TITLE: What algebraic structure is the set of natural numbers and addition?
QUESTION [7 upvotes]: 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 $\mathbb{N}$?

REPLY [9 votes]: 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.