There is a canonical bijection given by deleting the inside brackets. The diagonal function is given by .
Functions induce a function given by .
We noted that is uniquely determined. The function so determined is none other than the ternary diagonal . A monoid is a set M together with special purpose functions , such that the following diagrams commute. If we write 1 for the value of at the only element of 1 and we write x y for then the above diagrams translate to the equations This time functions and are not uniquely determined by the set M . Generally there is an unambiguous multiple product function determined by the binary .