Min Operation on Toset forms Semigroup

Theorem
Let $\left({S, \preceq}\right)$ be a totally ordered set. Then $(S, max)$ and $(S, min)$ are semigroups.

Proof
Let $x, y, z \in S$. Since Max and Min are Associative,
 * $max(x,max(y,z)) = max(max(x,y),z)$

and
 * $min(x,min(y,z)) = min(min(x,y),z)$.