Maximal Element - Existence and Uniqueness

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 .

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