Max Semigroup is Commutative

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

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

Proof
Let $x, y \in S$.

From Max Operation is Commutative:
 * $\map \max {x, y} = \map \max {y, x}$

Hence the result, by definition of commutative semigroup.

Also see

 * Min Semigroup is Commutative