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On the Distributivity Equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms

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Modeling Decisions for Artificial Intelligence (MDAI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8825))

Abstract

In this paper we continue investigations connected with distributivity of implication operations over decomposable (t-representable) operations. Our main goal is to show the general method of solving the following distributivity equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\), when \(\mathcal{U}_1\), \(\mathcal{U}_2\) are decomposable uninorms (in interval-valued fuzzy sets theory) generated from two conjunctive representable uninorms. As a byproduct result we show all solutions of some functional equation related to this case.

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Authors and Affiliations

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

    Michał Baczyński & Wanda Niemyska

Authors
  1. Michał Baczyński
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  2. Wanda Niemyska
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Editors and Affiliations

  1. CSIC, Consejo Superior de Investigaciones Cientificas, 08193, Bellaterra, Spain

    Vicenç Torra

  2. Toho Gakuen, 186-0004, Tokyo, Japan

    Yasuo Narukawa

  3. Faculty of Engineering, Information and Systems, University of Tsukuba, 305-8573, Tsukuba, Ibaraki, Japan

    Yasunori Endo

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Baczyński, M., Niemyska, W. (2014). On the Distributivity Equation \(\mathcal{I}(x,\mathcal{U}_1(y,z)) = \mathcal{U}_2(\mathcal{I}(x,y),\mathcal{I}(x,z))\) for Decomposable Uninorms (in Interval-Valued Fuzzy Sets Theory) Generated from Conjunctive Representable Uninorms. In: Torra, V., Narukawa, Y., Endo, Y. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2014. Lecture Notes in Computer Science(), vol 8825. Springer, Cham. https://doi.org/10.1007/978-3-319-12054-6_3

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  • DOI: https://doi.org/10.1007/978-3-319-12054-6_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-12053-9

  • Online ISBN: 978-3-319-12054-6

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