# What algebraic structure is the set of natural numbers and addition?

What algebraic structure is the set of natural numbers and addition?

I understand that

$$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$$

and $\mathbb{Z}$ and $\mathbb{Q}$ are rings and $\mathbb{R}$ and $\mathbb{C}$ are fields with normal addition and multiplication operations (right?)

So what algebraic structure is $\mathbb{N}$?

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Monoid – Calvin Lin Jan 23 '13 at 6:35
@Calvin: without $0$ you only get a semigroup (since $0$ is the additive identity). – Qiaochu Yuan Jan 23 '13 at 7:23
@QiaochuYuan My assumption was that the natural numbers included 0. I don't particularly like the phrase natural numbers due to the ambiguity involved. I prefer to say positive integers / non-negative integers. – Calvin Lin Jan 23 '13 at 7:33
Wikipedia adopts the Bourbaki definition of a ring that includes a multiplicative identity. – Steven Gamer Apr 5 '13 at 20:21

The natural numbers assuming $0$ is included are a monoid under multiplication and also a monoid under addition. Under both addition and multiplication $\mathbb N$ is whats called a semiring.
Edit: I might add that $\mathbb N$ under multiplication is actually rather interesting. In particular $(\mathbb N,\cdot)$ is the free abelian monoid on countably many generators.
The natural numbers (excluding $0$) under addition would be a semigroup (set with an associative operation, but no inverses and no identity). Under multiplciation, it would still be a monoid (even without $0$, since the identity element under multiplication is $1$). – Charles Boyd Jan 23 '13 at 6:50