Idempotent Semigroup/Examples/Relation induced by Inverse Element
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Example of Idempotent Semigroup
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 $a$.
Properties
The following properties can be deduced:
Property $1$
Let $x \circ y = y$ and $y \circ x = x$.
Then for all $z \in S$:
- $\paren {z \circ x} \mathrel \RR \paren {z \circ y}$
and:
- $\paren {x \circ z} \mathrel \RR \paren {y \circ z}$
Property $2$
Let $x \circ y = x$ and $y \circ x = y$.
Then for all $z \in S$:
- $\paren {x \circ z} \mathrel \RR \paren {y \circ z}$
and:
- $\paren {z \circ x} \mathrel \RR \paren {z \circ y}$
Property $3$
Let:
\(\ds x \circ y\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds y \circ x\) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds y \circ z\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds z \circ y\) | \(=\) | \(\ds z\) |
Then:
- $x \mathrel \RR z$
Property $4$
$\RR$ is an equivalence relation.
Property $5$
$\RR$ is a congruence relation for $\circ$.
Property $6$
The quotient structure $\struct {S / \RR, \circ_\RR}$ is a commutative idempotent semigroup.
The equivalence classes under $\RR$ are anticommutative subsemigroups of $\struct {S, \circ}$.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Exercise $11.19$