Inverse Completion of Natural Numbers

Theorem
There exists an inverse completion of the natural numbers under addition.

Proof
The set of natural numbers under addition can be denoted $$\left ({\mathbb{N}, +}\right)$$.

From Natural Numbers under Addition is Monoid, the algebraic structure $$\left ({\mathbb{N}, +}\right)$$ is a commutative monoid (and therefore a commutative semigroup) all of whose elements are cancellable.

The result follows from the Inverse Completion Theorem.