I have a function $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$. Let's say that I plot the function as a heat map (that is, I color each pixel in the square $[0,1]\times[0,1]$ according to the value of the function at that point)
I'm interested in knowing what the plot would look like given that the function is associative: $f(x,f(y,z))=f(f(x,y),z)$ for all $x,y,z$.
Commutative functions $f(x,y)=f(y,x)$ have a plot which is mirror-symmetric along the diagonal of the square. Is there, similarly, some simple visual property that characterizes all associative functions (and only them)?
(To make it simpler, we may assume that we are talking about a "regular-enough" function, e.g. piecewise continuous or something like that, so that it would make sense to speak about its plot)