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:

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

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 $a$.

Also see

• Properties of Idempotent Semigroup

• Definition:Semilattice

• Results about idempotent semigroups can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.19$