| 1. | Ordinal numbers are then introduced as the order types of well-ordered sets.
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| 2. | Every relatively atomic partially ordered set with a least element is atomic.
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| 3. | The maximal totally ordered sets are horizontal lines in ! 2.
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| 4. | If you have an already partially ordered set, you could use quicksort.
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| 5. | Let and be partially ordered sets and let be an order-preserving map.
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| 6. | Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
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| 7. | This definition is extendable to subsets of any partially ordered set.
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| 8. | It is readily verified that this yields a partially ordered set.
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| 9. | Mathematically, then, a relation is simply an " ordered set ".
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| 10. | Order theory is the study of partially ordered sets, both finite and infinite.
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