Associativity equation

Remarks

Example: $F(x, y) = x + y + xy$ on $X = (-1, \infty)$ is a solution, since both sides equal $(1 + x)(1 + y)(1 + z) - 1$. Non-example: $F(x, y) = x - y$ on $\R$ is not a solution, since $F(F(0, 0), 1) = -1$ while $F(0, F(0, 1)) = 1$. Every operation $F(x, y) = g^{-1}(g(x) + g(y))$, with $g \colon X \to \R$ injective and $g(X)$ closed under addition, is a solution. On a real interval, the continuous solutions that are strictly increasing in each variable are all of this form, with $g$ continuous and strictly monotone.