Min Semigroup is Commutative

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

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

Proof
Let $x, y \in S$.

We have that the min operation is commutative:
 * $\min \left({x, y}\right) = \min \left({y, x}\right)$

Hence the result, by definition of commutative semigroup.

Also see

 * Max Semigroup is Commutative