| 1. | This leaves open the possibility that there are many maximal elements.
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| 2. | The infinite decimal numbers are the maximal elements within this order.
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| 3. | A set can have several maximal elements without having a greatest element.
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| 4. | Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
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| 5. | The existence of a maximal elements is proved via Zorn's lemma.
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| 6. | Example 3 is an instance of existence of many maximal elements and no maximum.
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| 7. | The Smith set is the maximal element of the beat-or-tie order.
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| 8. | The converse is not true : there can be maximal elements despite there being no maximum.
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| 9. | The set of maximal elements of a social preference is called the " core ".
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| 10. | While this relation is not necessarily transitive, it does always contain at least one maximal element.
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