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 the example of the real numbers strictly smaller than 1 shows. This also demonstrates that the existence of a least upper bound (the number 1 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.

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Examples

Reference

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 'R'² with the product order, the set of (''x'', ''y'') with 0 < ''x'' < 1 has no upper bound.

★ In 'R'² with the lexicographical order, this set has upper bounds, e.g. (1, 0). It has no least upper bound.

Reference

★ 'Introduction to Lattices and Order', Davey, B.A., and Priestley, H. A., , , Cambridge University Press, 2002, ISBN 0-521-78451-4