property of a function

Hi,

I have, maybe a somehow strange question:
Let M, A be 2 sets with same order, i.e. |M| = |A|.
Let further be f: M^n -> A a function with f(m_1,...,m_n)=a, m_i \in M, a \in A.

I know that f is specified unexactly. The reason for this, is that I'm searching now for a property description. With theis property description I could check different function if they satisfy my special property. So in fact, I am searching for something like "A function is commutative, if f(x,y) = f(y,x). So then for example I can check if f(x,y) =x+y is commutative, which is surely true.

I will describe the property, for which I am searching a formal description (like in the case, where f is commutative) now with an example:
Let n=4 and f_1(m_1,m_2,m_3,m_4) =m_1+m_2+m_3 +m_4= a.
Now I can devide the function f_1 in 2 "part-functions", each computing a "part-result", e.g. the "part-functions" are:
a_1 = f'_1(m_1,m_2) = m_1 + m_2 = f_1(m_1,m_2,*,*)
a_2 = f'_1(m_3,m_4) = m_3 + m_4 = f_1(*,*,m_3,m_4)
Obviously I then have:
a_1+a_2 = f_1(m_1,m_2, m_3, m_4) = a.
That means, at first I did a "part-calculation" of the "part-result" a_1 and a_2 and then I could calculate the final result with the base-operation of f (here +) in doing: a_1+a_2.

The same is possible for a function f_2, where I only multiply several values.

But for example, for the function f_3(m_1,m_2,m_3,m_4) = m_1*m_2*m_3+m_4, cause there are 2 different base operations.

Sorry, if this is hard to understand, but maybe you have any ideas or it is a well known property of a function in math.

Thanks in advance,
Malice
.

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