Definition:Ordering on Integers

Definition

Definition 1

The integers are ordered on the relation $\le$ as follows:

$\forall x, y \in \Z: x \le y$
$\exists c \in P: x + c = y$

where $P$ is the set of positive integers.

That is, $x$ is less than or equal to $y$ if and only if $y - x$ is non-negative.

Definition 2

The integers are ordered on the relation $\le$ as follows:

Let $x$ and $y$ be defined as from the formal definition of integers:

$x = \eqclass {x_1, x_2} {}$ and $y = \eqclass {y_1, y_2} {}$ where $x_1, x_2, y_1, y_2 \in \N$.

Then:

$x < y \iff x_1 + y_2 \le x_2 + y_1$

where:

$+$ denotes natural number addition
$\le$ denotes natural number ordering.

Also see

• Results about orderings on integers can be found here.