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I'm looking for a definition for a particular function(-input) property. Considering a function $f$ that takes as input a list of elements and produces in output just one element, which is the property that says that the result of $f$ for a given list of elements is EQUAL to the result of the same function $f$ that takes as input a list of elements which are the results of $f$ on subset to the given list?

I provide here some example because I understand the description is quite messy: Given the list [1,2,3,4,5] the first two examples abide by the property meanwhile the third does not:

$max([1,2,3,4,5]) = 5 = max([max([1,2]),max([3,4,5])])$ $avg([1,2,3,4,5]) = 3 = avg([avg([1,2]),avg([3]),avg([4,5])])$ $avg([1,2,3,4,5]) = 3 \neq 2,75 = avg([avg([1,2]),avg([3,4,5])])$

Personally, I think is a sort of functions associativity and I think the second case is just a fortuitous one.

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Properties like the one described in the OP are studied in the context of aggregation functions (I refer, in particular to pag.32-33 and pag.34 for decomposability). As motivating example, I would consider means, generalized means and the celebrated quasi-arithmetic or Kolmogorov-Nagumo means.

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  • $\begingroup$ Thanks, right what I was looking for! $\endgroup$ – Nicola Sep 9 '14 at 14:11
  • $\begingroup$ You are welcome :) The book is quite nice. It contains many examples coming from applications, like copulae. $\endgroup$ – Avitus Sep 9 '14 at 14:15

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