Min Semigroup is Idempotent
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Theorem
Let $\struct {S, \preceq}$ be a totally ordered set.
Then the semigroup $\struct {S, \min}$ is an idempotent semigroup.
Proof
The fact that $\struct {S, \min}$ is a semigroup is demonstrated in Min Operation on Toset forms Semigroup.
Then the min operation is idempotent:
- $\forall x \in S: \min \set {x, x} = x$
The result follows by the definition of idempotent semigroup.
$\blacksquare$