Maximal Element - Directed Sets

Directed Sets

In a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. This observation does not only apply to totally ordered subsets of any poset, but also to their order theoretic generalization via directed sets. In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. It is easy to see that any maximal element of such a subset will be unique (unlike in a poset). Furthermore, this unique maximal element will also be the greatest element.

Similar conclusions are true for minimal elements.

Further introductory information is found in the article on order theory.

Read more about this topic:  Maximal Element

Famous quotes containing the words directed and/or sets:

    Religion, or the duty which we owe our Creator, and the manner of discharging it, can be directed only by reason and conviction, not by force and violence; and therefore all men are equally entitled to the free exercise of religion, according to the dictates of conscience.
    James Madison (1751–1836)

    Analysis as an instrument of enlightenment and civilization is good, in so far as it shatters absurd convictions, acts as a solvent upon natural prejudices, and undermines authority; good, in other words, in that it sets free, refines, humanizes, makes slaves ripe for freedom. But it is bad, very bad, in so far as it stands in the way of action, cannot shape the vital forces, maims life at its roots. Analysis can be a very unappetizing affair, as much so as death.
    Thomas Mann (1875–1955)