| 11. | If it does, it is a maximal or greatest element of " S ".
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| 12. | Least and greatest elements may fail to exist, as the example of the real numbers shows.
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| 13. | However, if it has a greatest element, it can't have any other maximal element.
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| 14. | In particular, it contains a least element and a greatest element ( also denoted " universe " ).
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| 15. | The union of a nonempty set of ordinals that has no greatest element is then always a limit ordinal.
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| 16. | In a bounded meet-semilattice, the identity 1 is the greatest element of " S ".
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| 17. | It should be remarked that the formal definition looks very much like that of a greatest element for an ordered set.
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| 18. | In the case of topologies, the greatest element is the discrete topology and the least element is the trivial topology.
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| 19. | For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
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| 20. | The dual notion, the empty lower bound, is the greatest element, top, or unit ( 1 ).
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