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In general (I didn't want to put this long formula in the title): $f(a_1,a_2,\dots,a_k)=g(x_1(a_{i_{1,1}},\dots,a_{i_{1,n_1}}),x_2(a_{i_{2,1}},\dots,a_{i_{2,n_2}}),\dots,x_l(a_{i_{l,1}},\dots,a_{i_{l,n_l}}))$, where $l and $a_{i_{1,1}}$ through $a_{i_{l,n_l}}$ are a permutation of $a_1$ through $a_l$.

Basically, $f$ can be "separated" into a function $g$ applied to the result of $l$ functions which take disjoint subsequences that form a partition of $f$'s arguments.

A concrete example: $f(a,b,c,d)=ab+cd$; here $g(x,y)=x+y$, $x(a,b)=ab$ and $y(c,d)=cd$. A function that wouldn't be separable like that would be $ab/c+cd$.

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    I don't think it really has a name.2012-08-16
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    What about some restricted forms? I.e., with $g$ being of a certain form, or for some specific $l$ and $k$?2012-08-16
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    You may be interested in [Hilbert's 13th](http://en.wikipedia.org/wiki/Hilbert's_thirteenth_problem)2012-08-16
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    The phrase "nomographic construction" has been associated with this type of formulation, but it would not be understood as such by most readers. See the Wikipedia link given by @LeonidKovalev. The narrow case of (summing) products of functions of one variable is of course the "separation of variables" technique taught, e.g. in introductory differential equations. In the work by Kolomogorov, Arnold, and Shimura, the phrase "superposition of functions" is used but is broader than the construction outlined here.2012-08-16

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