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General results on the decomposition of transitive fuzzy relations

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Abstract

We study the transitivity of fuzzy preference relations, often considered as a fundamental property providing coherence to a decision process. We consider the transitivity of fuzzy relations w.r.t. conjunctors, a general class of binary operations on the unit interval encompassing the class of triangular norms usually considered for this purpose. Having fixed the transitivity of a large preference relation w.r.t. such a conjunctor, we investigate the transitivity of the strict preference and indifference relations of any fuzzy preference structure generated from this large preference relation by means of an (indifference) generator. This study leads to the discovery of two families of conjunctors providing a full characterization of this transitivity. Although the expressions of these conjunctors appear to be quite cumbersome, they reduce to more readily used analytical expressions when we focus our attention on the particular case when the transitivity of the large preference relation is expressed w.r.t. one of the three basic triangular norms (the minimum, the product and the Łukasiewicz triangular norm) while at the same time the generator used for decomposing this large preference relation is also one of these triangular norms. During our discourse, we pay ample attention to the Frank family of triangular norms/copulas.

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Authors and Affiliations

  1. Department of Statistics and O.R., University of Oviedo, Oviedo, Spain

    Susana Díaz & Susana Montes

  2. Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Ghent, Belgium

    Bernard De Baets

Authors
  1. Susana Díaz
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  2. Bernard De Baets
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  3. Susana Montes
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Correspondence to Susana Díaz.

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Díaz, S., De Baets, B. & Montes, S. General results on the decomposition of transitive fuzzy relations. Fuzzy Optim Decis Making 9, 1–29 (2010). https://doi.org/10.1007/s10700-010-9074-1

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