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.
Read more about this topic: Maximal Element
Famous quotes containing the words elements, greatest and/or element:
“The popularity of that baby-faced boy, who possessed not even the elements of a good actor, was a hallucination in the public mind, and a disgrace to our theatrical history.”
—Thomas Campbell (17771844)
“Do not be deceived! The busiest people harbor the greatest weariness, their restlessness is weaknessthey no longer have the capacity for waiting and idleness.”
—Friedrich Nietzsche (18441900)
“All forms of beauty, like all possible phenomena, contain an element of the eternal and an element of the transitoryof the absolute and of the particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction creamed from the general surface of different beauties. The particular element in each manifestation comes from the emotions: and just as we have our own particular emotions, so we have our own beauty.”
—Charles Baudelaire (18211867)