and I(U 1(x,y),z) = U 2(I(x,z),I(y,z)), when U 1, U 2 are given representable uninorms and I is an unknown function, in particular a fuzzy implication. Unfortunately, we have realized that results presented for the second equation, in section 6, are incorrect. Our mistake arises from the absence of the proofs in this section – we have incorrectly considered the vertical section instead of horizontal section. In this paper we will show the correct results and will also present the sketches of proofs. We would like to note that all other results presented in article [1] are correct.

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References

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Silesia, 40-007, Katowice, ul. Bankowa 14, Poland

    Michał Baczyński

Authors
  1. Michał Baczyński
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Editor information

Editors and Affiliations

  1. Faculty of Economics, University of Catania, Corso Italia, 55, 95129, Catania, Italy

    Salvatore Greco  & Benedetto Matarazzo  & 

  2. Université Pierre et Marie Curie - Paris6, CNRS UMR 7606, DAPA, LIP6 8, rue du Capitaine Scott, F-75015, Paris, France

    Bernadette Bouchon-Meunier

  3. Dip. Matematica e Informatica, Università di Perugia, 06123, Perugia, Italy

    Giulianella Coletti

  4. Department of Computer and Management Science, University of Trento, Via Inama 5, 38122, Trento, Italy

    Mario Fedrizzi

  5. Machine Intelligence Institute, IONA College, 10801, New Rochelle, NY, USA

    Ronald R. Yager

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© 2012 Springer-Verlag Berlin Heidelberg

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Baczyński, M. (2012). A Note on the Distributivity of Fuzzy Implications over Representable Uninorms. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds) Advances in Computational Intelligence. IPMU 2012. Communications in Computer and Information Science, vol 298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31715-6_40

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  • DOI: https://doi.org/10.1007/978-3-642-31715-6_40

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31714-9

  • Online ISBN: 978-3-642-31715-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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