For a set of values N, I wanted to find out how many functions existed with the following properties:
f(f(a, b), c) = f(f(a, c), b)
∃ g : g(a, f(a, b)) = b
where a, b, c ← N
Now I have an answer, but maybe not enough time to explain it. The number of such functions is:
(|N|)! * (|N|-1)! * (|N|-2)! * ... * (1)! / (|N|-1)
So, for example, I determined experimentally that there were 96 such functions for |N| = 4, and this works out as 4! * 3! * 2! * 1! / 3 = 96.
In the process of working this out, I realized that these functions have the interesting property that there is always a value i such that f(x, i) = x. I also worked out (for the heck of it) a way to derive the truth table for the nth function. I ought to write it down somewhere, or I will almost certainly forget it, but I don't have time at the moment.
The reason I was interested in this question is that this type of function could be used in establishing a shared secret for cryptography. For example, I could publish a function f, a value a, and a value f(a, b), then you could publish f(a, c). I could take your value and generate f(f(a, c), b), while you could take mine and generate the same value as f(f(a, b), c). The resulting value would be our shared secret, provided the inverse function g is difficult to discover.