## Transitive decomposition of fuzzy preference relations: The case of nilpotent minimum.(English)Zbl 1249.91024

Summary: Transitivity is a fundamental notion in preference modelling. In this work, we study this property in the framework of additive fuzzy preference structures. In particular, we depart from a large preference relation that is transitive w.r.t. the nilpotent minimum $$t$$-norm and decompose it into an indifference and strict preference relation by means of generators based on $$t$$-norms, i.e., using a Frank $$t$$-norm as indifference generator. We identify the strongest type of transitivity these indifference and strict preference components show, both in general and for the important class of weakly complete large preference relations.

### MSC:

 91B08 Individual preferences 06F05 Ordered semigroups and monoids 03E72 Theory of fuzzy sets, etc.
Full Text:

### References:

 [1] Butnariu D., Klement E. P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer, Dordrecht 1993 · Zbl 0804.90145 [2] Dasgupta M., Deb R.: Transitivity and fuzzy preferences. Social Choice and Welfare 13 (1996), 305-318 · Zbl 1075.91526 [3] Dasgupta M., Deb R.: Factoring fuzzy transitivity. Fuzzy Sets and Systems 118 (2001), 489-502 · Zbl 1017.91012 [4] Baets B. De, Fodor J.: Twenty years of fuzzy preference structures (1978-1997). JORBEL 37 (1997), 61-82 · Zbl 0926.91012 [5] Baets B. De, Fodor J.: Generator triplets of additive fuzzy preference structures. Proc. Sixth Internat. Workshop on Relational Methods in Computer Science, Tilburg, The Netherlands, 2001, pp. 306-315 [6] Baets B. De, Walle, B. Van De, Kerre E.: Fuzzy preference structures without incomparability. Fuzzy Sets and Systems 76 (1995), 333-348 · Zbl 0858.90001 [7] Díaz S., Baets, B. De, Montes S.: On the transitivity of fuzzy indifference relations. Fuzzy Sets and Systems - IFSA 2003 (T. Bilgiç, B. DeBaets, and O. Kayak, Lecture Notes in Computer Science 2715.) Springer-Verlag, Berlin 2003, pp. 87-94 · Zbl 1132.68768 [8] Díaz S., Baets, B. De, Montes S.: $$T$$-Ferrers relations versus $$T$$-biorders. Fuzzy Sets and Systems - IFSA 2003 (T. Bilgiç, B. DeBaets, and O. Kayak, Lecture Notes in Computer Science 2715.) Springer-Verlag, Berlin 2003, pp. 269-276 · Zbl 1132.68769 [9] Fodor J.: Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems 69 (1995), 141-156 · Zbl 0845.03007 [10] Fodor J., Roubens M.: Valued preference structures. European J. Oper. Res. 79 (1994), 277-286 · Zbl 0812.90005 [11] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht 1994 · Zbl 0827.90002 [12] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (I) Rotation construction. J. Appl. Non-Classical Logics 10 (2000), 83-92 · Zbl 1050.03505 [13] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (II) Rotation-annihilation construction. J. Appl. Non-Classical Logics 11 (2001), 351-366 · Zbl 1050.03505 [14] Jenei S.: Structure of left-continuous triangular norms with strong induced negations. (III) Construction and decomposition. Fuzzy Sets and Systems 128 (2002), 197-208 · Zbl 1050.03505 [15] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer, Dordrecht 2000 · Zbl 1087.20041 [16] Perny P.: Modélisation, agrégation et expoitation des préférences floues dans une problématique de rangement. Ph.D. Thesis, Université Paris Dauphine, Paris 1992 [17] Perny P., Roy B.: The use of fuzzy outranking relations in preference modelling. Fuzzy Sets and Systems 49 (1992), 33-53 · Zbl 0765.90003 [18] Roubens M., Vincke, Ph.: Preference Modelling. Springer-Verlag, Berlin 1985 · Zbl 0612.92020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.