Comment by qbit42

I think there is still an implicit restriction on the complexity of the operator for this to be interesting. Otherwise you could design an operator which accepts a pair x,y and performs one of 2^k elementary binary operations by reading off the first k bits of x and applying the specified operation on the remainder of x and y. (This is kind of like how real-valued computational models become too powerful for complexity theory to work if you allow bitwise operations.)