Definition:Idempotent

From ProofWiki

Jump to: navigation, search

Definition

Let $S$ be a set.

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 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:

$f \circ f = f$

Let $\circ: S \times S \to S$ 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.

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

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 iff:

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

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