Idempotent

Idempotent functions

In the monoid ${\displaystyle (E^{E},\circ )}$ of the functions from a set ${\displaystyle E}$ to itself (see set exponentiation) with function composition${\displaystyle \circ }$, idempotent elements are the functions ${\displaystyle f\colon E\to E}$ such that ${\displaystyle f\circ f=f}$, that is such that ${\displaystyle f(f(x))=f(x)}$ for all ${\displaystyle x\in E}$ (in other words, the image ${\displaystyle f(x)}$ of each element ${\displaystyle x\in E}$ is a fixed point of ${\displaystyle f}$). For example: