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 form 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.