Definition:Integer/Formal Definition

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Definition

Let $\left ({\N, +}\right)$ be the commutative semigroup of natural numbers under addition.

From Inverse Completion of Natural Numbers, we can create $\left({\N', +'}\right)$, an inverse completion of $\left ({\N, +}\right)$.

From Construction of Inverse Completion, this is done as follows:


Let $\boxtimes$ be the cross-relation defined on $\N \times \N$ by:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

From Cross-Relation is Congruence Relation, $\boxtimes$ is a congruence relation.


Let $\left({\N \times \N, \oplus}\right)$ be the external direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$:

$\left({x_1, y_1}\right) \oplus \left({x_2, y_2}\right) = \left({x_1 + x_2, y_1 + y_2}\right)$

Let the quotient structure defined by $\boxtimes$ be $\left({\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}\right)$

where $\oplus_\boxtimes$ is the operation induced on $\dfrac {\N \times \N} \boxtimes$ by $\oplus$.

Let us use $\N'$ to denote the quotient set $\dfrac {\N \times \N} \boxtimes$.

Let us use $+'$ to denote the operation $\oplus_\boxtimes$.


Thus $\left({\N', +'}\right)$ is the Inverse Completion of Natural Numbers.

As the Inverse Completion is Unique up to isomorphism, it follows that we can define the structure $\left({\Z, +}\right)$ which is isomorphic to $\left({\N', +'}\right)$.

An element of $\N'$ is therefore an equivalence class of the congruence relation $\boxtimes$.

So an element of $\Z$ is the isomorphic image of an element $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxtimes$ of $\dfrac {\N \times \N} \boxtimes$.


The set of elements $\Z$ is called the integers.


Natural Number Difference

In the context of the natural numbers, the difference is defined as:

$n - m = p \iff m + p = n$

from which it can be seen that the above congruence can be understood as:

$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1 \iff x_1 - y_1 = x_2 - y_2$

Thus this congruence defines an equivalence between pairs of elements which have the same difference.


Notation

We have that $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\eqclass {a, b} \boxminus$, or $\eqclass {a, b} {}$.

This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\eqclass {a, b} {}$ at the start of any exposition.


Sources