We are familiar with the algebraic meaning of associativity. But suppose we have a real function $f(x, y)$. What constraints does $$ f(f(x,y),z) = f(x,f(y,z)) \label{1}\tag{1} $$ put on $f$? Can they be visualised as certain symmetries of (the graph of) $f$, or understood in any other way which isn’t (obviously) just a restatement of \eqref{1}?
For instance, commutativity $f(x,y) = f(y,x)$ is reflection-symmetry of the graph of $f$ across the diagonal $x=y$. Linearity of $f$ means that the graph is a (sloped, non-vertical) plane crossing the origin.