Greatest element

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 but no 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.

References

* cite book
author = Davey, B.A., and Priestley, H. A.
year = 2002
title = Introduction to Lattices and Order
edition = Second Edition
publisher = Cambridge University Press
id = ISBN 0-521-78451-4