Max Semigroup is Idempotent

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

Then the semigroups $\left({S, \max}\right)$ and $\left({S, \min}\right)$ are idempotent semigroups.

Proof
The fact that $\left({S, \max}\right)$ and $\left({S, \min}\right)$ are semigroups is demonstrated in Max and Min are Semigroups.

Then from Max and Min are Idempotent:
 * $\forall x \in S: \max \left({x, x}\right) = x$

and
 * $\forall x \in S: \min \left({x, x}\right) = x$

The result follows by the definition of idempotent semigroup.