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These connectives model fuzzy set intersection and union (or conjunction and disjunction). 7) max(x) = max xi . ,n 18 √ Take f (x1√ , x2 ) = x1 x2 , which is continuous for x1 , x2 ≥ 0, and let x2 = 1. f (t, 1) = t is continuous but not Lipschitz. To see this, let t = 0 and u > 0. √ √ √ 1 Then | 0 − u| = u > M u = M |0 − u|, or u− 2 > M , for whatever choice of M , if we make u sufficiently small. Hence the Lipschitz condition fails. 24 1 Introduction The minimum and maximum are conjunctive and disjunctive extended aggregation functions respectively, and simultaneously limiting cases of averaging aggregation functions.

10 (Weighted harmonic mean). Given a weighting vector w, the weighted harmonic mean is the function n Hw (x) = i=1 wi xi −1 . 11. , W = n i=1 wi = 1, then one can either normalize it first by dividing each component by W , or use the alternative expressions for weighted geometric and harmonic means 1/W n i xw i Gw (x) = , i=1 n Hw (x) = W i=1 wi xi −1 . Fig. 2. 3D plots of weighted geometric means G( 1 , 1 ) and G( 1 , 4 ) . 2 2 5 5 Fig. 3. 3D plots of weighted harmonic means H( 1 , 1 ) and H( 1 , 4 ) .

Xn , a, . . , a ). 4 Ordered weighted averaging Ordered weighted averaging functions (OWA) are also averaging aggregation functions, which associate weights not with a particular input, but rather with its value. They have been introduced by Yager [263] and have become very popular in the fuzzy sets community. Let x be the vector obtained from x by arranging its components in non-increasing order x(1) ≥ x(2) ≥ . . ≥ x(n) . 72 (OWA). Given a weighting vector w, the OWA function is n OW Aw (x) = wi x(i) =< w, x >.

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