Max Semigroup on Toset forms Semilattice

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

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

Proof
The Max Semigroup is Commutative and idempotent.

Hence the result, by definition of a semilattice.

Also see

 * Min Semigroup on Toset is Semilattice