Min Semigroup is Commutative

Theorem
Let $\struct {S, \preceq}$ be a totally ordered set.

Then the semigroup $\struct {S, \min}$ is commutative.

Proof
Let $x, y \in S$.

From Min Operation is Commutative:
 * $\map \min {x, y} = \map \min {y, x}$

Hence the result, by definition of commutative semigroup.

Also see

 * Max Semigroup is Commutative