TITLE: What is the term given to two functions when their order of composition does not matter?
QUESTION [17 upvotes]: 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.

REPLY [14 votes]: 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.