Max Operation on Toset forms Semigroup

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

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

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

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

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

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

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

Hence the result, by definition of semigroup.

Also see

 * Min Operation on Toset is Semigroup