| 11. | If a poset has more than one maximal element, then these elements will not be mutually comparable.
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| 12. | However, if it has a greatest element, it can't have any other maximal element.
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| 13. | So, assume that P has at least one element, and let a be a maximal element of P.
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| 14. | Zorn emigrated to the chain of subsets to have one chain not contained in any other, called the maximal element.
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| 15. | In the second case the definition of maximal element requires m = s so we conclude that s \ leq m.
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| 16. | Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements.
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| 17. | That is, we should regard a rule as choosing the maximal elements ( " best " alternatives ) of some social preference.
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| 18. | For totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.
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| 19. | Every lower set L of a finite ordered set P is equal to the smallest lower set containing all maximal elements of L.
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| 20. | An important tool to ensure the existence of maximal elements under certain conditions is "'Zorn's Lemma " '.
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