Transitivity-preserving fuzzification schemes for cardinality-based similarity measures. (English) Zbl 1061.90080

Summary: A family of fuzzification schemes is proposed that can be used to transform cardinality-based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t-norms. The fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity measures for ordinary sets and it is demonstrated that transitivity is preserved in the fuzzification process.


90B99 Operations research and management science
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
Full Text: DOI


[1] Castiñeira, E.; Cubillo, S.; Trillas, E., On a similarity ratio, (Proceedings of the 1999 Eusflat-Estylf Joint Conference (1999), Palma: Palma Spain), 417-420
[3] De Baets, B.; De Meyer, H.; Naessens, H., A class of rational cardinality-based similarity measures, Journal of Computers and Applied Mathematics, 51-69 (2001) · Zbl 0985.03045
[4] De Baets, B.; Mesiar, R., Pseudo-metrics and \(T\)-equivalences, Journal of Fuzzy Mathematics, 5, 471-481 (1997) · Zbl 0883.04007
[5] De Sutter, R.; De Meyer, H.; De Baets, B.; Naessens, H., Fuzzy similarity measures and tree comparison, (Hu, C.; Wang, P.; Wang, T., Proceedings of the Atlantic Symposium on Computational Biology (2001), Genome Information Systems and Technology: Genome Information Systems and Technology Durham, North-Carolina, USA), 87-91
[6] Dubois, D.; Prade, H., A unifying view of comparison indices in a fuzzy set-theoretic framework, (Yager, R., Fuzzy Set and Possibility Theory: Recent Developments (1982), Pergamon Press: Pergamon Press New York), 3-13
[7] Fodor, J.; Roubens, M., Fuzzy Preference Modelling and Multicriteria Decision Support (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0827.90002
[8] Frank, M., On the simultaneous associativity of \(F(x,y)\) and \(x+y−F(x,y)\), Aequationes Mathematics, 19, 141-160 (1979) · Zbl 0444.39003
[9] Jaccard, P., Nouvelles recherches sur la distribution florale, Bulletin de la Société Vaudoise des Sciences Naturelles, 44, 223-270 (1908)
[10] Klement, E.; Mesiar, R.; Pap, E., (Triangular Norms, Trends in Logic, Studia Logica Library, vol. 8 (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht)
[11] Pykacz, J.; D’Hooghe, B., Bell-type inequalities in fuzzy probability calculus, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 9, 263-275 (2001) · Zbl 1113.03344
[13] Sarker, B., The resemblance coefficients in group technology: A survey and comparative study of relational metrics, Computers and Engineering, 30, 103-116 (1996)
[14] Sokal, R.; Michener, C., A statistical method for evaluating systematic relationships, University Kansas Science Bulletin, 38, 1409-1438 (1958)
[15] Trillas, E.; Valverde, Ll, An inquiry into indistinguishability operators, (Skala, H.; Termini, S.; Trillas, E., Aspects of Vagueness (1984), Reidel), 231-256 · Zbl 0564.03027
[16] Zadeh, L., Fuzzy sets, Information and Control, 8, 338-353 (1965) · Zbl 0139.24606
[17] Zadeh, L., Similarity relations and fuzzy orderings, Information Science, 3, 177-200 (1971) · Zbl 0218.02058
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.