# Max Semigroup is Commutative

## Theorem

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

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

## Proof

Let $x, y \in S$.

$\map \max {x, y} = \map \max {y, x}$

Hence the result, by definition of commutative semigroup.

$\blacksquare$