Min Semigroup on Toset forms Semilattice

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

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

Proof
The Min Semigroup is Commutative and idempotent.

Hence the result, by definition of a semilattice.

Also see

 * Max Semigroup on Toset is Semilattice