Set of mappings/Properties of composition/Example

Let ${}L$ be a set, and let

${}M=\operatorname {Map} \,{\left(L,L\right)}\,$

be the set of all mappings from ${}L$ to itself. The composition of mappings gives a binary operation on ${}M$, which is associative, due to fact. In general, it is not commutative. The identity on ${}L$ is the neutral element. A mapping ${}f\colon L\rightarrow L$ has an inverse element if and only if it is bijective; the inverse element is just the inverse mapping.