# Definition:Idempotence/Mapping

< Definition:Idempotence(Redirected from Definition:Idempotent Mapping)

## Definition

Let $S$ be a set.

Let $f: S \to S$ be a mapping.

Then $f$ is **idempotent** if and only if:

- $\forall x \in S: f \left({f \left({x}\right)}\right) = f \left({x}\right)$

That is, if and only if applying the same mapping a second time to an argument gives the same result as applying it once.

And of course, that means the same as applying it as many times as you want.

The condition for **idempotence** can also be written:

- $f \circ f = f$

where $\circ$ denotes composition of mappings.