In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. A bounded poset is a poset that has both a greatest element and a least element.
Formally, given a partially ordered set (P, ≤), then an element g of a subset S of P is the greatest element of S if
- s ≤ g, for all elements s of S.
Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S.
Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. A finite chain always has a greatest and a least element.
Greatest elements of a partially ordered subset must not be confused with maximal elements of such a set which are elements that are not smaller than any other element. A poset can have several maximal elements without having a greatest element.
In a totally ordered set both terms coincide; it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. The dual terms are minimum and absolute minimum. Together they are called the absolute extrema.
The least and greatest elements of the whole partially ordered set play a special role and are also called bottom and top or zero (0) and unit (1), respectively. The latter notation of 0 and 1 is only used when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1. The existence of least and greatest elements is a special completeness property of a partial order. Bottom and top are often represented by the symbols ⊥ and ⊤, respectively.
Further introductory information is found in the article on order theory.
Examples
- Z in R has no upper bound.
- Let the relation "≤" on {a, b, c, d} be given by a ≤ c, a ≤ d, b ≤ c, b ≤ d. The set {a, b} has upper bounds c and d, but no least upper bound.
- In Q, the set of numbers with their square less than 2 has upper bounds but no least upper bound.
- In R, the set of numbers less than 1 has a least upper bound, but no greatest element.
- In R, the set of numbers less than or equal to 1 has a greatest element.
- In R² with the product order, the set of (x, y) with 0 < x < 1 has no upper bound.
- In R² with the lexicographical order, this set has upper bounds, e.g. (1, 0). It has no least upper bound.
See also
References
- Davey, B.A., and Priestley, H. A. (2002). Introduction to Lattices and Order (Second Edition ed.). Cambridge University Press. ISBN 0-521-78451-4.
Open source encyclopedia content modification information:
This page was last modified on 5 March 2009 at 14:57.
Authorship and Review
Open source encyclopedia content provided here is not reviewed directly by MedLibrary.org. Content is sourced directly from Wikipedia and is authored by an open community of volunteers. It is not produced by or in any way affiliated with MedLibrary.org.
Usage Guidelines
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Greatest element", which is available in its original form here:
http://en.wikipedia.org/w/index.php?title=Greatest_element
All material adapted used from Wikipedia is available under the terms of the GNU Free Documentation License. Wikipedia® itself is a registered trademark of the Wikimedia Foundation, Inc.