Min Semigroup is Commutative

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