Maximal Elements and The Greatest Element
It looks like should be a greatest element or maximum but in fact it is not necessarily the case: the definition of maximal element is somewhat weaker. Suppose we find with, then, by the definition of greatest element, so that . In other words, a maximum, if it exists, is the (unique) maximal element.
The converse is not true: there can be maximal elements despite there being no maximum. Example 3 is an instance of existence of many maximal elements and no maximum. The reason is, again, that in general is only a partial order on . If is a maximal element and, it remains the possibility that neither nor .
If there are many maximal elements, they are in some contexts called a frontier, as in the Pareto frontier.
Of course, when the restriction of to is a total order, the notions of maximal element and greatest element coincide. Let be a maximal element, for any either or . In the second case the definition of maximal element requires so we conclude that . In other words, is a greatest element.
Finally, let us remark that being totally ordered is sufficient to ensure that a maximal element is a greatest element, but it is not necessary.
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