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