Min Semigroup on Toset forms Semilattice

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

Then the min semigroup $\struct {S, \min}$ is a semilattice.

Proof
We have:
 * Min Semigroup is Commutative
 * Min Semigroup is Idempotent.

Hence the result, by definition of a semilattice.

Also see

 * Max Semigroup on Toset forms Semilattice