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Enric Trillas: A Passion for Fuzzy Sets pp 151–165Cite as

Many-Valued Preorders II: The Symmetry Axiom and Probabilistic Geometry

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 322))

Abstract

This paper is a continuation of Part I (Höhle, Many-valued preorders I: the basis of many-valued mathematics (in this volume) [10]) and explains the symmetrization of many-valued preorders and the subsequent quotient construction. An application of these concepts to probabilistic geometry leads to \([0,1]\)-valued metric spaces which appear as quotient of Menger spaces.

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Notes

  1. 1.

    A \(Q\)-valued set with a \(Q\)-valued strict equivalence relation is called \(Q\)-set in [11].

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Acknowledgments

I am very grateful for the support I received from T. Kubiak during the preparation of both parts of this paper.

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Authors and Affiliations

  1. FB C Bergische Universität Wuppertal, 42097, Wuppertal, Germany

    Ulrich Höhle

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  1. Ulrich Höhle
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Correspondence to Ulrich Höhle .

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Editors and Affiliations

  1. European Centre for Soft Computing, Asturias, Spain

    Luis Magdalena

  2. Depto. de Ciencias de la Computación e Inteligencia Artificial, University of Granada, Granada, Spain

    Jose Luis Verdegay

  3. Spanish Council for Scientific Research (CSIC), Artificial Intelligence Research Institute (IIIA-CSIC), Catalonia, Spain

    Francesc Esteva

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Höhle, U. (2015). Many-Valued Preorders II: The Symmetry Axiom and Probabilistic Geometry. In: Magdalena, L., Verdegay, J., Esteva, F. (eds) Enric Trillas: A Passion for Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-16235-5_11

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  • DOI: https://doi.org/10.1007/978-3-319-16235-5_11

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  • Online ISBN: 978-3-319-16235-5

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