Are all n-ary operators simply compositions of binary operators? Take for example $A \times B \cdot C$ = $(A \times B) \cdot C$ where $A, B, C$ are 3-component real vectors.
We can define a 3-nary operator $\times - \cdot$ that is a composition of the two common binary operators $\times$ and $\cdot$.
The same thing happens with most functions (operators) - the way we calculate them is by doing smaller binary problems and adding together.
Every time I try to come up with an $(n > 2)$-ary operator my mind automatically looks for binary operators.
So, the question is, do there exist operators (of some weird kind in some branch of math) that cannot be decomposed into 2-ary and 1-ary operators?
Thanks.

A slightly different context provides a nice example. Let $X=\{0,1\}$ and let $\mu:X\times X\times X\to X$ be the function such that $\mu(x,y,z)$ is the element of $X$ which appears more times in the argument. This function $\mu$ cannot be written as a composition of binary operators.
