Natural Numbers under Addition form Commutative Semigroup

Theorem

The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative semigroup.

Proof

Consider the natural numbers $\N$ defined as the naturally ordered semigroup.

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

$\blacksquare$

Also see

• Natural Numbers under Addition form Commutative Monoid