Max Semigroup on Toset forms Semilattice

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

Then the semigroups $\left({S, \max}\right)$ and $\left({S, \min}\right)$ are semilattices.

Proof
The Max and Min Semigroups are Commutative and idempotent.

Hence the result, by definition of a semilattice.