maximal element in a sentence

Examples

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