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Characterization of invariant aggregation operators. (English) Zbl 1049.68133

Aggregation of numerical data not reflecting the ordinal structure of input data suffers from the chosen scale we deal with. If the ordinal structure of input data is dominant, the corresponding aggregation should be invariant under transformations preserving the ordinal structure. An aggregation operator \(R\) acting on \(n\) inputs is a mapping from the \(n\)-dimensional unit-cube to \([0,1]\) which is nondecreasing, and \(R(0,\dots,0)=0\), \(R(1,\dots,1)=1\).
The aim of the paper is the complete characterization of all (\(n\)-ary) invariant aggregation operators. For this, a recent characterization of invariant \(n\)-ary functions by Bartlomiejczyk and Drewniak is used which is based on minimal invariant subsets of the unit-cube.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
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