Definition:Additive Group of Integers

Theorem
The set of integers under addition $$\left({\mathbb{Z}, +}\right)$$forms an abelian group.

Thus it follows that integer addition is:


 * well-defined on $$\mathbb{Z}$$;
 * closed on $$\mathbb{Z}$$;
 * associative on $$\mathbb{Z}$$;
 * commutative on $$\mathbb{Z}$$;
 * The identity of $$\left({\mathbb{Z}, +}\right)$$ is $$0$$;
 * Each element of $$\left({\mathbb{Z}, +}\right)$$ has an inverse.

Proof
From the definition of the integers, the algebraic structure $$\left({\mathbb{Z}, +}\right)$$ is an isomorphic copy of the inverse completion of $$\left ({\mathbb{N}, +}\right)$$.

As the Natural Numbers are a Naturally Ordered Semigroup, it follows that:
 * $$\left ({\mathbb{N}, +}\right)$$ is a commutative semigroup;
 * all elements of $$\left ({\mathbb{N}, +}\right)$$ are cancellable.

The result follows from Inverse Completion a Group.


 * From Theorem 11 of Construction of Inverse Completion, the identity of $$\left({\mathbb{Z}, +}\right)$$ is $$\left[\left[{\left({c, c}\right)}\right]\right]_\boxminus$$ for any $$c \in \mathbb{N}$$:

$$\forall a, b, c \in \mathbb{N}: \left[\left[{\left({a, b}\right)}\right]\right]_\boxminus + \left[\left[{\left({c, c}\right)}\right]\right]_\boxminus = \left[\left[{\left({a, b}\right)}\right]\right]_\boxminus = \left[\left[{\left({c, c}\right)}\right]\right]_\boxminus + \left[\left[{\left({a, b}\right)}\right]\right]_\boxminus$$

which is the equivalence class of pairs of elements $$\mathbb{N} \times \mathbb{N}$$ whose difference is Zero.

Thus the identity of $$\left({\mathbb{Z}, +}\right)$$ is seen to be $$0$$.


 * From Theorem 12 of Construction of Inverse Completion, we see that every element of $$\left({\mathbb{Z}, +}\right)$$ has an inverse.

We can see that:

The above construction is valid because $$a$$ and $$b$$ are both in $$\mathbb{N}$$ and hence cancellable.

From Theorem 11 of Construction of Inverse Completion, $$\left[\left[{\left({a + b, a + b}\right)}\right]\right]_\boxminus$$ is a member of the equivalence class which is the identity of $$\left({\mathbb{Z}, +}\right)$$.

Thus the inverse of $$\left[\left[{\left({a, b}\right)}\right]\right]_\boxminus$$ is $$\left[\left[{\left({b, a}\right)}\right]\right]_\boxminus$$.