# Category:Orderings on Integers

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This category contains results about **Orderings on Integers**.

Definitions specific to this category can be found in Definitions/Orderings on Integers.

### 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.

## Pages in category "Orderings on Integers"

The following 12 pages are in this category, out of 12 total.