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:
“A lover is never a completely self-reliant person viewing the world through his own eyes, but a hostage to a certain delusion. He becomes a perjurer, all his thoughts and emotions being directed with reference, not to an accurate and just appraisal of the real world but rather to the safety and exaltation of his loved one, and the madness with which he pursues her, transmogrifying his attention, blinds him like a victim.”
—Alexander Theroux (b. 1940)
“Drink, sir, is a great provoker of three things ... nose-painting, sleep, and urine. Lechery, sir, it provokes and unprovokes: it provokes the desire but it takes away the performance. Therefore much drink may be said to be an equivocator with lechery: it makes him and it mars him; it sets him on and it takes him off.”
—William Shakespeare (1564–1616)