Two integrals and some modified versions - critical remarks. (English) Zbl 0595.28012
Summary: The aim of this paper is to discuss different constructions of integrals based on \(\perp\)-decomposable measures. According to the classification of the continuous t-conorms \(\perp\) in essentially two types namely \(\vee\) and Archimedean t-conorms, there exist mainly two types of integrals namely the constructions of Sugeno and of the author. Further constructions corresponding to the Archimedean case result to be special cases or not well defined. In all cases a crucial property is some restricted distribution law for the pair (\(\perp,\square)\) with an appropriate operation \(\square\). Some applications shall illustrate the use of the two integrals.
Keywords:
distributivity; Sugeno’s fuzzy integral; Weber’s fuzzy integral; constructions of integrals; decomposable measures; Archimedean t-conormsReferences:
| [1] | Aczél, J., Lectures on Functional Equations and their Applications (1966), Academic Press: Academic Press Amsterdam · Zbl 0139.09301 |
| [2] | Batle, N.; Trillas, E., Entropy and fuzzy integral, J. Math. Anal. Appl., 69, 469-474 (1979) · Zbl 0421.28015 |
| [3] | Barnard, G. A., Statistical inference, J. Roy. Statist. Soc. Ser. B, 11, 115-149 (1949) · Zbl 0039.35401 |
| [4] | Dubois, D.; Prade, H., A class of fuzzy measures based on triangular norms, Internat. J. General Systems, 8, 43-61 (1982) · Zbl 0473.94023 |
| [5] | Frank, M. J., On the simultaneous associativity of F (x ,y) and x + y −F (x y), Aequationes Mathematicae, 19, 194-226 (1979) · Zbl 0444.39003 |
| [6] | Höhle, U., A mathematical theory of uncertainty, (Yager, R. R., Fuzzy Sets and Possibility Theory (1982), Pergamon Press: Pergamon Press New York), 344-355 |
| [7] | de Fériet, J. Kampé; Forte, B., Information et probabilité, C.R. Acad. Sci. Paris, 265, 110-114 (1967) · Zbl 0201.53002 |
| [8] | Kruse, R., Fuzzy integrals conditional fuzzy measures, Fuzzy Sets and Systems, 10, 309-313 (1983) · Zbl 0525.28001 |
| [9] | Ling, C. H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401 |
| [10] | Mostert, P. S.; Shields, A. L., On the structure of semigroups on a compact manifold with boundary, Ann. Math., 65, 117-143 (1957) · Zbl 0096.01203 |
| [11] | Puri, M. L.; Ralescu, D., A possibility measure is not a fuzzy measure, Fuzzy Sets and Systems, 7, 311-313 (1982) · Zbl 0543.28002 |
| [12] | Richter, H., Zur Grundlegung der Wahrscheinlichkeitstheorie II, Math. Annalen, 125, 223-234 (1953) · Zbl 0051.10202 |
| [13] | Schweizer, B.; Sklar, A., Associative functions and statistical triangle inequalities, Publ. Math. Debrecen, 8, 169-186 (1961) · Zbl 0107.12203 |
| [14] | Schwyhla, W., About the isomorphism between some Sugeno-measures and classical measures, (Klement, E. P.; E. P., Klement, Proc. 2nd Int. Sem. on Fuzzy Set Theory. Proc. 2nd Int. Sem. on Fuzzy Set Theory, Linz (1980)), 143-176 · Zbl 0475.28003 |
| [15] | Sugeno, M., Theory of fuzzy integrals and its applications, (Thesis (1974), Tokyo Institute of Technology: Tokyo Institute of Technology New York) · Zbl 0316.60005 |
| [16] | Weber, S., ⊥-decomposable measures integrals for Archimedean t-conorms ⊥, J. Math. Anal. Appl., 101, 114-138 (1984) · Zbl 0614.28019 |
| [17] | Weber, S., A general concept of fuzzy connectives negations implications based on t-norms and t-conorms, Fuzzy Sets and Systems, 11, 115-134 (1983) · Zbl 0543.03013 |
| [18] | Weber, S., Measures of fuzzy sets measures of fuzziness, Fuzzy Sets and Systems, 13, 247-271 (1984) · Zbl 0576.28011 |
| [19] | Weber, S., Decomposable measures and measures of information for crisp and fuzzy sets, (Sanchez, E., Fuzzy Information, Knowledge Representation and Decision Analysis (1983), Pergamon Press), 321-327 · Zbl 0575.94006 |
| [20] | Weber, S., How to measure fuzzy sets, (Klement, E. P., Proc. 5th Int. Sem. on Fuzzy Set Theory. Proc. 5th Int. Sem. on Fuzzy Set Theory, Linz (1983)) |
| [21] | Weber, S., Measure theory for fuzzy sets based on Archimedean semigroups, (1st IFSA Congress. 1st IFSA Congress, Palma de Mallorca (1985)) |
| [22] | S. Weber, Generalized probabilities, to appear.; S. Weber, Generalized probabilities, to appear. |
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