Inverse Completion of Natural Numbers

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

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

From Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\left ({\N, +}\right)$ is a commutative monoid.

Therefore by definition of commutative monoid, $\left ({\N, +}\right)$ is a commutative semigroup.

From Natural Number Addition is Cancellable, all of the elements of $\left ({\N, +}\right)$ are cancellable.

The result follows from the Inverse Completion Theorem.

$\blacksquare$