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:
“Views of women, on one side, as inwardly directed toward home and family and notions of men, on the other, as outwardly striving toward fame and fortune have resounded throughout literature and in the texts of history, biology, and psychology until they seem uncontestable. Such dichotomous views defy the complexities of individuals and stifle the potential for people to reveal different dimensions of themselves in various settings.”
—Sara Lawrence Lightfoot (20th century)
“We love the indomitable bellicose patriotism that sets you apart; we love the national pride that guides your muscularly courageous race; we love the potent individualism that doesn’t prevent you from opening your arms to individualists of every land, whether libertarians or anarchists.”
—Tommaso Marinetti (1876–1944)