## Fuzzy preference structures without incomparability.(English)Zbl 0858.90001

Summary: We establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability. This study is carried out for the fuzzy preference structures introduced recently by the authors [J. Fuzzy Math. 3, No. 2, 373-381 (1995; Zbl 0839.90002)]. A set of remarkable theorems gives detailed insight in the relationships between the sup-$${\mathcal T}$$ transitivity of the fuzzy preference and indifference relations and the sup-$${\mathcal T}$$ transitivity of the fuzzy large preference relation. Several paths of thought, involving $$t$$-norms with or without zero-divisors, are explored and, where required, illustrative counterexamples confirm the falsity of certain implications. Finally, we introduce the $$({\mathcal T},{\mathcal N})$$-Ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this Ferrers property.

### MSC:

 91B06 Decision theory 03E72 Theory of fuzzy sets, etc.

Zbl 0839.90002
Full Text:

### References:

 [1] Abbas, M.; Vincke, P., Preference structures and threshold models, Internat. J. multicriteria decision aid, 1, 327-336, (1993) [2] B. De Baets, B. Van de Walle and E. Kerre, Fuzzy preference structures and their characterization, J. Fuzzy Math., submitted. · Zbl 0839.90002 [3] B. De Baets and B. Van de Walle, Weak and strong fuzzy interval orders, Fuzzy Sets and Systems to appear. · Zbl 0870.90010 [4] Fodor, J., An axiomatic approach to fuzzy preference modelling, Fuzzy sets and systems, 52, 47-52, (1992) · Zbl 0787.90004 [5] Montjardet, B., Axiomatiques et propriétés des quasi-ordres, Math. sci. hum., 63, 51-82, (1978) · Zbl 0417.06005 [6] Ovchinnikov, S., Modelling valued preference relations, (), 64-70 · Zbl 0738.90003 [7] Ovchinnikov, S.; Roubens, M., On fuzzy strict preference, indifference and incomparability relations, Fuzzy sets and systems, 49, 15-20, (1992) · Zbl 0768.90005 [8] Roubens, M.; Vincke, P., Preference modelling, () · Zbl 0612.92020 [9] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (1983), Elsevier New York · Zbl 0546.60010 [10] Vincke, P., Multicriteria decision-aid, (1992), Wiley Chichester
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.