Non-Zero Natural Numbers under Multiplication form Commutative Semigroup

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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.


Proof

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.

$\Box$


Associativity

Natural Number Multiplication is Associative.

$\Box$


Commutativity

Natural Number Multiplication is Commutative.

$\Box$


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.

$\blacksquare$


Sources