If m is a maximal element and s \ in S, it remains the possibility that neither s \ leq m nor m \ leq s.
32.
However, when \ preceq is only a preorder, an element x with the property above behaves very much like a maximal element in an ordering.
33.
Zorn's lemma provides us with at least one maximal element M, which, from our reverse ordering, is actually minimal under standard inclusion.
34.
For, if every element of a partially ordered set is either a minimal element or a maximal element, then the corresponding comparability graph is bipartite.
35.
As the example shows, there can be many maximal elements and some elements may be both maximal and minimal ( e . g . 5 above ).
36.
For instance, a minimal element of } } will be a maximal element of } } } } : minimality and maximality are dual concepts in order theory.
37.
;" The priority queue maintenance problem " : It is a simplified version of this dynamic problem, where one requires to delete only the maximal element.
38.
The collection \ mathcal { S } _ 2 of these subsets cannot contain an infinite ascending chain, so has a maximal element " F " 2.
39.
The elements of an abstract polyhedron are its body ( the maximal element ), its faces, edges, vertices and the " null polytope " or empty set.
40.
This corresponds algebraically to the universal perfect central extension ( called " covering group ", by analogy ) as the maximal element, and a group mod its center as minimal element.
