# zbMATH — the first resource for mathematics

Fuzzy t-conorm integral with respect to fuzzy measures: Generalization of Sugeno integral and Choquet integral. (English) Zbl 0733.28014
Summary: This article discusses integrals with respect to Sugeno’s fuzzy measures. Sugeno integrals and Choquet integrals with respect to general fuzzy measures are generalized in the setting of a t-conorm algebraic system. First t-conorm integrals with respect to decomposable measures based on t-conorm algebraic system are defined. They are a modification of the Weber integral. Secondly they are extended to fuzzy t-conorm integrals with respect to general fuzzy measures. The extension is made in the same way as the extension of the Lebesgue integral to the Choquet integral. The class of the fuzzy t-conorm integrals includes Choquet integrals and Sugeno integrals. Properties of the integrals, especially their representation, are discussed.

##### MSC:
 2.8e+11 Fuzzy measure theory
Full Text:
##### References:
 [1] Choquet, G., Theory of capacities, Ann. inst. Fourier, 5, 131-295, (1953) · Zbl 0064.35101 [2] Frank, M.J., On the simultaneous associativity of F(x,y) and x + y − F(x,y), Aequationes math., 19, 194-226, (1979) · Zbl 0444.39003 [3] Kruse, R., Fuzzy integrals and conditional fuzzy measures, Fuzzy sets and systems, 10, 309-313, (1983) · Zbl 0525.28001 [4] Menger, K., Statistical metric spaces, (), 535-537 · Zbl 0063.03886 [5] Murofushi, T.; Sugeno, M., An interpretation of fuzzy measures and Choquet’s integral as an integral with respect to a fuzzy measure, Fuzzy sets and systems, 29, 201-227, (1989) · Zbl 0662.28015 [6] Murofushi, T., Fuzzy measures and Choquet’s integral, (), 58-69 [7] Sugeno, M., Theory of fuzzy integrals and its applications, () · Zbl 0316.60005 [8] Sugeno, M.; Murofushi, T., Pseudo-additive measures and integrals, J. math. anal. appl., 122, 197-222, (1987) · Zbl 0611.28010 [9] Weber, S., ⊥-decomposable measures and integrals for Archimedean t-conorms ⊥, J. math. anal. appl., 101, 114-138, (1984) · Zbl 0614.28019 [10] Weber, S., Two integrals and some modified versions—critical remarks, Fuzzy sets and systems, 20, 97-105, (1986) · Zbl 0595.28012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.