Natural Numbers under Multiplication form Ordered Commutative Semigroup
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Theorem
Let $\N$ be the natural numbers.
Let $\times$ be multiplication.
Let $\le$ be the ordering on $\N$.
Then $\struct {\N, \times, \le}$ is an ordered commutative semigroup.
Proof
By Natural Numbers under Multiplication form Semigroup, $\struct {\N, \times, \le}$ 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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.13$: Corollary