# Natural Numbers under Addition form Commutative Monoid

## Theorem

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

## Proof

Consider the natural numbers $\N$ defined as the 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.

$\blacksquare$