Definition:Additive Group of Integers

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

Thus it follows that integer addition is:


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

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

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

The result follows from Inverse Completion a Group.

Let us define $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ as in the formal definition of integers.

That is, $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $$\boxminus$$.

$$\boxminus$$ is the congruence relation defined on $$\N \times \N$$ by $$\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$$.

In order to streamline the notation, we will use $$\left[\!\left[{a, b}\right]\!\right]$$ to mean $$\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$$, as suggested.


 * From Construction of Inverse Completion: Identity of Quotient Structure, the identity of $$\left({\Z, +}\right)$$ is $$\left[\!\left[{c, c}\right]\!\right]$$ for any $$c \in \N$$:

$$\forall a, b, c \in \N: \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c, c}\right]\!\right] = \left[\!\left[{a, b}\right]\!\right] = \left[\!\left[{c, c}\right]\!\right] + \left[\!\left[{a, b}\right]\!\right]$$

which is the equivalence class of pairs of elements $$\N \times \N$$ whose difference is zero.

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

Note that a perfectly good representative of $$\left[\!\left[{c, c}\right]\!\right]$$ is $$\left[\!\left[{0, 0}\right]\!\right]$$. This usually keeps to a minimum the complexity of any arithmetic that is needed.


 * From Construction of Inverse Completion: Invertible Elements in Quotient Structure, we see that every element of $$\left({\Z, +}\right)$$ has an inverse.

We can see that:

$$ $$

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

From Construction of Inverse Completion: Identity of Quotient Structure, $$\left[\!\left[{a + b, a + b}\right]\!\right]$$ is a member of the equivalence class which is the identity of $$\left({\Z, +}\right)$$.

Thus the inverse of $$\left[\!\left[{a, b}\right]\!\right]$$ is $$\left[\!\left[{b, a}\right]\!\right]$$.


 * Finally we note that from Integers are Countable, the set of integers can be placed in one-to-one correspondence with the set of natural numbers.