Max Semigroup on Toset forms Semilattice

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

Then the max semigroup $\struct {S, \max}$ 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 forms Semilattice