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.

There are two trivial senses in which the answer is "yes", you can always reduce it to binary functions.
One of them is the pairing operator -- the function that takes any two objects and returns the ordered pair containing them. So, for example, given any function $f$ of four variables, we can construct a new function $g$ (of 1 variable of type "ordered pair of ordered pairs of objects") by
$$ g( ((a,b), (c, d)) ) = f(a, b, c, d) $$
It might be instructive to see this restated in terms of an ordered-pair variable. If $x$ is an ordered pair, then the function $L(x)$ is the left coordinate, and $R(x)$ is the right coordinate. Then, $g$ is defined by
$$ g(x) = f(L(L(x)), R(L(x)), L(R(x)), R(R(x)) $$
(with a lot of pain, one could write this explicitly as composition of functions, but it is painful. We use the above notations for a reason!)
Dually, there is the transpose operator. Again, if $f$ is a function of four variables, then I can define a new function $h$ that is a function of one variable, whose values are themselves functions of 3 variables, by
$$ h(a)(b, c, d) = f(a, b, c, d)$$
($h(a)$ is a function, so it makes sense to evaluate it, as above. The above defines $h(a)$ pointwise, and thus $h$ pointwise)
This can be iterated: you can have a function of one variable whose values are of type "Function of one variable whose values are of type {Function of two variables}", defined by:
$$ k(a)(b)(c,d) = f(a, b, c, d)$$
