Min Semigroup is Commutative

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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.

$\blacksquare$

Also see

• Max Semigroup is Commutative