# Definition:Idempotence/Element

## Definition

Let $S$ be a set.

Let $\circ: S \times S \to S$ be a binary operation on $S$.

Let $x \in S$ have the property that $x \circ x = x$.

Then $x \in S$ is described as idempotent under the operation $\circ$.

## Examples

### Zero is Idempotent for Addition

$0$ is idempotent under the operation of addition in the set of integers $\Z$, but no other element of $\Z$ is so.

## Also see

• Results about idempotence can be found here.