# Definition:Idempotent Semigroup

## Definition

An idempotent semigroup is a semigroup whose operation is idempotent.

That is, a semigroup $\struct {S, \circ}$ is idempotent if and only if:

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

## Properties

### Property $1$

Let $x \circ y = y$ and $y \circ x = x$.

Then for all $z \in S$:

$z \circ x \circ z \circ y = z \circ y$

and:

$z \circ y \circ z \circ x = z \circ x$

### Property $2$

Let $x \circ y = x$ and $y \circ x = y$.

Then for all $z \in S$:

$x \circ z \circ y \circ z = x \circ z$

and:

$y \circ z \circ x \circ z = y \circ z$

## Also known as

There are extensive bodies of research covering idempotent semigroups.

In this literature, band is a common synonym for idempotent semigroup.

## Examples

### Idempotent Semigroup with Relation induced by Inverse

Let $\struct {S, \circ}$ be an idempotent semigroup.

Let $\RR$ be the relation on $S$ defined as:

$\forall a, b \in S: a \mathrel \RR b \iff \paren {a \circ b \circ a = a \land b \circ a \circ b = b}$

That is, such that $a$ is the inverse of $b$ and $b$ is the inverse of $a$.

Then the following properties can be deduced:

## Also see

• Results about idempotent semigroups can be found here.