Article
Full entry |
PDF
(2.5 MB)
Feedback
PDF
(2.5 MB)
Feedback
Keywords:
aggregation operator; homogeneity; kernel property
aggregation operator; homogeneity; kernel property
Summary:
Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneous aggregation operators. We discuss a relationship between homogeneity, kernel property and shift-invariance of aggregation operators. Several examples are included.
References:
[1] Aczél J., Gronau, D., Schwaiger J.: Increasing solutions of the homogeneity equation and similar equations. J. Math. Anal. Appl. 182 (1994), 436–464 DOI 10.1006/jmaa.1994.1097 | MR 1269471
[2] Calvo T., Mesiar R.: Stability of aggregation operators. In: Proc. EUSFLAT’2001, Leicester 2001, pp. 475–478 MR 1821982
[3] Calvo T., Kolesárová A., Komorníková, M., Mesiar R.: Aggregation operators: Basic concepts, issues and properties. In: Aggregation Operators. New Trends and Applications (T. Calvo, G. Mayor, and R. Mesiar, eds.), Physica–Verlag, Heidelberg 2002, pp. 3–105 MR 1936383
[4] Calvo T., Mesiar, R., Yager R. R.: Quantitative weights and aggregation. IEEE Trans. Fuzzy Systems 12 (2004), 1, 62–69 DOI 10.1109/TFUZZ.2003.822679 | MR 2073568
[5] Dujmovic J. J.: Weighted conjuctive and disjunctive means and their application in system evaluation. Univ. Beograd Publ. Elektrotehn. Fak. 483 (1974), 147–158 MR 0378884
[6] Grabisch M.: Symmetric and asymmetric integrals: the ordinal case. In: Proc. IIZUKA’2000, Iizuka 2000, CD-rom
[7] Grabisch M., Murofushi, T., Sugeno M., (eds.) M.: Fuzzy Measures and Integrals. Theory and Applications. Physica–Verlag, Heidelberg 2000 MR 1767776 | Zbl 0935.00014
[8] Klir G. J., Folger T. A.: Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey 1988 MR 0930102 | Zbl 0675.94025
[9] Kolesárová A., Mordelová J.: 1-Lipschitz and kernel aggregation operators. In: Proc. AGOP’2001, Oviedo 2001, pp. 71–75
[10] Lázaro J., Rückschlossová, T., Calvo T.: Shift invariant binary aggregation operators. Fuzzy Sets and Systems 142 (2004), 51–62 DOI 10.1016/j.fss.2003.10.031 | MR 2045342 | Zbl 1081.68106
[11] Mesiar R., Rückschlossová T.: Characterization of invariant aggregation operators. Fuzzy Sets and Systems 142 (2004), 63–73 DOI 10.1016/j.fss.2003.10.032 | MR 2045343 | Zbl 1049.68133
[12] Nagumo M.: Über eine Klasse der Mittelwerte. Japan. J. Math. 7 (1930), 71–79
[13] Rückschlossová T.: Aggregation Operators and Invariantness. Ph.D. Thesis, Slovak University of Technology, Bratislava 2004
[14] Zadeh L. A.: Fuzzy sets. Inform. and Control 8 (1965), 338–353 DOI 10.1016/S0019-9958(65)90241-X | MR 0219427 | Zbl 0139.24606
Search
Partner of
