Associativity equation

Summary

The associativity equation is the functional equation $F(F(x, y), z) = F(x, F(y, z))$ for an unknown binary operation $F$ on a set. Its solutions are exactly the associative binary operations, so it turns the algebraic law of associativity into a problem about unknown functions. On a real interval, its continuous solutions that are strictly increasing in each variable are the operations $F(x, y) = g^{-1}(g(x) + g(y))$ with $g$ continuous and strictly monotone. So, up to a change of scale, every such operation on an interval is ordinary addition.