Definition:Idempotence

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 $$\mathbb{Z}$$, but no other element of $$\mathbb{Z}$$ 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 $$\cap$$.