# Non-Zero Natural Numbers under Multiplication form Commutative Monoid

## Theorem

Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \left\{{0}\right\}$.

The structure $\left({\N_{>0}, \times}\right)$ forms a commutative monoid.

## Proof

From Non-Zero Natural Numbers under Multiplication form Commutative Semigroup, $\left({\N_{>0}, \times}\right)$ forms a commutative semigroup.

From Identity Element of Natural Number Multiplication is One, $\left({\N_{>0}, \times}\right)$ has an identity element which is $1$.

Hence the result, by definition of commutative monoid.

$\blacksquare$