# Addition of Order Types/Examples/Example Ordering on Integers

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## Examples of Addition of Order Types

Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:

- $a \preccurlyeq b$ if and only if $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

where $\le$ is the usual ordering on $\Z$.

Then the order type of $\struct {\Z, \preccurlyeq}$ is:

- $\map \ot {\Z, \preccurlyeq} = \omega + \omega$

where $\omega$ denotes the order type of the natural numbers.

## Proof

Consider the following mappings:

- $i_\N: \N \to \Z_{\ge 0}: x \mapsto x$

- $\phi: \N \to \Z_{<0}: x \mapsto -\paren {x + 1}$

These are seen to be order isomorphisms.

We have that:

- $\struct {\N, \le} \oplus \struct {\N, \le} \cong \struct {\Z, \preccurlyeq}$

This needs considerable tedious hard slog to complete it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations