Order Type of Integers under Usual Ordering

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Theorem

Consider the ordered structure $\struct {\Z, \le}$ that is the set of integers under the usual ordering.


Then:

$\map \ot {\Z, \le} = \omega^* + \omega$

where:

$\ot$ denotes order type
$\omega$ denotes the order type of the natural numbers $\N$
$\omega^*$ denotes the dual of $\omega$
$+$ denotes addition of order types.


Proof




Sources