## A note on the super-additive and sub-additive transformations of aggregation functions: the multi-dimensional case.(English)Zbl 1424.26032

Summary: 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.

### MSC:

 26B40 Representation and superposition of functions 26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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