# Natural Numbers under Addition form Commutative Monoid

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## Contents

## 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.

$\blacksquare$

## Also see

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*: Chapter $\text{I}$: $\S 1$ Examples: $\text{(1)}$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$: Example $7.1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups: Exercise $1$