# Definition:Idempotence

## Definition

Idempotence is a property of an algebraic system, or an element of an algebraic system such that:

$E \circledcirc E = E$

for an object $E$ and an operation $\circledcirc$.

Let $S$ be a set.

### Idempotent Relation

Let $\mathcal R \subseteq S \times S$ be a relation on $S$.

Then $\mathcal R$ is idempotent if and only if:

$\mathcal R \circ \mathcal R = \mathcal R$

where $\circ$ denotes composition of relations.

### Idempotent Mapping

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.

### Idempotent Element

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 $\circ$.

### Idempotent Operation

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:

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

### Idempotent Algebraic Structure

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $\circ$ be an idempotent operation on $S$.

Then $\left({S, \circ}\right)$ is an idempotent algebraic structure.

## Also known as

The concept of idempotence can also be referred to as idempotency.

## Historical Note

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