# Natural Numbers under Multiplication form Semigroup

## Theorem

Let $\N$ be the set of natural numbers.

Let $\times$ denote the operation of multiplication on $\N$.

The structure $\struct {\N, \times}$ forms a semigroup.

## Proof

### Semigroup Axiom $\text S 0$: Closure

We have that Natural Number Multiplication is Closed.

That is, $\struct {\N, \times}$ is closed.

$\Box$

### Semigroup Axiom $\text S 1$: Associativity

We have that Natural Number Multiplication is Associative.

$\Box$

Thus the criteria are fulfilled for $\struct {\N, \times}$ to form a semigroup.

$\blacksquare$