Idempotency

Notes

1. ^
2. ^ and ${\displaystyle g}$ commute under composition (i.e. if ${\displaystyle f\circ g=g\circ f}$) then idempotency of both ${\displaystyle f}$ and ${\displaystyle g}$ implies that of ${\displaystyle f\circ g}$, since ${\displaystyle (f\circ g)\circ (f\circ g)=f\circ (g\circ f)\circ g=f\circ (f\circ g)\circ g=(f\circ f)\circ (g\circ g)=f\circ g}$, using the associativity of composition.
3. ^, but ${\displaystyle f(g(1))=f(5)=2\neq 1}$
4. ^ and ${\displaystyle g}$ is not a necessary condition for idempotency preservation.