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Distributivity of Implication Functions over Decomposable Uninorms Generated from Representable Uninorms in Interval-Valued Fuzzy Sets Theory

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 610))

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Abstract

In this work we investigate two distributivity equations \(\mathcal {I}(x,\mathcal {U}_1(y,z)) = \mathcal {U}_2(\mathcal {I}(x,y),\mathcal {I}(x,z))\), \(\mathcal {I}(\mathcal {U}_1(x,y),z) = \mathcal {U}_2(\mathcal {I}(x,z),\mathcal {I}(y,z))\) for implication operations and uninorms in interval-valued fuzzy sets theory. We consider decomposable (t-representable) uninorms generated from two conjunctive or disjunctive representable uninorms. Our method reduces to solve the following functional equation \(f(u_1+v_1,u_2+v_2) = f(u_1,u_2) + f(v_1,v_2)\), thus we present new solutions for this equation.

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Authors and Affiliations

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

    Michał Baczyński & Wanda Niemyska

Authors
  1. Michał Baczyński
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  2. Wanda Niemyska
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Corresponding author

Correspondence to Michał Baczyński .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Machine Intelligence Institute, Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Baczyński, M., Niemyska, W. (2016). Distributivity of Implication Functions over Decomposable Uninorms Generated from Representable Uninorms in Interval-Valued Fuzzy Sets Theory. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_35

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  • DOI: https://doi.org/10.1007/978-3-319-40596-4_35

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