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

Question

When functions $f$ and $g$ have the property that $f(g(x)) = g(f(x))$ for all $x$ in the domains I call this property 'commutativity'. (usually both functions map from $\mathbb{R}$ to $\mathbb{R}$, so the problem of domain/range doesn't matter)

However, commutativity is actually when: $a*b=b*a$

I use it knowing that it probably isn't the right term... but I've never found out what I should call it.

Answer 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.

Answer 2

Composition is a binary operation on functions that is not necessarily commutative. However, if $f \circ g = g \circ f$, then $f$ and $g$ are said to “commute” with respect to $\circ$.

Answer 3

Indeed, commute is the correct terminology: See the Wikipedia entry for function composition. To be more precise (or verbose), you might call it "commute with respect to composition" but when we are speaking of functions $f$ and $g$, then it is clear what is meant by the phrase "$f$ and $g$ commute" as the operation that is implied is composition, not multiplication--functions operate on each other as compositions.

Answer 4

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.