Max Semigroup is Commutative

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

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

Proof
Let $x, y \in S$.

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

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

Hence the result, by definition of commutative semigroup.