Structure of transitive valued binary relations

Abstract The paper is concerned with the transitivity property of valued binary relations. A general representation theorem is established for any T -transitive valued binary relation, where T is a left-continuous triangular norm. The study is carried out by using traces of valued binary relations. Some results that have already been cited in the literature are shown to be consequences of the representation theorem. Maximal T -transitive relations contained in a given relation are also investigated and a procedure is developed to construct them.

[1]  H. J. Skala,et al.  Aspects of vagueness , 1984 .

[2]  L. Valverde.  On the structure of F-indistinguishability operators , 1985 .

[3]  Georges Bordes.  On the possibility of reasonable consistent majoritarian choice: Some positive results , 1983 .

[4]  S. Ovchinnikov.  Representations of Transitive Fuzzy Relations , 1984 .

[5]  C. J. Hearne.  Non-conventional Preference Relations in Decision Making , 1989 .

[6]  Philippe Vincke,et al.  Multicriteria Decision-Aid , 1992 .

[7]  S. Ovchinnikov.  STRUCTURE OF FUZZY BINARY RELATIONS , 1981 .

[8]  P. Fishburn.  Binary choice probabilities: on the varieties of stochastic transitivity , 1973 .

[9]  J. Fodor.  On fuzzy implication operators , 1991 .

[10]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[11]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[12]  Lotfi A. Zadeh,et al.  Similarity relations and fuzzy orderings , 1971, Inf. Sci..

[13]  J. Fodor.  Traces of fuzzy binary relations , 1992 .

[14]  E. Szpilrajn.  Sur l'extension de l'ordre partiel , 1930 .

[15]  Bruno Leclerc,et al.  Caractérisation, construction et dénombrement des ultramétriques supérieures minimales , 1986 .

[16]  R. Luce.  A PROBABILISTIC THEORY OF UTILITY , 1958 .

[17]  E. Jacquet-Lagreze,et al.  Représentation de quasi-ordres et de relations probabilistes transitives sous forme standard et méthodes d'approximation , 1978 .

[18]  A. Sen,et al.  Choice Functions and Revealed Preference , 1971 .

[19]  P. Vincke,et al.  Biorder families, valued relations and preference modelling , 1986 .

[20]  S. Ovchinnikov.  On the transitivity property , 1986 .

[21]  S. Weber.  A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms , 1983 .

[22]  Bernard Monjardet,et al.  A Generalisation of Probabilistic Consistency: Linearity Conditions for Valued Preference Relations , 1988 .

[23]  Bernard Roy,et al.  Aide multicritère à la décision : méthodes et cas , 1993 .

[24]  Liu Dsosu,et al.  Fuzzy random measure and its extension theorem , 1983 .

[25]  Nicholas R. Miller.  A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting , 1980 .

[26]  S. Ovchinnikov.  Similarity relations, fuzzy partitions, and fuzzy orderings , 1991 .

[27]  Fred S. Roberts,et al.  Homogeneous families of semiorders and the theory of probabilistic consistency , 1971 .