| 1. | Every relatively atomic partially ordered set with a least element is atomic.
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| 2. | But some subsets of the real numbers do not have least elements.
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| 3. | Similarly, an identity element in a join semilattice is a least element.
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| 4. | Next we might try specifying the least element from each set.
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| 5. | A finite chain always has a greatest and a least element.
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| 6. | Likewise, the join of the empty set yields the least element.
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| 7. | Other common names for the least element are bottom and zero ( 0 ).
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| 8. | Let a nonempty be given and assume " X " has no least element.
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| 9. | Suppose there exists a non-empty set, " S ", of naturals with no least element.
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| 10. | For example, functions that preserve the empty supremum are those that preserve the least element.
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