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
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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: Exercise $28 \ \text {(a)}$