Max Semigroup is Idempotent

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

Then every element of the semigroups $\left({S, \max}\right)$ and $\left({S, \min}\right)$ is idempotent.

Proof
Let $x \in S$.

We have that max and min are idempotent:
 * $\max \left({x, x}\right) = x$

and
 * $\min \left({x, x}\right) = x$

Hence the result.