Natural Numbers under Multiplication form Ordered Commutative Semigroup

Theorem

Let $\N$ be the natural numbers.

Let $\times$ be multiplication.

Let $\le$ be the ordering on $\N$.

Then $\left({\N, \times, \le}\right)$ is an ordered commutative semigroup.

Proof

By Natural Numbers under Multiplication form Semigroup, $\left({\N, \times, \le}\right)$ is a semigroup.

By Natural Number Multiplication is Commutative, $\times$ is commutative.

By Ordering on Natural Numbers is Compatible with Multiplication, $\le$ is compatible with $\times$.

The result follows.

$\blacksquare$