Meta-theorems on inequalities for scalar fuzzy set cardinalities. (English) Zbl 1106.03046

De Baets, B.; Janssens, S.; De Meyer, H.

Fuzzy Sets Syst. 157, No. 11, 1463-1476 (2006).

Summary: We present meta-theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification, when adopting a scalar approach to fuzzy set cardinality. The conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection. In particular, this conjunctor should fulfil a number of Bell-type inequalities. The advantage of these meta-theorems is that repetitious calculations can be avoided. This is illustrated in the demonstration of the Łukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the Jaccard coefficient, or equivalently, the triangle inequality of the corresponding dissimilarity measures.

MSC:

03E72

Theory of fuzzy sets, etc.

Keywords:

Bell inequalities; conjunctor; quasi-copula; Frank t-norm; scalar cardinality; sigma count; simple matching coefficient; Jaccard coefficient

References:

[1]	Bell, J.S., On the einstein – podolsky – rosen paradox, Physics, 1, 195-200, (1964)
[2]	Blanchard, N., Cardinal and ordinal theories about fuzzy sets, (), 149-157
[3]	Bouchon-Meunier, B.; Rifqi, M.; Bothorel, S., Towards general measures of comparison of objects, Fuzzy sets and systems, 84, 143-153, (1996) · Zbl 0917.94028
[4]	Casasnovas, J.; Rossello, F., Probabilities of fuzzy events based on scalar cardinalities, (), 92-97
[5]	Casasnovas, J.; Torrens, J., Scalar cardinalities of finite fuzzy sets for t-norms and t-conorms, Internat. J. uncertainty, fuzziness knowledge-based systems, 11, 599-614, (2003) · Zbl 1073.03032
[6]	De Baets, B.; De Meyer, H., Transitivity-preserving fuzzification schemes for cardinality-based similarity measures, European J. oper. res., 160, 726-740, (2005) · Zbl 1061.90080
[7]	De Baets, B.; De Meyer, H.; Naessens, H., A class of rational cardinality-based similarity measures, J. comput. appl. math., 132, 51-69, (2001) · Zbl 0985.03045
[8]	De Baets, B.; De Meyer, H.; Naessens, H., On rational cardinality-based inclusion measures, Fuzzy sets and systems, 128, 169-183, (2002) · Zbl 1001.03048
[9]	De Baets, B.; Mesiar, R., Metrics and \(T\)-equalities, J. math. anal. appl., 267, 531-547, (2002) · Zbl 0996.03035
[10]	Delgado, M.; Sánchez, D.; Martin Bautista, M.; Vila, M.A., A probabilistic definition of a nonconvex fuzzy cardinality, Fuzzy sets and systems, 126, 177-190, (2002) · Zbl 1002.03048
[11]	De Luca, A.; Termini, S., A definition of non-probabilistic entropy in the setting of fuzzy sets theory, Inform. and control, 20, 301-312, (1972) · Zbl 0239.94028
[12]	Dubois, D.; Prade, H., Fuzzy cardinality and the modelling of imprecise quantification, Fuzzy sets and systems, 16, 199-230, (1985) · Zbl 0601.03006
[13]	Frank, M., On the simultaneous associativity of \(f(x, y)\) and \(x + y - f(x, y)\), Aequationes math., 19, 141-160, (1979) · Zbl 0444.39003
[14]	Genest, C.; Molina, L.; Lallena, L.; Sempi, C., A characterization of quasi-copulas, J. multivariate anal., 69, 193-205, (1999) · Zbl 0935.62059
[15]	Gottwald, S., A note on fuzzy cardinals, Kybernetica, 16, 156-158, (1980) · Zbl 0434.03038
[16]	Jaccard, P., Nouvelles recherches sur la distribution florale, Bull. soci. vaudoise sci. naturelles, 44, 223-270, (1908)
[17]	Janssens, S.; De Baets, B.; De Meyer, H., Bell-type inequalities for parametric families of triangular norms, Kybernetika, 40, 89-106, (2004) · Zbl 1249.54015
[18]	Janssens, S.; De Baets, B.; De Meyer, H., Bell-type inequalities for quasi-copulas, Fuzzy sets and systems, 148, 263-278, (2004) · Zbl 1057.81011
[19]	Kaufmann, A., Introduction à la théorie des sous-ensembles flous, (1977), Masson Paris · Zbl 0346.94002
[20]	Klement, E.P.; Mesiar, R.; Pap, E., Triangular norms, trends in logic, studia logica library, vol. 8, (2000), Kluwer Dordrecht
[21]	Klement, E.P.; Mesiar, R.; Pap, E., Invariant copulas, Kybernetika, 38, 275-285, (2002) · Zbl 1264.62045
[22]	R. Nelsen, An Introduction to Copulas, Lecture Notes in Statistics, vol. 139, Springer, New York, 1999. · Zbl 0909.62052
[23]	Pappis, C.; Karacapilidis, N., A comparative assessment of measures of similarity of fuzzy values, Fuzzy sets and systems, 56, 171-174, (1993) · Zbl 0795.04007
[24]	I. Pitowsky, Quantum Probability — Quantum Logic, Lecture Notes in Physics, vol. 321, Springer, Berlin, New York, 1989. · Zbl 0668.60096
[25]	Pykacz, J.; D’Hooghe, B., Bell-type inequalities in fuzzy probability calculus, Internat. J. uncertainty, fuzziness knowledge-based systems, 9, 263-275, (2001) · Zbl 1113.03344
[26]	Ralescu, D., Cardinality, quantifiers and the aggregation of fuzzy criteria, Fuzzy sets and systems, 69, 355-365, (1995) · Zbl 0844.04007
[27]	Sokal, R.; Michener, C., A statistical method for evaluating systematic relationships, Univ. kansas sci. bull., 38, 1409-1438, (1958)
[28]	Tolias, Y.; Panas, S.; Tsoukalas, L., Generalized fuzzy indices for similarity matching, Fuzzy sets and systems, 120, 255-270, (2001) · Zbl 1013.68189
[29]	Tversky, A., Features of similarity, Psychological rev., 84, 327-352, (1977)
[30]	Wang, X.; De Baets, B.; Kerre, E., A comparative study of similarity measures, Fuzzy sets and systems, 73, 259-268, (1995) · Zbl 0852.04011
[31]	Wygralak, M., Fuzzy cardinals based on the generalized equality of fuzzy subsets, Fuzzy sets and systems, 18, 338-353, (1986) · Zbl 0597.03034
[32]	Wygralak, M., Triangular operations, negations, and scalar cardinality of a fuzzy set, (), 326-341 · Zbl 0948.03053
[33]	Wygralak, M., An axiomatic approach to scalar cardinalities of fuzzy sets, Fuzzy sets and systems, 110, 175-179, (2000) · Zbl 0945.03080
[34]	Wygralak, M., Cardinalities of fuzzy sets, (2003), Springer Berlin · Zbl 1028.03043
[35]	Zadeh, L., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606