| Title:
|
Multiplication, distributivity and fuzzy-integral. II (English) |
| Author:
|
Sander, Wolfgang |
| Author:
|
Siedekum, Jens |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
41 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
[469]-496 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Based on results of generalized additions and generalized multiplications, proven in Part I, we first show a structure theorem on two generalized additions which do not coincide. Then we prove structure and representation theorems for generalized multiplications which are connected by a strong and weak distributivity law, respectively. Finally – as a last preparation for the introduction of a framework for a fuzzy integral – we introduce generalized differences with respect to t-conorms (which are not necessarily Archimedean) and prove their essential properties. (English) |
| Keyword:
|
fuzzy measures |
| Keyword:
|
distributivity law |
| Keyword:
|
restricted domain |
| Keyword:
|
pseudo- addition |
| Keyword:
|
pseudo-multiplication |
| Keyword:
|
Choquet integral |
| Keyword:
|
Sugeno integral |
| MSC:
|
20M30 |
| MSC:
|
28A12 |
| MSC:
|
28A25 |
| MSC:
|
28E10 |
| idZBL:
|
Zbl 1249.28029 |
| idMR:
|
MR2180358 |
| . |
| Date available:
|
2009-09-24T20:10:33Z |
| Last updated:
|
2015-03-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135670 |
| . |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/135664 |
| Related article:
|
http://dml.cz/handle/10338.dmlcz/135671 |
| . |
| Reference:
|
[1] Aczél J.: Lectures on Functional Equations and Their Applications.Academic Press, New York – London 1966 MR 0208210 |
| Reference:
|
[2] Benvenuti P., Mesiar, R., Vivona D.: Monotone set-functions-based integrals.In: Handbook of Measure Theory, Vol. II (E. Pap, ed.), Elsevier, Amsterdam 2002, pp. 1329–1379 Zbl 1099.28007, MR 1954643 |
| Reference:
|
[3] Benvenuti P., Vivona, D., Divari M.: The Cauchy equation on I-semigroups.Aequationes Math. 63 (2002), 220–230 MR 1904716, 10.1007/s00010-002-8020-x |
| Reference:
|
[4] Bertoluzza C., Cariolaro D.: On the measure of a fuzzy set based on continuous t-conorms.Fuzzy Sets and Systems 88 (1997), 355–362 Zbl 0923.94049, MR 1456033, 10.1016/S0165-0114(96)00053-X |
| Reference:
|
[5] deCampos L. M., Bolaños M. J.: Characterization and comparison of Sugeno and Choquet integral.Fuzzy Sets and Systems 52 (1992), 61–67 MR 1195202, 10.1016/0165-0114(92)90037-5 |
| Reference:
|
[6] Denneberg D.: Non-additive Measure and Integral.Kluwer, Dordrecht 1994 Zbl 0968.28009, MR 1320048 |
| Reference:
|
[7] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support.Kluwer Academic Publishers, Dordrecht 1994 Zbl 0827.90002 |
| Reference:
|
[8] Grabisch M., Murofushi, T., (eds.) M. Sugeno: Fuzzy Measures and Integrals.Theory and Applications. Physica–Verlag, Heidelberg 2000 Zbl 0935.00014, MR 1767776 |
| Reference:
|
[9] Grabisch M., Nguyen H. T., Walker E. A.: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference.Kluwer Academic Publishers, Dordrecht 1995 Zbl 0817.94036, MR 1472733 |
| Reference:
|
[10] Kruse R. L., Deeley J. J.: Joint continuity of monotonic functions.Amer. Math. Soc. 76 (1969), 74–76 10.2307/2316804 |
| Reference:
|
[11] Klement E. P., Mesiar, R., Pap E.: Triangular Norms.(Trends in Logic, Volume 8.) Kluwer Academic Publishers, Dordrecht 2000 Zbl 1087.20041, MR 1790096 |
| Reference:
|
[12] Ling C. H.: Representations of associative functions.Publ. Math. Debrecen 12 (1965), 189–212 MR 0190575 |
| Reference:
|
[13] Mesiar R.: Choquet-like integrals.J. Math. Anal. Appl. 194 (1995), 477–488 MR 1345050, 10.1006/jmaa.1995.1312 |
| Reference:
|
[14] Mostert P. S., Shields A. L.: On the structure of semigroups on a compact manifold with boundary.Ann. of Math. 65 (1957), 117–143 MR 0084103, 10.2307/1969668 |
| Reference:
|
[15] Murofushi T., Sugeno M.: Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral.Fuzzy Sets and Systems 42 (1991), 57–71 Zbl 0733.28014, MR 1123577 |
| Reference:
|
[16] Pap E.: Null-Additive Set Functions.Kluwer Academic Publishers. Dordrecht 1995 Zbl 1003.28012, MR 1368630 |
| Reference:
|
[17] Sander W.: Associative aggregation operators.In: Aggregation Operators. New Trends and Applications (T. Calvo, G. Mayor, and R. Mesiar, eds.), Physica–Verlag, Heidelberg – New York 2002, pp. 124–158 Zbl 1025.03054, MR 1936386 |
| Reference:
|
[18] Shilkret N.: Maxitive measures and integration.Indag. Math. 33 (1971), 109–116 MR 0288225, 10.1016/S1385-7258(71)80017-3 |
| Reference:
|
[19] Siedekum J.: Multiplikation und t-Conorm Integral.Ph.D. Thesis. Braunschweig 2002 Zbl 1196.28033 |
| Reference:
|
[20] Sugeno M., Murofushi T.: Pseudo-additive measures and integrals.J. Math. Anal. Appl. 122 (1987), 197–222 Zbl 0611.28010, MR 0874969, 10.1016/0022-247X(87)90354-4 |
| Reference:
|
[21] Wang Z., Klir G. J.: Fuzzy Measure Theory.Plenum Press, New York 1992 Zbl 0812.28010, MR 1212086 |
| Reference:
|
[22] Weber S.: $\perp $-decomposable measures and integrals for Archimedean t-conorms $\perp $.J. Math. Anal. Appl. 101 (1984), 114–138 Zbl 0614.28019, MR 0746230, 10.1016/0022-247X(84)90061-1 |
| . |
