Non-Zero Natural Numbers under Addition form Semigroup

Theorem

Let $\N_{>0}$ be the set of natural numbers without zero, that is:

$\N_{>0} = \N \setminus \set 0$

Let $+$ denote natural number addition.

The structure $\struct {\N_{>0}, +}$ forms a semigroup.

Proof

This is a specific instance of Natural Numbers Bounded Below under Addition form Commutative Semigroup.

$\blacksquare$

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Chapter $\text{I}$: Semi-Groups and Groups: $1$: Definition and examples of semigroups: Example $1$