Max Operation on Toset forms Semigroup

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $\max \left({x, y}\right)$ denote the max operation on $x, y \in S$.

Then $\left({S, \max}\right)$ is a semigroup.

Proof
By the definition of the max operation, either:
 * $\max \left({x, y}\right) = x$

or
 * $\max \left({x, y}\right) = y$

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

Then we have that the max operation is associative:
 * $\forall x, y, z \in S: \max \left({x, \max \left({y, z}\right)}\right) = \max \left({\max \left({x, y}\right), z}\right)$

Hence the result, by definition of semigroup.

Also see

 * Min Operation on Toset is Semigroup