By Harold Hotelling

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That is, given a strong negation N , the N -dual of a weighted quasi-arithmetic mean Mw,g is in turn a weighted quasi-arithmetic mean, given by Mw,g◦N . , if N (t) = g −1 (g(0) + g(1) − g(t)) for any t ∈ [0, 1]. , if g(t) is a generating function of some weighted quasi-arithmetic mean, then ag(t) + b, a, b ∈ , a = 0 is also a generating function of the same mean 5 , provided Ran(g) = [−∞, ∞]; There are incomparable quasi-arithmetic means. Two quasi-arithmetic means Mg and Mh satisfy Mg ≤ Mh if and only if either the composite g ◦ h−1 is convex and g is decreasing, or g ◦ h−1 is concave and g increasing; The only homogeneous weighted quasi-arithmetic means are weighted power means; Weighted quasi-arithmetic means do not have a neutral element6 .

It is expressed as g ≤ f . When f is stronger that g, it is equivalently said that g is weaker than f . Not all aggregation functions are comparable. It may happen that f is stronger than g only on some part of the domain, and the opposite is true on the rest of the domain. In this case we say that f and g are incomparable. 57. 9). Any disjunctive aggregation function is stronger than an averaging function, and any averaging function is stronger than a conjunctive one. 5 Continuity and stability We will be mostly interested in continuous aggregation functions, which intuitively are such functions that a small change in the input results in a small change in the output.

1 − xn ) and we will simply say that fd is the dual of f . It is evident that the dual of a conjunctive aggregation function is disjunctive, and vice versa, regardless of what strong negation is used. Some functions are self-dual. 55 (Self-dual aggregation function). Given a strong negation N , an aggregation function f is self-dual with respect to N (for short, N -self-dual or N -invariant), if f (x) = N (f (N (x))), where N (x) = (N (x1 ), . . , N (xn )). For the standard negation we have f (x) = 1 − f (1 − x), and it is simply said that f is self-dual.