Natural Numbers under Addition form Commutative Monoid

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

• Natural Numbers under Addition do not form Group

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 $1$