# Article

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Keywords:
fuzzy measures; distributivity law; restricted domain; pseudo- addition; pseudo-multiplication; Choquet integral; Sugeno integral
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.
References:
[1] Aczél J.: Lectures on Functional Equations and Their Applications. Academic Press, New York – London 1966 MR 0208210
[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 MR 1954643 | Zbl 1099.28007
[3] Benvenuti P., Vivona, D., Divari M.: The Cauchy equation on I-semigroups. Aequationes Math. 63 (2002), 220–230 MR 1904716
[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 MR 1456033 | Zbl 0923.94049
[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
[6] Denneberg D.: Non-additive Measure and Integral. Kluwer, Dordrecht 1994 MR 1320048 | Zbl 0968.28009
[7] Fodor J., Roubens M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht 1994 Zbl 0827.90002
[8] Grabisch M., Murofushi, T., (eds.) M. Sugeno: Fuzzy Measures and Integrals. Theory and Applications. Physica–Verlag, Heidelberg 2000 MR 1767776 | Zbl 0935.00014
[9] Grabisch M., Nguyen H. T., Walker E. A.: Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer Academic Publishers, Dordrecht 1995 MR 1472733 | Zbl 0817.94036
[10] Kruse R. L., Deeley J. J.: Joint continuity of monotonic functions. Amer. Math. Soc. 76 (1969), 74–76
[11] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. (Trends in Logic, Volume 8.) Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[12] Ling C. H.: Representations of associative functions. Publ. Math. Debrecen 12 (1965), 189–212 MR 0190575
[13] Mesiar R.: Choquet-like integrals. J. Math. Anal. Appl. 194 (1995), 477–488 MR 1345050
[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
[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 MR 1123577 | Zbl 0733.28014
[16] Pap E.: Null-Additive Set Functions. Kluwer Academic Publishers. Dordrecht 1995 MR 1368630 | Zbl 1003.28012
[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 MR 1936386 | Zbl 1025.03054
[18] Shilkret N.: Maxitive measures and integration. Indag. Math. 33 (1971), 109–116 MR 0288225
[19] Siedekum J.: Multiplikation und t-Conorm Integral. Ph.D. Thesis. Braunschweig 2002 Zbl 1196.28033
[20] Sugeno M., Murofushi T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122 (1987), 197–222 MR 0874969 | Zbl 0611.28010
[21] Wang Z., Klir G. J.: Fuzzy Measure Theory. Plenum Press, New York 1992 MR 1212086 | Zbl 0812.28010
[22] Weber S.: \$\perp \$-decomposable measures and integrals for Archimedean t-conorms \$\perp \$. J. Math. Anal. Appl. 101 (1984), 114–138 MR 0746230 | Zbl 0614.28019

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