# Min Operation on Toset is Semigroup

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## Theorem

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

Let $\map \min {x, y}$ denote the min operation on $x, y \in S$.

Then $\struct {S, \min}$ is a semigroup.

## Proof

By the definition of the min operation, either:

- $\map \min {x, y}= x$

or

- $\map \min {x, y}= y$

So $\min$ is closed on $S$.

From Min Operation is Associative:

- $\forall x, y, z \in S: \map \min {x, \map \min {y, z} } = \map \min {\map \min {x, y}, z}$

Hence the result, by definition of semigroup.

$\blacksquare$