Max Semigroup is Commutative

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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.

$\blacksquare$


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