Addition of Order Types/Examples

Examples of Addition of Order Types

Example Ordering on Integers

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.