Existence and Uniqueness
Maximal elements need not exist.
- Example 1: Let, for all we have but (that is, but not ).
- Example 2: Let and recall that .
In general is only a partial order on . If is a maximal element and, it remains the possibility that neither nor . This leaves open the possibility that there are many maximal elements.
- Example 3: In the fence, all the are maximal, and all the are minimal.
- Example 4: Let be a set with at least two elements and let be the subset of the power set consisting of singletons, partially ordered by . This is the discrete poset—no two elements are comparable—and thus every element is maximal (and minimal) and for any neither nor .
Read more about this topic: Maximal Element
Famous quotes containing the words existence and/or uniqueness:
“Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.”
—Charles Sanders Peirce (1839–1914)
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)