Abstract
A (crisp) binary relation is transitive if and only if its dual relation is negatively transitive. In preference modelling, if a weak preference relation is complete, the associated strict preference relation is its dual relation. It follows from here this well-known result: given a complete weak preference relation, it is transitive if and only if its strict preference relation is negatively transitive.
In the context of fuzzy relations, transitivity is traditionally defined by a t-norm and negative transitivity, by a t-conorm. In this setting, it is also well known that a (valued) binary relation is T-transitive if and only if its dual relation is negatively S-transitive where S stands for the dual t-conorm of the t-norm T. However, in this context there are several proposals to get the strict preference relation from the weak preference relation. Also, there are different definitions of completeness. In this contribution we depart from a reflexive fuzzy relation. We assume that this relation is transitive with respect to a conjunctor (a generalization of t-norms). We consider almost all the possible generators and therefore all the possible strict preference relations obtained from the reflexive relation and we provide a general expression for the negative transitivity that those relations satisfy.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arrow, K.J.: Social Choice and Individual Values. Wiley, Chichester (1951)
Bodenhofer, U., Klawonn, F.: A formal study of linearity axioms for fuzzy orderings. Fuzzy Sets and Systems 145, 323–354 (2004)
De Baets, B., Fodor, J.: Twenty years of fuzzy preference structures (1978-1997. Belg. J. Oper. Res. Statist. Comput. Sci. 37, 61–82 (1997)
De Baets, B., Fodor, J.: Additive fuzzy preference structures: the next generation. In: De Baets, B., Fodor, J. (eds.) Principles of Fuzzy Preference Modelling and Decision Making, pp. 15–25. Academic Press, London (2003)
De Baets, B., Van de Walle, B.: Minimal definitions of classical and fuzzy preference structures. In: Proceedings of the Annual Meeting of the North American Fuzzy Information Processing Society, USA, Syracuse, New York, pp. 299–304 (1997)
Díaz, S., De Baets, B., Montes, S.: Additive decomposition of fuzzy pre-orders. Fuzzy Sets and Systems 158, 830–842 (2007)
Díaz, S., De Baets, B., Montes, S.: On the compositional characterization of complete fuzzy pre-orders. Fuzzy Sets and Systems 159, 2221–2239 (2008)
Díaz, S., De Baets, B., Montes, S.: General results on the decomposition of transitive fuzzy relations. Fzzy Optim. Decis. Making 9, 1–29 (2010)
Díaz, S., Montes, S., De Baets, B.: Transitive decomposition of fuzzy preference relations: the case of nilpotent minimum. Kybernetika 40, 71–88 (2004)
Díaz, S., Montes, S., De Baets, B.: Transitivity bounds in additive fuzzy preference structures. IEEE Trans. on Fuzzy Systems 15, 275–286 (2007)
Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)
Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)
Roubens, M., Vincke, P.: Preference Modelling. Lecture Notes in Economics and Mathematical Systems, vol. 76. Springer, Heidelberg (1998)
Van de Walle, B., De Baets, B., Kerre, E.: A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. Part 1: General argumentation. Fuzzy Sets and Systems 97, 349–359 (1998)
Van de Walle, B., De Baets, B., Kerre, E.: Characterizable fuzzy preference structures. Annals of Operations Research 80, 105–136 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Díaz, S., De Baets, B., Montes, S. (2011). Transitivity and Negative Transitivity in the Fuzzy Setting. In: Melo-Pinto, P., Couto, P., Serôdio, C., Fodor, J., De Baets, B. (eds) Eurofuse 2011. Advances in Intelligent and Soft Computing, vol 107. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24001-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-24001-0_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24000-3
Online ISBN: 978-3-642-24001-0
eBook Packages: EngineeringEngineering (R0)