Editing Infimum and supremum (section)

== Relation to maximum and minimum elements ==

The infimum of a subset <math>S</math> of a partially ordered set <math>P,</math> assuming it exists, does not necessarily belong to <math>S.</math> If it does, it is a [[Minimal element|minimum or least element]] of <math>S.</math> Similarly, if the supremum of <math>S</math> belongs to <math>S,</math> it is a [[Maximal element|maximum or greatest element]] of <math>S.</math>

For example, consider the set of negative real numbers (excluding zero).  This set has no greatest element, since for every element of the set, there is another, larger, element.  For instance, for any negative real number <math>x,</math> there is another negative real number <math>\tfrac{x}{2},</math> which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set.  Hence, <math>0</math> is the least upper bound of the negative reals, so the supremum is 0. This set has a supremum but no greatest element.

However, the definition of [[Maximal element|maximal and minimal elements]] is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

Whereas maxima and minima must be members of the subset that is under consideration, the infimum and supremum of a subset need not be members of that subset themselves.

=== Minimal upper bounds ===
Finally, a partially ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no strictly smaller element that also is an upper bound. This does not say that each minimal upper bound is smaller than all other upper bounds, it merely is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a [[Totally ordered set|total]] one. In a totally ordered set, like the real numbers, the concepts are the same.

As an example, let <math>S</math> be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from <math>S</math> together with the set of [[integer]]s <math>\Z</math> and the set of positive real numbers <math>\R^+,</math> ordered by subset inclusion as above. Then clearly both <math>\Z</math> and <math>\R^+</math> are greater than all finite sets of natural numbers. Yet, neither is <math>\R^+</math> smaller than <math>\Z</math> nor is the converse true: both sets are minimal upper bounds but none is a supremum.

=== Least-upper-bound property ===
{{main|Least-upper-bound property}}

The {{em|least-upper-bound property}} is an example of the aforementioned [[Completeness (order theory)|completeness properties]] which is typical for the set of real numbers. This property is sometimes called {{em|Dedekind completeness}}.

If an ordered set <math>S</math> has the property that every nonempty subset of <math>S</math> having an upper bound also has a least upper bound, then <math>S</math> is said to have the least-upper-bound property.  As noted above, the set <math>\R</math> of all real numbers has the least-upper-bound property.  Similarly, the set <math>\Z</math> of integers has the least-upper-bound property; if <math>S</math> is a nonempty subset of <math>\Z</math> and there is some number <math>n</math> such that every element <math>s</math> of <math>S</math> is less than or equal to <math>n,</math> then there is a least upper bound <math>u</math> for <math>S,</math> an integer that is an upper bound for <math>S</math> and is less than or equal to every other upper bound for <math>S.</math> A [[well-order]]ed set also has the least-upper-bound property, and the empty subset has also a least upper bound: the minimum of the whole set.

An example of a set that {{em|lacks}} the least-upper-bound property is <math>\Q,</math> the set of rational numbers.  Let <math>S</math> be the set of all rational numbers <math>q</math> such that <math>q^2 < 2.</math> Then <math>S</math> has an upper bound (<math>1000,</math> for example, or <math>6</math>) but no least upper bound in <math>\Q</math>: If we suppose <math>p \in \Q</math> is the least upper bound, a contradiction is immediately deduced because between any two reals <math>x</math> and <math>y</math> (including [[square root of 2|<math>\sqrt{2}</math>]] and <math>p</math>) there exists some rational <math>r,</math> which itself would have to be the least upper bound (if <math>p > \sqrt{2}</math>) or a member of <math>S</math> greater than <math>p</math> (if <math>p < \sqrt{2}</math>). Another example is the [[hyperreals]]; there is no least upper bound of the set of positive infinitesimals.

There is a corresponding {{em|greatest-lower-bound property}}; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

If in a partially ordered set <math>P</math> every bounded subset has a supremum, this applies also, for any set <math>X,</math> in the function space containing all functions from <math>X</math> to <math>P,</math> where <math>f \leq g</math> if and only if <math>f(x) \leq g(x)</math> for all <math>x \in X.</math> For example, it applies for real functions, and, since these can be considered special cases of functions, for real <math>n</math>-tuples and sequences of real numbers.

The [[least-upper-bound property]] is an indicator of the suprema.