Geodesic
On $\star $- associated comonotone functions
Kybernetika, Tome 54 (2018) no. 2, pp. 268-278
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.14736/kyb-2018-2-0268
Classification : 26A48, 28E10
Keywords: comonotone functions; binary operation; $\star $-associatedness; Sugeno integral 
@article{10_14736_kyb_2018_2_0268,
     author = {Hutn{\'\i}k, Ondrej and P\'ocs, Jozef},
     title = {On $\star $- associated comonotone functions},
     journal = {Kybernetika},
     pages = {268--278},
     year = {2018},
     volume = {54},
     number = {2},
     doi = {10.14736/kyb-2018-2-0268},
     mrnumber = {3807714},
     zbl = {06890419},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0268/}
}
TY  - JOUR
AU  - Hutník, Ondrej
AU  - Pócs, Jozef
TI  - On $\star $- associated comonotone functions
JO  - Kybernetika
PY  - 2018
SP  - 268
EP  - 278
VL  - 54
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0268/
DO  - 10.14736/kyb-2018-2-0268
LA  - en
ID  - 10_14736_kyb_2018_2_0268
ER  - 
%0 Journal Article
%A Hutník, Ondrej
%A Pócs, Jozef
%T On $\star $- associated comonotone functions
%J Kybernetika
%D 2018
%P 268-278
%V 54
%N 2
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2018-2-0268/
%R 10.14736/kyb-2018-2-0268
%G en
%F 10_14736_kyb_2018_2_0268
Hutník, Ondrej; Pócs, Jozef. On $\star $- associated comonotone functions. Kybernetika, Tome 54 (2018) no. 2, pp. 268-278. doi: 10.14736/kyb-2018-2-0268

[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 | MR

[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 | MR

[3] Denneberg, D.: Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht/Boston/London, 1994. | DOI | MR

[4] Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Theory and Decision Library C 46, Springer International Publishing 2016. | MR

[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

[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 | MR

[7] Kandel, A., Byatt, W. J.: Fuzzy sets, fuzzy algebra and fuzzy statistics. Proc. IEEE 66 (1978), 1619-1639. | DOI | MR

[8] Klement, E. P., Mesiar, R., Pap, E.: Triangular Norms. Trends in Logic. Studia Logica Library 8, Kluwer Academic Publishers, 2000. | DOI | MR | Zbl

[9] Mitrinović, D. S., Pečarić, J. E., Fink, A. M.: Classical and New Inequalities in Analysis. Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. | DOI | MR

[10] Puccetti, G., Scarsini, M.: Multivariate comonotonicity. J. Multivariate Anal. 101 (2010), 1, 291-304. | DOI | MR

[11] Suárez-García, F., Álvarez-Gil, P.: Two families of fuzzy integrals. | DOI

[12] Sugeno, M., Murofushi, T.: Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122 (1987), 197-222. | DOI | MR | Zbl

[13] Wang, Z., Klir, G.: Generalized Measure Theory. IFSR International Series on Systems Science and Engineering, Vol. 25, Springer, 2009. | DOI | MR | Zbl

Cité par Sources :