Given a group $(G,f)$, I'm trying to characterize the graph of $f$ maybe in a more visual or geometric way. I'll explain a bit more:
1) because $f$ admits the zero element axiom of a group operation, we must have: $f$ restricted to the line $x=0$ is the identity function on $M$, and same for the line $y=0$. (we can also replace zero by other unique element)
2) because $f$ admits the inverse axiom of a group operation, we must have: $f$ restricted to every line $x=a$ is a bijection $M \to M$ (quiet easy to see).
My question is, can you find a similar characterization for the third axiom, the associativity axiom. all I can get so far is that $f(f(a,b),c) = f(a,f(b,c))$), and that's the definition. Maybe I'm looking for a small insight which is more visual, geometric, or just more developed than the definition. Perhaps something about the structure of the restrictions of this function to its fibers, or something else that I don't see and might help.
Will appreciate your help!
I'm all for exlaining math in avisual way, but in this case I'm not sure you're going to succeed. On the one hand, explaining associativity requires taking a result (i.e. a $z$ value of your plot) and feeding it back as an input (an $x$ or $y$ value). While this can be explained in terms of some obscure projection sequence, this is hardly intuitive.
On the other hand, I don't fully agree with what you have so far. For example over the reals, the function $f:(x,y)\mapsto\frac{x+y}2$ is bijective in every restriction, but there still is no universal inverse for every element. I know this function violates the first axiom, but even I'm not sure just now whether your form of the first two axioms actually implies the common first two axioms.