# Definition:Idempotent Semigroup

(Redirected from Definition:Band)

## Definition

An idempotent semigroup is a semigroup whose operation is idempotent.

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

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

## Also known as

There are extensive bodies of research covering idempotent semigroups.

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