Min Semigroup is Idempotent

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then the semigroup $\left({S, \min}\right)$ is an idempotent semigroup.

Proof
The fact that $\left({S, \min}\right)$ is a semigroup is demonstrated in Min Operation on Toset is Semigroup.

Then the min operation is idempotent:
 * $\forall x \in S: \min \left({x, x}\right) = x$

The result follows by the definition of idempotent semigroup.

Also see

 * Max Semigroup is Idempotent