Abstract
The aim of the paper is to propose a graded dominance for fuzzy sets that assigns to each pair of fuzzy sets a degree in which one fuzzy set has less cardinality than another one or the cardinalities of both fuzzy sets are approximately equal. The graded dominance for fuzzy sets is a natural generalization of the dominance relation for sets. The graded dominance is then used for the introduction of a fuzzy class equivalence that satisfies a graded version of the Cantor-Bernstein theorem.
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Notes
- 1.
More precisely, Wygralak studied in [9] more general objects called “vaguely defined objects”, where fuzzy sets form a special case.
- 2.
See, Theorem 4 on p. 518 in this paper.
- 3.
See, Example 2 on p. 518 in this paper.
- 4.
For simplicity, we use in the definition the term of “fuzzy sets”, although, a more convenient denotation should be \(\mathbf {L}\)-fuzzy sets with reference to the lattice \(\mathbf {L}\).
- 5.
The proof can be designed similarly to the proof of Theorem 5.6 in [5] for finite fuzzy sets.
- 6.
We assume \({\mathscr {D}(A)}\ne \emptyset \) to avoid the empty function.
References
Bělohlávek, R.: Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publisher, New York (2002)
Bodenhofer, U.: A similarity-based generalization of fuzzy orderings preserving the classical axioms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 08(05), 593–610 (2000)
Gottwald, S.: Set theory for fuzzy sets of higher level. Fuzzy Sets Syst. 2, 125–151 (1979)
Höhle, U.: M-valued sets and sheaves over integral commutative CL-monoid. In: Rodabaugh, S.E., Klement, E.P., Höhle, U. (eds.) Applications to Category Theory to Fuzzy Subsets, pp. 33–72. Kluwer Academic Publishers, Dordrecht (1992)
Holčapek, M.: A graded approach to cardinal theory of finite fuzzy sets, part I: Graded equipollence. Fuzzy Sets Syst. (2015). http://dx.doi.org/10.1016/j.fss.2015.08.010
Holčapek, M.: Graded equipollence and fuzzy c-measures of finite fuzzy sets. In: Proceedings of 2011 IEEE International Conference on Fuzzy Systems, pp. 2375–2382, DnE Taiwan (2011)
Holčapek, M., Turčan, M.: Graded equipollence of fuzzy sets. In: Carvalho, J.P., Kaymak, D.U., Sousa, J.M.C. (eds.) Proceedings of IFSA/EUSFLAT 2009, pp. 1565–1570. European Soc Fuzzy logic & Technology, Johannes Kepler university, Austria (2009)
Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publisher, Boston (1999)
Wygralak, M.: Vaguely defined objects. Representations, fuzzy sets and nonclassical cardinality theory. Theory and Decision Library. Mathematical and Statistical Methods, vol. 33. Kluwer Academic Publisher, Dordrecht (1996)
Acknowledgments
This work was supported by the project LQ1602 IT4Innovations excellence in science.
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Holčapek, M. (2016). Graded Dominance and Cantor-Bernstein Equipollence of Fuzzy Sets. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 611. Springer, Cham. https://doi.org/10.1007/978-3-319-40581-0_41
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