Geodesic
A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case
Kybernetika, Tome 53 (2017) no. 1, pp. 129-136
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

For an aggregation function $A$ we know that it is bounded by $A^*$ and $A_*$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^*$ is directionally convex, then $A=A^*$ and $A_*$ is linear; similarly, if $A_*$ is directionally concave, then $A=A_*$ and $A^*$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively.
For an aggregation function $A$ we know that it is bounded by $A^*$ and $A_*$ which are its super-additive and sub-additive transformations, respectively. Also, it is known that if $A^*$ is directionally convex, then $A=A^*$ and $A_*$ is linear; similarly, if $A_*$ is directionally concave, then $A=A_*$ and $A^*$ is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively.
DOI : 10.14736/kyb-2017-1-0129
Classification : 47H04, 47S40
Keywords: aggregation function; overrunning and underrunning property; sub-additive and super-additive transformation 
@article{10_14736_kyb_2017_1_0129,
     author = {Kouchakinejad, Fateme and \v{S}ipo\v{s}ov\'a, Alexandra},
     title = {A note on the super-additive and sub-additive transformations of aggregation functions: {The} multi-dimensional case},
     journal = {Kybernetika},
     pages = {129--136},
     year = {2017},
     volume = {53},
     number = {1},
     doi = {10.14736/kyb-2017-1-0129},
     mrnumber = {3638560},
     zbl = {06738598},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-1-0129/}
}
TY  - JOUR
AU  - Kouchakinejad, Fateme
AU  - Šipošová, Alexandra
TI  - A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case
JO  - Kybernetika
PY  - 2017
SP  - 129
EP  - 136
VL  - 53
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-1-0129/
DO  - 10.14736/kyb-2017-1-0129
LA  - en
ID  - 10_14736_kyb_2017_1_0129
ER  - 
%0 Journal Article
%A Kouchakinejad, Fateme
%A Šipošová, Alexandra
%T A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case
%J Kybernetika
%D 2017
%P 129-136
%V 53
%N 1
%U http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-1-0129/
%R 10.14736/kyb-2017-1-0129
%G en
%F 10_14736_kyb_2017_1_0129
Kouchakinejad, Fateme; Šipošová, Alexandra. A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case. Kybernetika, Tome 53 (2017) no. 1, pp. 129-136. doi: 10.14736/kyb-2017-1-0129

[1] Arlotto, A., Scarsini, M.: Hessian orders and multinormal distributions. J. Multivariate Anal. 100 (2009), 2324-2330. | DOI | MR | Zbl

[2] Beliakov, G., Pradera, A., Calvo, T.: Aggregation Functions: A Guide for Practitioners. Springer-Verlag, Berlin 2007. | DOI

[3] Bernstein, F., Doetsch, G.: Zur Theorie der konvexen Functionen. Math. Annalen 76 (1915), 514-526. | DOI | MR

[4] Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation Functions (Encyklopedia of Mathematics and its Applications). Cambridge University Press, 2009. | DOI | MR

[5] Greco, S., Mesiar, R., Rindone, F., Šipeky, L.: The superadditive and the subadditive transformations of integrals and aggregation functions. Fuzzy Sets and Systems 291 (2016), 40-53. | DOI | MR

[6] Kouchakinejad, F., Šipošová, A.: A note on the super-additive and sub-additive transformations of aggregation functions: The one-dimensional case. In: Mathematics, Geometry and their Applications, STU Bratislava 2016, pp. 15-19. | MR

[7] Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Second edition. Birkhäuser, 2009. | DOI | MR

[8] Marinacci, M., Montrucchio, L.: Ultramodular functions. Math. Oper. Res. 30 (2005), 311-332. | DOI | MR | Zbl

[9] Murenko, A.: A generalization of Bernstein-Doetsch theorem. Demonstration Math. XLV 1 (2012), 35-38. | MR | Zbl

[10] Šipošová, A.: A note on the superadditive and the subadditive transformations of aggregation functions. Fuzzy Sets and Systems 299 (2016), 98-104. | DOI | MR

[11] Šipošová, A., Šipeky, L.: On aggregation functions with given superadditive and subadditive transformations. In: Congress on Information Technology, Computational and Experimental Physics, Krakow (Poland) 2015, pp. 199-202.

[12] Šipošová, A., Šipeky, L., Širáň, J.: On the existence of aggregation functions with given super-additive and sub-additive transformations. Fuzzy Sets and Systems (in press). | DOI | MR

Cité par Sources :