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.
$\blacksquare$