Natural Numbers under Addition form Commutative Monoid

Theorem
The algebraic structure $$\left({\N, +}\right)$$ consisting of the set of natural numbers $$\N$$ under addition $$+$$ forms a commutative monoid whose identity is zero, and all of whose elements are cancellable.

Proof
From Natural Numbers are a Naturally Ordered Semigroup, the algebraic structure $$\left ({\N, +}\right)$$ is a naturally ordered semigroup.

From the definition of the naturally ordered semigroup, it follows that $$\left ({\N, +}\right)$$ is a commutative semigroup.

From the definition of zero, $$\left({\N, +}\right)$$ has $$0 \in \N$$ as the identity, hence is a monoid.

From Naturally Ordered Semigroup: NO 2, every element of $$\left ({\N, +}\right)$$ is cancellable.

Note
Because the general element of $$\left({\N, +}\right)$$ has no inverse, it is not a group.