Min Operation on Toset forms Semigroup

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

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

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

Proof
By the definition of the min operation, either:
 * $\min \set{x, y}= x$

or
 * $\min \set{x, y}= y$

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

From Min Operation is Associative:
 * $\forall x, y, z \in S: \min \set{x, \min \set{y, z}} = \min \set{\min \set{x, y}, z}$

Hence the result, by definition of semigroup.

Also see

 * Max Operation on Toset is Semigroup