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Formally, the minimum and maximum functions can be considered as averaging, however they are the limiting cases, right on the border with conjunctive and disjunctive functions, and will be treated in Chapter 3. There are also some types of mixed aggregation functions, such as uninorms or nullnorms, that include averaging functions as particular cases; these will be treated in Chapter 4. Measure of orness The measure of orness, also called the degree of orness or attitudinal character, is an important numerical characteristic of averaging aggregation functions.

Xn−1 ), xn ), starting with f2 = f22 . Each extended aggregation function built from an associative bivariate aggregation function is recursive by Montero, but the converse is not true. 41. Define f2n (t1 , t2 ) = (n−1)t1 +t2 . n Then fn (x) = 1 n n xi , the i=1 arithmetic mean (which is not associative). 42 (Decomposable extended aggregation function). An extended aggregation function F is decomposable if for all m, n = 1, 2, . . and for all x ∈ [0, 1]m , y ∈ [0, 1]n : fm+n (x1 , . . , xm , y1 , .

37 (One divisor). An element a ∈]0, 1[ is a one divisor of an aggregation function f if for all i = 1, . . , the equality f (x1 , . . , xi−1 , a, xi+1 , . . , xn ) = 1, can hold for some x < 1 with a at any position. The interpretation is similar: the value of any inputs larger than a can make the output f (x) = 1, even if none of the inputs is actually 1. On the other hand, if b is not a one divisor, then the output cannot be one if all the inputs are no larger than b. The following property is useful for construction of n-ary aggregation functions from a single two-variable function.

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Collusion-Secure Fingerprinting for Digital Data by Boneh Shaw

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