# Definition:Idempotence/Operation

## Definition

Let $S$ be a set.

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

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

The binary operation $\circ$ is idempotent if and only if:

$\forall x \in S: x \circ x = x$

## Also see

Examples of idempotent operations: