Addition of Order Types/Examples/Example Ordering on Integers

Examples of Addition of Order Types
Let $\preccurlyeq$ denote the relation on the set of integers $\Z$ defined as:
 * $a \preccurlyeq b$ $0 \le a \le b \text { or } b \le a < 0 \text { or } a < 0 \le b$

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