Greatest element and least element

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In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually, that is, it is an element of that is smaller than every other element of

Hasse diagram of the set of divisors of 60, partially ordered by the relation " divides ". The red subset has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.

DefinitionsEdit

Let   be a preordered set and let   An element   is said to be a greatest element of   if   and if it also satisfies:

  for all  

By using   instead of   in the above definition, the definition of a least element of   is obtained. Explicitly, an element   is said to be a least element of   if   and if it also satisfies:

  for all  

If   is a partially ordered set then   can have at most one greatest element and it can have at most one least element. Whenever a greatest element of   exists and is unique then this element is called the greatest element of  . The terminology the least element of   is defined similarly.

If   has a greatest element (resp. a least element) then this element is also called a top (resp. a bottom) of  

Relationship to upper/lower boundsEdit

Greatest elements are closely related to upper bounds.

Let   be a preordered set and let   An upper bound of   in   is an element   such that   and   for all   Importantly, an upper bound of   in   is not required to be an element of  

If   then   is a greatest element of   if and only if   is an upper bound of   in   and   In particular, any greatest element of   is also an upper bound of   (in  ) but an upper bound of   in   is a greatest element of   if and only if it belongs to   In the particular case where   the definition of "  is an upper bound of   in  " becomes:   is an element such that   and   for all   which is completely identical to the definition of a greatest element given before. Thus   is a greatest element of   if and only if   is an upper bound of   in  .

If   is an upper bound of   in   that is not an upper bound of   in   (which can happen if and only if  ) then   can not be a greatest element of   (however, it may be possible that some other element is a greatest element of  ). In particular, it is possible for   to simultaneously not have a greatest element and for there to exist some upper bound of   in  .

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 maximumsEdit

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.

Let   be a preordered set and let   An element   is said to be a maximal element of   if the following condition is satisfied:

whenever   satisfies   then necessarily  

If   is a partially ordered set then   is a maximal element of   if and only if there does not exist any   such that   and   A maximal element of   is defined to mean a maximal element of the subset  

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.[1] The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions hold for least elements.

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

One of the most important differences between a greatest element   and a maximal element   of a preordered set   has to do with what elements they are comparable to. Two elements   are said to be comparable if   or  ; they are called incomparable if they are not comparable. Because preorders are reflexive (which means that   is true for all elements  ), every element   is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are distinct pairs. In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.

By definition, an element   is a greatest element of   if   for every  ; so by its very definition, a greatest element of   must, in particular, be comparable to every element in   This is not required of maximal elements. Maximal elements of   are not required to be comparable to every element in   This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important if statement. The defining condition for   to be a maximal element of   can be reworded as:

For all   IF   (so elements that are incomparable to   are ignored) then  
Example where all elements are maximal but none are greatest

Suppose that   is a set containing at least two (distinct) elements and define a partial order   on   by declaring that   if and only if   If   belong to   then neither   nor   holds, which shows that all pairs of distinct (i.e. non-equal) elements in   are incomparable. Consequently,   can not possibly have a greatest element (because a greatest element of   would, in particular, have to be comparable to every element of   but   has no such element). However, every element   is a maximal element of   because there is exactly one element in   that is both comparable to   and   that element being   itself (which of course, is  ).[note 1]

In contrast, if a preordered set   does happen to have a greatest element   then   will necessarily be a maximal element of   and moreover, as a consequence of the greatest element   being comparable to every element of   if   is also partially ordered then it is possible to conclude that   is the only maximal element of   However, the uniqueness conclusion is no longer guaranteed if the preordered set   is not also partially ordered. For example, suppose that   is a non-empty set and define a preorder   on   by declaring that   always holds for all   The directed preordered set   is partially ordered if and only if   has exactly one element. All pairs of elements from   are comparable and every element of   is a greatest element (and thus also a maximal element) of   So in particular, if   has at least two elements then   has multiple distinct greatest elements.

PropertiesEdit

Throughout, let   be a partially ordered set and let  

  • A set   can have at most one greatest element.[note 2] Thus if a set has a greatest element then it is necessarily unique.
  • If it exists, then the greatest element of   is an upper bound of   that is also contained in  
  • If   is the greatest element of   then   is also a maximal element of  [note 3] and moreover, any other maximal element of   will necessarily be equal to  [note 4]
    • Thus if a set   has several maximal elements then it cannot have a greatest element.
  • If   satisfies the ascending chain condition, a subset   of   has a greatest element if, and only if, it has one maximal element.[note 5]
  • When the restriction of   to   is a total order (  in the topmost picture is an example), then the notions of maximal element and greatest element coincide.[note 6]
    • However, this is not a necessary condition for whenever   has a greatest element, the notions coincide, too, as stated above.
  • If the notions of maximal element and greatest element coincide on every two-element subset   of   then   is a total order on  [note 7]

Sufficient conditionsEdit

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

Top and bottomEdit

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.

ExamplesEdit

 
Hasse diagram of example 2
  • The subset of integers has no upper bound in the set   of real numbers.
  • Let the relation   on   be given by         The set   has upper bounds   and   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.
  • In   the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
  • In   the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
  • In   with the product order, the set of pairs   with   has no upper bound.
  • In   with the lexicographical order, this set has upper bounds, e.g.   It has no least upper bound.

See alsoEdit

NotesEdit

  1. ^ Of course, in this particular example, there exists only one element in   that is comparable to   which is necessarily   itself, so the second condition "and  " was redundant.
  2. ^ If   and   are both greatest, then   and   and hence   by antisymmetry.
  3. ^ If   is the greatest element of   and   then   By antisymmetry, this renders (  and  ) impossible.
  4. ^ If   is a maximal element, then   since   is greatest, hence   since   is maximal.
  5. ^ Only if: see above. — If: Assume for contradiction that   has just one maximal element,   but no greatest element. Since   is not greatest, some   must exist that is incomparable to   Hence   cannot be maximal, that is,   must hold for some   The latter must be incomparable to   too, since   contradicts  's maximality while   contradicts the incomparability of   and   Repeating this argument, an infinite ascending chain   can be found (such that each   is incomparable to   and not maximal). This contradicts the ascending chain condition.
  6. ^ Let   be a maximal element, for any   either   or   In the second case, the definition of maximal element requires that   so it follows that   In other words,   is a greatest element.
  7. ^ If   were incomparable, then   would have two maximal, but no greatest element, contradicting the coincidence.

ReferencesEdit

  1. ^ The notion of locality requires the function's domain to be at least a topological space.
  • Davey, B. A.; Priestley, H. A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 978-0-521-78451-1.