Multiplicatively separable function

Definition

For a function of two variables

Suppose $F$ is a function of two variables $x$ and $y$. We say that $F$ is multiplicatively separable if there exist functions $f,g$ of one variable such that:

$F(x,y)=f(x)g(y)$

on the entire domain of $F$.

Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.

For a function of many variables

Suppose $F$ is a function of $n$ variables $x_1,x_2,\dots ,x_n$. We say that $F$ is completely multiplicatively separable if there exist functions $f_1,f_2,\dots ,f_n$, each a function of one variable, such that:

$F(x_1,x_2,\dots ,x_n)=f_1(x_1)f_2(x_2)\dots f_n(x_n)$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially multiplicatively separable: if we express the set $\{1,2,\dots ,n\}$ as a union of two disjoint subsets $A,B$, $F$ is multiplicatively separable with respect to the partition if there exist functions $f_A,f_B$ such that:

$F(x_1,x_2,\dots ,x_n)=f_A(\text{only the variables}{x}_i,i\in A)f_B(\text{only the variables}{x}_i,i\in B)$