| 11. | Thus the example Z above is an example of a GO-space that is not a linearly ordered topological space.
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| 12. | Every cyclically ordered group can also be expressed as a subgroup of a product, where is a linearly ordered group.
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| 13. | This result is analogous to Otto H�lder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of.
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| 14. | The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers.
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| 15. | In Easton's model, " V " cannot be linearly ordered, so it cannot be well-ordered.
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| 16. | There are in fact many ways to construct such a linearly ordered set of numbers, but fundamentally, there are two different approaches:
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| 17. | A convex set is linearly ordered by the cut for any not in the set; this ordering is independent of the choice of.
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| 18. | For example, given two linearly ordered sets and, one may form a circle by joining them together at positive and negative infinity.
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| 19. | Every cyclically ordered group can be expressed as a quotient, where is a linearly ordered group and is a cyclic cofinal subgroup of.
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| 20. | Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.
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