Min Semigroup is Idempotent

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 is 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.

Also see

 * Max Semigroup is Idempotent