Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
fuzzy sets; convexity and its generalizations; aggregation functions; fusion operators; triangular norms
fuzzy sets; convexity and its generalizations; aggregation functions; fusion operators; triangular norms
Summary:
We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F: [0,1] \times [0,1] \rightarrow [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x, y, z \in \mathbb{R}$ such that $x \le y \le z$, it holds that $\mu_X(y) \ge F(\mu_X(x),\mu_X(z))$, where $\mu_X: \mathbb{R} \rightarrow [0,1]$ stands here for the membership function that defines the fuzzy set $X$. We study the existence of such sets paying attention to different classes of aggregation operators (that is, the corresponding functions $F$, as above), and preserving $F$-convexity under aggregation of fuzzy sets. Among those typical classes, triangular norms $T$ will be analyzed, giving rise to the concept of norm convexity or $T$-convexity, as a particular case of $F$-convexity. Other different kinds of generalized convexities will also be discussed as a by-product.
References:
[1] Ammar, E., Metz, J.: On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets and Systems 49 (1992), 135-141. DOI 10.1016/0165-0114(92)90319-y | MR 1179741
[2] Beckenbach, E. F.: Generalized convex functions. Bull. Amer. Math. Soc. 43 (1937), 363-371. DOI 10.1090/s0002-9904-1937-06549-9 | MR 1563543
[3] Chalco-Cano, Y., Rufián-Lizana, A., Román-Flores, H., Osuna-Gómez, R.: A note on generalized convexity for fuzzy mappings through a linear ordering. Fuzzy Sets and Systems 231 (2013), 70-83. DOI 10.1016/j.fss.2013.07.001 | MR 3111894
[4] S., S. Díaz, Induráin, E., Janiš, V., Montes, S.: Aggregation of convex intuitionistic fuzzy sets. Inform. Sci. 308 (2015), 61-71. DOI 10.1016/j.ins.2015.03.003 | MR 3327115
[5] Dugundji, J.: Topology. Allyn and Bacon, Boston 1966. MR 0193606
[6] Froda, A.: Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles. Thèse, Hermann, Paris 1929. MR 3532965
[7] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions. Encyclopedia of Mathematics and its Applications, Cambridge University Press 2009. DOI 10.1017/cbo9781139644150 | MR 2538324 | Zbl 1206.68299
[8] Horvath, Ch. D.: Contractibility and generalized convexity. J. Math. Anal. Appl. 156 (1991), 2, 341-357. DOI 10.1016/0022-247x(91)90402-l | MR 1103017
[9] Iglesias, T., Montes, I., Janiš, V., Montes, S.: $T$-convexity for lattice-valued fuzzy sets. In: Proc. ESTYLF Conference, 2012.
[10] Janiš, V., Král', P., Renčová, M.: Aggregation operators preserving quasiconvexity. Inform. Sci. 228 (2013), 37-44. DOI 10.1016/j.ins.2012.12.003 | MR 3018702
[11] Janiš, V., Montes, S., Iglesias, T.: Aggregation of weakly quasi-convex fuzzy sets. In: Communications in Computer and Information Science: 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU 2012, Catania 2012, pp. 353-359. DOI 10.1007/978-3-642-31718-7_37
[12] Liu, X.-W., He, D.: Equivalent conditions of generalized convex fuzzy mappings. The Scientific World Journal 2014 (2014), 1-5. DOI 10.1155/2014/412534
[13] Llinares, J. V.: Abstract convexity, some relations and applications. Optimization 51 (2002), 6, 797-818. DOI 10.1080/0233193021000015587 | MR 1941715
[14] Saminger-Platz, S., Mesiar, R., Bodenhofer, U.: Domination of aggregation operators and preservation of transitivity. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(2002), Suppl., 11-35. DOI 10.1142/s0218488502001806 | MR 1962666
[15] Saminger-Platz, S., Mesiar, R., Dubois, D.: Aggregation operators and commuting. IEEE Trans. Fuzzy Systems 15 (2007), 6, 1032-1045. DOI 10.1109/tfuzz.2006.890687
[16] Syau, Y. R.: Some properties of weakly convex fuzzy mappings. Fuzzy Sets and Systems 123 (2001), 203-207. DOI 10.1016/s0165-0114(00)00090-7 | MR 1849405
[17] Wu, S.-Y., Cheng, W.-H.: A note on fuzzy convexity. Appl. Math. Lett. 17 (2004), 1124-1133. DOI 10.1016/j.aml.2003.11.003 | MR 2091846
[18] Yuan, X.-H., Lee, E. S.: The definition of convex fuzzy set. Comp. Math. Appl. 47 (2004), 1, 101-113. DOI 10.1016/s0898-1221(04)90009-0 | MR 2062729
[19] Zadeh, L. A.: Fuzzy sets. Inform. Control 8 (1965), 338-353. DOI 10.1016/s0019-9958(65)90241-x | MR 0219427 | Zbl 0942.00007
Search
Partner of
