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Wikipedia / Greatest element and least element

In mathematics, especially in order theory, the greatest element of a partially ordered set. The term least element is defined dually

Definitions

be a preordered set is said to be and if it also satisfies:

is said to be and if it also satisfies:

is also a partially ordered set exists and is unique then this element is called . The terminology is defined similarly.

has a greatest element (resp. a least element) then this element is also called (resp.

Relationship to upper/lower bounds

Greatest elements are closely related to upper bounds.

be a preordered set An is

if and only if it which is .

can (however, it may be possible that some other element to simultaneously have a greatest element .

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.

Contrast to maximal elements and local/absolute maximums

A greatest element of a subset of a preordered set should not be confused with a maximal element of the set, which are elements that are not strictly smaller than any other element in the set.

be a preordered set is said to be a if the following condition is satisfied:

is a partially ordered set if and only if there does A

A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.

In a totally ordered set the maximal element and the greatest element coincide; and 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. Similar conclusions hold for least elements.

  1. Role of (in)comparability in distinguishing greatest vs. maximal elements

are said to be ; they are called if they are not comparable. Because preorders are reflexive is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are pairs. In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.

must, in particular, be comparable to are This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important can be reworded as:

  2. Example where all elements are maximal but none are greatest

is a set containing are would, in particular, have to be comparable to has no such element). However, ).

In contrast, if a preordered set being comparable to is the is The directed are comparable and has multiple greatest elements.

Properties

be a partially ordered set

  • can have at most greatest element. Thus if a set has a greatest element then it is necessarily unique.

  • is an upper bound

  • has several maximal elements then it cannot have a greatest element.

  • satisfies the ascending chain condition has a greatest element if, and only if, it has one maximal element.

is a total order in the topmost picture is an example), then the notions of maximal element and greatest element coincide.

  • has a greatest element, the notions coincide, too, as stated above.

Sufficient conditions

  • A finite chain always has a greatest and a least element.

Top and bottom

The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order.

Further introductory information is found in the article on order theory.

Examples

  • The subset of integers of real numbers.

  • but no least upper bound, and no greatest element (cf. picture).

  • In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.

  • the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.

  • the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.

  • with the product order has no upper bound.

  • with the lexicographical order It has no least upper bound.

See also

Notes

References