Natural Numbers are a Naturally Ordered Semigroup

Theorem

The natural numbers under addition form an algebraic structure $\left({\N, +, \le}\right)$ which is a naturally ordered semigroup.

As all naturally ordered semigroups are isomorphic, it is usual to refer to $\left({\N, +, \le}\right)$ as the "archetypal" naturally ordered semigroup.

All results that are valid for a naturally ordered semigroup are also valid for $\N$.

Proof

The naturally ordered semigroup $\left({S, \circ, \preceq}\right)$ possesses all the properties that define the natural numbers:

• NO 1: The set $\N$ is well-ordered by $\le$
• NO 2: $\forall m, n, p \in \N: m + p = n + p \iff m = n$
• NO 3: $\forall m, n \in \N: m \le n \implies \exists p \in \N: m + p = n$
• NO 4: $\exists m, n \in \N: m \ne n$

It has also been demonstrated that the naturally ordered semigroup is unique up to isomorphism.

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 29 \ (1)$