What is the term given to two functions when their order of composition does not matter?

Solution 1:

The term is OK, and in fact refers to the same property: The set $\Bbb R^{\Bbb R}$ of all maps $\Bbb R\to\Bbb R$ is endowed with a binary operation $\circ$, the composition of functions: If $f$ and $g$ are maps $\Bbb R\to\Bbb R$, then so is $f\circ g$ (which is defined by $(f\circ g)(x):=f(g(x))$). This binary operation has many interesting properties, such as

• there is a neutral element, namely the identity function $\operatorname{id}\colon \Bbb R\to\Bbb R, x\mapsto x$. We have $\operatorname{id}\circ f=f\circ\operatorname{id}=f$ for all $f$
• it is associative: We have $f\circ(g\circ h)=(f\circ g)\circ h$. In fact, in many cases we use this associativity to prove the associativity of any other given operation.
• Sometimes, we have $f\circ g=g\circ f$. We say that these elements commute, just as we would with any other algebraic structure.

Solution 2:

The terminology "$f$ and $g$ commute" is perfectly fine and commonly used. For example, in linear algebra, it is a useful fact that two diagonalizable operators can be simultaneously diagonalized if and only if the operators commute. Another example is given by the Lie bracket of two vector fields, a very important object in geometry that measures the extent to which the flows of the vector fields commute locally.