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The Quest for Transitivity, a Showcase of Fuzzy Relational Calculus

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Advances in Computational Intelligence (WCCI 2012)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 7311))

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Abstract

We present several relational frameworks for expressing similarities and preferences in a quantitative way. The main focus is on the occurrence of various types of transitivity in these frameworks. The first framework is that of fuzzy relations; the corresponding notion of transitivity is C-transitivity, with C a conjunctor. We discuss two approaches to the measurement of similarity of fuzzy sets: a logical approach based on biresidual operators and a cardinal approach based on fuzzy set cardinalities. The second framework is that of reciprocal relations; the corresponding notion of transitivity is cycle-transitivity. It plays a crucial role in the description of different types of transitivity arising in the comparison of (artificially coupled) random variables in terms of winning probabilities. It also embraces the study of mutual rank probability relations of partially ordered sets.

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Author information

Authors and Affiliations

  1. Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000, Gent, Belgium

    Bernard De Baets

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  1. Bernard De Baets
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Editor information

Editors and Affiliations

  1. Key Lab of Intelligent Perception and Image Understanding of Ministry of Education of China, Xidian University, 710071, Xi’an, Shaanxi, China

    Jing Liu

  2. Dipartimento di Elettronica e Informazione, Politecnico di Milano, 20133, Milano, Italy

    Cesare Alippi

  3. Université Pierre et Marie Curie - Paris 6, LIP6, 4 place Jussieu, 75005, Paris, France

    Bernadette Bouchon-Meunier

  4. Department of Electrical & Computer Engineering, Portland State University, 97207, Portland, OR, USA

    Garrison W. Greenwood

  5. School of Engineering and Information Technology, University of New South Wales, 2600, Canberra, ACT, Australia

    Hussein A. Abbass

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De Baets, B. (2012). The Quest for Transitivity, a Showcase of Fuzzy Relational Calculus. In: Liu, J., Alippi, C., Bouchon-Meunier, B., Greenwood, G.W., Abbass, H.A. (eds) Advances in Computational Intelligence. WCCI 2012. Lecture Notes in Computer Science, vol 7311. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30687-7_8

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  • DOI: https://doi.org/10.1007/978-3-642-30687-7_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30686-0

  • Online ISBN: 978-3-642-30687-7

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