Definition:Ordering on Integers/Definition 1

Definition

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

$\forall x, y \in \Z: x \le y$

if and only if:

$\exists c \in P: x + c = y$

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.

Also see

• Equivalence of Definitions of Ordering on Integers

• Definition:Strict Ordering on Integers

• Definition:Ordering on Natural Numbers

Sources

• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization