Definition:Integer/Formal Definition

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Definition

Let $\struct {\N, +}$ be the commutative semigroup of natural numbers under addition.

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

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


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

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

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


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

$\tuple {x_1, y_1} \oplus \tuple {x_2, y_2} = \tuple {x_1 + x_2, y_1 + y_2}$

Let the quotient structure defined by $\boxtimes$ be $\struct {\dfrac {\N \times \N} \boxtimes, \oplus_\boxtimes}$

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 $\struct {\N', +'}$ is the Inverse Completion of Natural Numbers.

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

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 $\eqclass {\tuple {a, b} } \boxtimes$ 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:

$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1 \iff x_1 - y_1 = x_2 - y_2$

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