# A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Kybernetika (2017)

• Volume: 53, Issue: 1, page 129-136
• ISSN: 0023-5954

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## Abstract

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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.

## How to cite

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Kouchakinejad, Fateme, and Šipošová, Alexandra. "A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case." Kybernetika 53.1 (2017): 129-136. <http://eudml.org/doc/287943>.

abstract = {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.},
author = {Kouchakinejad, Fateme, Šipošová, Alexandra},
journal = {Kybernetika},
keywords = {aggregation function; overrunning and underrunning property; sub-additive and super-additive transformation},
language = {eng},
number = {1},
pages = {129-136},
publisher = {Institute of Information Theory and Automation AS CR},
title = {A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case},
url = {http://eudml.org/doc/287943},
volume = {53},
year = {2017},
}

TY - JOUR
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
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 1
SP - 129
EP - 136
AB - 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.
LA - eng
KW - aggregation function; overrunning and underrunning property; sub-additive and super-additive transformation
UR - http://eudml.org/doc/287943
ER -

## References

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1. Arlotto, A., Scarsini, M., 10.1016/j.jmva.2009.03.009, J. Multivariate Anal. 100 (2009), 2324-2330. Zbl1177.60020MR2560373DOI10.1016/j.jmva.2009.03.009
2. Beliakov, G., Pradera, A., Calvo, T., 10.1007/978-3-540-73721-6_5, Springer-Verlag, Berlin 2007. DOI10.1007/978-3-540-73721-6_5
3. Bernstein, F., Doetsch, G., 10.1007/bf01458222, Math. Annalen 76 (1915), 514-526. MR1511840DOI10.1007/bf01458222
4. Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E., 10.1017/cbo9781139644150, Cambridge University Press, 2009. MR2538324DOI10.1017/cbo9781139644150
5. Greco, S., Mesiar, R., Rindone, F., Šipeky, L., 10.1016/j.fss.2015.08.006, Fuzzy Sets and Systems 291 (2016), 40-53. MR3463652DOI10.1016/j.fss.2015.08.006
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. MR3510879
7. Kuczma, M., 10.1007/978-3-7643-8749-5, Birkhäuser, 2009. MR2467621DOI10.1007/978-3-7643-8749-5
8. Marinacci, M., Montrucchio, L., 10.1287/moor.1040.0143, Math. Oper. Res. 30 (2005), 311-332. Zbl1082.52006MR2142035DOI10.1287/moor.1040.0143
9. Murenko, A., A generalization of Bernstein-Doetsch theorem., Demonstration Math. XLV 1 (2012), 35-38. Zbl1260.26014MR2934395
10. Šipošová, A., 10.1016/j.fss.2015.10.008, Fuzzy Sets and Systems 299 (2016), 98-104. MR3510879DOI10.1016/j.fss.2015.10.008
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., 10.1016/j.fss.2016.11.009, Fuzzy Sets and Systems (in press). MR3510879DOI10.1016/j.fss.2016.11.009

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