Symmetry vs. commutativity and more.. I was reading the following segment of an article about commutativity:
"Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seen in the image on the right.
For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then a R b ⇔ b R a."
...and  the following questions came up:
If the above statements are true then can I apply this thinking to 


*

*the distributive property of an operator and a linear function  

*an associative operator and homogeneous functions

*any other property of an operator and property of a function


and if so can it go the other way around?
 In other words if I have a symmetric function for example then can I define an operator with commutativity?
I am interested in the broader interpretation involving relations and Cartesian products too.
link of the article:
https://en.wikipedia.org/wiki/Commutative_property
 A: A usual binary operation on $X$ is something like $\star$ where $x_1\star x_2$ produces a new element of $X$. It is common to define these in terms of a function $f:X\times X\to X$, by $x_1\star x_2=f(x_1,x_2)$. We could also generalize this to functions $f:X\times X\to Y$ to include things like the "dot product" of vectors. For lack of a standard term, I'll call these "generalized binary operations".
For a (generalized or not) binary operation $\star$, being commutative means $x_1\star x_2=x_2\star x_1$ for all pairs; i.e. $f(x_1,x_2)=f(x_2,x_1)$ for all pairs.
I think by "symmetric function" you meant a function $f:X\times X\to Y$ such that $f(x_1,x_2)=f(x_2,x_1)$ for all pairs. So yes, every symmetric function corresponds directly to a commutative (generalized) binary operation.
When talking about relations, one way to reinterpret a relation on $X$ that makes this connection clearer is as a function from $X\times X\to\{T,F\}$. Then a relation is symmetric precisely when its corresponding generalized binary operation is commutative.

I'm not certain what you meant by "Can I apply this thinking to..." Are you asking if other properties like distributivity and associativity can be seen in graphs of functions? 
If a graph shows you all the values of a function, then you can in-principle confirm associativity by checking all the relations. But there won't be any nice/obvious property of the picture of the graph/Cayley table (this is mentioned on the Wikipedia page for Cayley table. For a vaguely related idea, see Light's associativity test. 
Distributivity is sort of worse since we usually talk about one operation distributing over another. But if you had the entire graphs/tables for both operations, in principle you could confirm distributivity, but again there wouldn't really be an obvious property of those pictures to be inspected.
