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