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.

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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.
DOI : 10.14736/kyb-2017-1-0129
Classification : 47H04, 47S40
Keywords: aggregation function; overrunning and underrunning property; sub-additive and super-additive transformation 
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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. http://geodesic.mathdoc.fr/articles/10.14736/kyb-2017-1-0129/

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