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Keywords:
comonotone functions; binary operation; $\star $-associatedness; Sugeno integral
comonotone functions; binary operation; $\star $-associatedness; Sugeno integral
Summary:
We give a positive answer to two open problems stated by Boczek and Kaluszka in their paper [1]. The first one deals with an algebraic characterization of comonotonicity. We show that the class of binary operations solving this problem contains any strictly monotone right-continuous operation. More precisely, the comonotonicity of functions is equivalent not only to $+$-associatedness of functions (as proved by Boczek and Kaluszka), but also to their $\star$-associatedness with $\star$ being an arbitrary strictly monotone and right-continuous binary operation. The second open problem deals with an existence of a pair of binary operations for which the generalized upper and lower Sugeno integrals coincide. Using a fairly elementary observation we show that there are many such operations, for instance binary operations generated by infima and suprema preserving functions.
References:
[1] Boczek, M., Kaluszka, M.: On the Minkowski-Hölder type inequalities for generalized Sugeno integrals with an application. Kybernetika 52 (2016), 3, 329-347. DOI 10.14736/kyb-2016-3-0329 | MR 3532510
[2] Boczek, M., Kaluszka, M.: On conditions under which some generalized Sugeno integrals coincide: A solution to Dubois problem. Fuzzy Sets and Systems 326 (2017), 81-88. DOI 10.1016/j.fss.2017.06.004 | MR 3694474
[3] Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht/Boston/London, 1994. DOI 10.1007/978-94-017-2434-0 | MR 1320048
[4] Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Theory and Decision Library C 46, Springer International Publishing 2016. MR 3524619
[5] Halaš, R., Mesiar, R., Pócs, J.: Congruences and the discrete Sugeno integrals on bounded distributive lattices. Inform. Sci. 367-368 (2016), 443-448. DOI 10.1016/j.ins.2016.06.017
[6] Halaš, R., Mesiar, R., Pócs, J.: Generalized comonotonicity and new axiomatizations of Sugeno integrals on bounded distributive lattices. Int. J. Approx. Reason. 81 (2017), 183-192. DOI 10.1016/j.ijar.2016.11.012 | MR 3589740
[7] Kandel, A., Byatt, W. J.: Fuzzy sets, fuzzy algebra and fuzzy statistics. Proc. IEEE 66 (1978), 1619-1639. DOI 10.1109/proc.1978.11171 | MR 0586279
[8] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic. Studia Logica Library 8, Kluwer Academic Publishers, 2000. DOI 10.1007/978-94-015-9540-7 | MR 1790096 | Zbl 1087.20041
[9] Mitrinović, D. S., Pečarić, J. E., Fink, A. M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. DOI 10.1007/978-94-017-1043-5 | MR 1220224
[10] Puccetti, G., Scarsini, M.: Multivariate comonotonicity. J. Multivariate Anal. 101 (2010), 1, 291-304. DOI 10.1016/j.jmva.2009.08.003 | MR 2557634
[11] Suárez-García, F., Álvarez-Gil, P.: Two families of fuzzy integrals. DOI 10.1016/0165-0114(86)90028-x
[12] Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122 (1987), 197-222. DOI 10.1016/0022-247x(87)90354-4 | MR 0874969 | Zbl 0611.28010
[13] Wang, Z., Klir, G.: Generalized Measure Theory. IFSR International Series on Systems Science and Engineering, Vol. 25, Springer, 2009. DOI 10.1007/978-0-387-76852-6 | MR 2453907 | Zbl 1184.28002
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