# Natural Numbers under Addition form Commutative Monoid

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## Theorem

The algebraic structure $\struct {\N, +}$ 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 $\struct {\N, +}$ is a commutative semigroup.

From the definition of zero, $\struct {\N, +}$ has $0 \in \N$ as the identity, hence is a monoid.

$\blacksquare$

## Also see

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.1$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids - 1999: J.C. Rosales and P.A. García-Sánchez:
*Finitely Generated Commutative Monoids*... (previous) ... (next): Chapter $1$: Basic Definitions and Results

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