Definition:Idempotent Semigroup
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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:
Properties
Also see
- 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 $11.19$