Non-Zero Natural Numbers under Multiplication form Commutative Semigroup

Theorem
Let $\N_{>0}$ be the set of natural numbers without zero, that is, $\N_{>0} = \N \setminus \set 0$.

The structure $\left({\N_{>0}, \times}\right)$ forms an infinite commutative semigroup.

Closure
From Natural Numbers have No Proper Zero Divisors:
 * $\forall m, n \in \N: m \times n = 0 \iff m = 0 \lor n = 0$

It follows that:
 * $\forall m, n \in \N_{>0}: m \times n \ne 0$

and so:
 * $\forall m, n \in \N_{>0}: m \times n \in \N_{>0}$

That is, $\struct {\N_{>0}, \times}$ is closed.

Associativity
Natural Number Multiplication is Associative.

Commutativity
Natural Number Multiplication is Commutative.

Infinite
We have that the Natural Numbers are Infinite.

Then we have that Infinite if Injection from Natural Numbers.

The mapping $s: \N \to \N_{>0}: \map s n = n + 1$ is such an injection.

Hence $\N_{>0}$ is infinite.