# Definition:Integer/Formal Definition

## 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

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 5$: The system of integers - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 20$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: The rational integers