Definition:Idempotence

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

Then $f$ is idempotent iff:
 * $\forall x \in S: f \left({f \left({x}\right)}\right) = f \left({x}\right)$

That is, iff applying the same operation 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.

Operation
Let $\circ$ be a binary operation.

The element $x \in S$ is idempotent under the operation $\circ$ iff $x \circ x = x$.

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

If all the elements of $S$ are idempotent, then the term can be applied to the operation itself:

The binary operation $\circ$ is idempotent iff $\forall x \in S: x \circ x = x$.

Examples of idempotent operations are set union $\cup$ and set intersection $\cap$.

Historical Note
The concept of idempotence was introduced in 1870 by Benjamin Peirce.