In this work, we prove that quadratic transformations of aggregation functions must come from quadratic aggregation functions. We also show that this is different from quadratic transformations of (multivariate) semi-copulas and quasi-copulas. In the latter case, those two classes are actually the same and consists of convex combinations of the identity map and another fixed quadratic transformation. In other words, it is a convex set with two extreme points. This result is different from the bivariate case in which the two classes are different and both are convex with four extreme points.
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