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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
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@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

2 Answers 2

up vote 6 down vote accepted

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.

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What if 0 is excluded? –  Problemaniac Jan 23 '13 at 6:44
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Then you don't get a semiring because you don't have a zero, you also don't have a monoid under addition, it's what's called a semigroup now. But you still have a monoid under multiplication. –  JSchlather Jan 23 '13 at 6:48
    
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
    
Could someone recommend a listing of abstract algebraic structures with its features and examples? I have the feeling someone must have tabulated that. –  Problemaniac Jan 23 '13 at 20:02
    
@Problemania There's this wiki page. –  JSchlather Jan 23 '13 at 22:22

Wikipedia adopts the Bourbaki definition of a ring that includes a multiplicative identity.

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