are equal.
[[Image:Supremum illustration.svg|250px|A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond).
In [[mathematics, the infimum (abbreviated inf; plural infima) of a [[subset $S$ of a [[partially ordered set $T$ is the [[greatest element in $T$ that is less than or equal to all elements of $S,$ if such an element exists. Consequently, the term ''greatest lower bound'' (abbreviated as ''GLB'') is also commonly used.
The supremum (abbreviated sup; plural suprema) of a subset $S$ of a partially ordered set $T$ is the least element in $T$ that is greater than or equal to all elements of $S,$ if such an element exists. Consequently, the supremum is also referred to as the ''least upper bound'' (or ''LUB'').
The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
The concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have . For instance, the set of positive real numbers $\backslash mathbb^$ (not including 0) does not have a minimum, because any given element of $\backslash mathbb^$ could simply be divided in half resulting in a smaller number that is still in $\backslash mathbb^$. There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.

** Formal definition **

A of a subset $S$ of a partially ordered set $(P,\; \backslash leq)$ is an element $a$ of $P$ such that
* $a\; \backslash leq\; x$ for all $x\; \backslash in\; S.$
A lower bound $a$ of $S$ is called an (or , or ) of $S$ if
* for all lower bounds $y$ of $S$ in $P,$ $y\; \backslash leq\; a$ ($a$ is larger than or equal to any other lower bound).
Similarly, an of a subset $S$ of a partially ordered set $(P,\; \backslash leq)$ is an element $b$ of $P$ such that
* $b\; \backslash geq\; x$ for all $x\; \backslash in\; S.$
An upper bound $b$ of $S$ is called a (or , or ) of $S$ if
* for all upper bounds $z$ of $S$ in $P,$ $z\; \backslash geq\; b$ ($b$ is less than any other upper bound).

** Existence and uniqueness **

Infima and suprema do not necessarily exist. Existence of an infimum of a subset $S$ of $P$ can fail if $S$ has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique.
Consequently, partially ordered sets for which certain infima are known to exist become especially interesting. For instance, a lattice is a partially ordered set in which all subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which subsets have both a supremum and an infimum. More information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties.
If the supremum of a subset $S$ exists, it is unique. If $S$ contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to $S$ (or does not exist). Likewise, if the infimum exists, it is unique. If $S$ contains a least element, then that element is the infimum; otherwise, the infimum does not belong to $S$ (or does not exist).

** Relation to maximum and minimum elements **

The infimum of a subset $S$ of a partially ordered set $P,$ assuming it exists, does not necessarily belong to $S.$ If it does, it is a minimum or least element of $S.$ Similarly, if the supremum of $S$ belongs to $S,$ it is a maximum or greatest element of $S.$
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 $x,$ there is another negative real number $\backslash tfrac$, 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, 0 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 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 total one. In a totally ordered set, like the real numbers, the concepts are the same.
As an example, let $S$ be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from $S$ together with the set of integers $\backslash mathbb$ and the set of positive real numbers $\backslash mathbb^$, ordered by subset inclusion as above. Then clearly both $\backslash mathbb$ and $\backslash mathbb^$ are greater than all finite sets of natural numbers. Yet, neither is $\backslash mathbb^$ smaller than $\backslash mathbb$ nor is the converse true: both sets are minimal upper bounds but none is a supremum.

** Least-upper-bound property **

The is an example of the aforementioned completeness properties which is typical for the set of real numbers. This property is sometimes called .
If an ordered set $S$ has the property that every nonempty subset of $S$ having an upper bound also has a least upper bound, then $S$ is said to have the least-upper-bound property. As noted above, the set $\backslash mathbb$ of all real numbers has the least-upper-bound property. Similarly, the set $\backslash mathbb$ of integers has the least-upper-bound property; if $S$ is a nonempty subset of $\backslash mathbb$ and there is some number $n$ such that every element $s$ of $S$ is less than or equal to $n,$ then there is a least upper bound $u$ for $S,$ an integer that is an upper bound for $S$ and is less than or equal to every other upper bound for $S.$ A well-ordered 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 the least-upper-bound property is $\backslash mathbb$, the set of rational numbers. Let $S$ be the set of all rational numbers $q$ such that $q^2\; <\; 2.$ Then $S$ has an upper bound ($1000,$ for example, or $6$) but no least upper bound in $\backslash mathbb$: If we suppose $p\; \backslash in\; \backslash mathbb$ is the least upper bound, a contradiction is immediately deduced because between any two reals $x$ and $y$ (including $\backslash sqrt$ and $p$) there exists some rational $p$, which itself would have to be the least upper bound (if $p\; >\; \backslash sqrt$) or a member of $S$ greater than $p$ (if $p\; <\; \backslash sqrt$). Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.
There is a corresponding ; 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 $P$ every bounded subset has a supremum, this applies also, for any set $X,$ in the function space containing all functions from $X$ to $P,$ where $f\; \backslash leq\; g$ if and only if $f(x)\; \backslash leq\; g(x)$ for all $x$ in $X.$ For example, it applies for real functions, and, since these can be considered special cases of functions, for real $n$-tuples and sequences of real numbers.
The least-upper-bound property is an indicator of the suprema.

** Infima and suprema of real numbers **

In analysis, infima and suprema of subsets $S$ of the real numbers are particularly important. For instance, the negative real numbers do not have a greatest element, and their supremum is 0 (which is not a negative real number).
The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset $S$ of the real numbers has an infimum and a supremum. If $S$ is not bounded below, one often formally writes $\backslash inf\_\; S\; =\; -\backslash infty.$ If $S$ is empty, one writes $\backslash inf\_\; S\; =\; +\backslash infty.$

** Properties **

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets $A,\; B\; \backslash subseteq\; \backslash mathbb,$ and scalar $r\; \backslash in\; \backslash mathbb.$ Define
* $A\; \backslash neq\; \backslash varnothing$ if and only if $\backslash sup\; A\; \backslash geq\; \backslash inf\; A,$ and otherwise $-\backslash infty\; =\; \backslash sup\; \backslash varnothing\; <\; \backslash inf\; \backslash varnothing\; =\; \backslash infty.$
* $r\; A\; =\; \backslash $; the scalar product of a set is just the scalar multiplied by every element in the set.
* $A\; +\; B\; =\; \backslash $; called the Minkowski sum, it is the arithmetic sum of two sets is the sum of all possible pairs of numbers, one from each set.
* $A\; \backslash cdot\; B\; =\; \backslash $; the arithmetic product of two sets is all products of pairs of elements, one from each set.
In those cases where the infima and suprema of the sets $A$ and $B$ exist, the following identities hold:
* $p\; =\; \backslash inf\; A$ if and only if for every $\backslash epsilon\; >\; 0$ there is an $x\; \backslash in\; A$ with $x\; <\; p\; +\; \backslash epsilon,$ and $y\; \backslash geq\; p$ for every $y\; \backslash in\; A.$
* $p\; =\; \backslash sup\; A$ if and only if for every $\backslash epsilon\; >\; 0$ there is an $x\; \backslash in\; A$ with $x\; >\; p\; -\; \backslash epsilon,$ and $y\; \backslash leq\; p$ for every $y\; \backslash in\; A.$
* If $A\; \backslash subseteq\; B$ and then $\backslash inf\; A\; \backslash geq\; \backslash inf\; B$ and $\backslash sup\; A\; \backslash leq\; \backslash sup\; B.$
* If $r\; >\; 0$ then $\backslash inf\; (\; r\; \backslash cdot\; A\; )\; =\; r\; \backslash left(\; \backslash inf\; A\; \backslash right)$ and $\backslash sup\; (\; r\; \backslash cdot\; A\; )\; =\; r\; \backslash left(\; \backslash sup\; A\; \backslash right).$
* If $r\; \backslash leq\; 0$ then $\backslash inf\; (\; r\; \backslash cdot\; A\; )\; =\; r\; \backslash left(\; \backslash sup\; A\; \backslash right)$ and $\backslash sup\; (\; r\; \backslash cdot\; A\; )\; =\; r\; \backslash left(\; \backslash inf\; A\; \backslash right).$
* $\backslash inf\; (\; A\; +\; B\; )\; =\; \backslash left(\; \backslash inf\; A\; \backslash right)\; +\; \backslash left(\; \backslash inf\; B\; \backslash right)$ and $\backslash sup\; (\; A\; +\; B\; )\; =\; \backslash left(\; \backslash sup\; A\; \backslash right)\; +\; \backslash left(\; \backslash sup\; B\; \backslash right).$
* If $A$ and $B$ are nonempty sets of positive real numbers then $\backslash inf\; (\; A\; \backslash cdot\; B\; )\; =\; \backslash left(\; \backslash inf\; A\; \backslash right)\; \backslash cdot\; \backslash left(\; \backslash inf\; B\; \backslash right)$ and similarly for suprema $\backslash sup\; (\; A\; \backslash cdot\; B\; )\; =\; \backslash left(\; \backslash sup\; A\; \backslash right)\; \backslash cdot\; \backslash left(\; \backslash sup\; B\; \backslash right)$.

** Duality **

If one denotes by $P^$ the partially-ordered set $P$ with the opposite order relation, i.e.
* $x\; \backslash leq\; y$ in $P^$ if and only if $x\; \backslash geq\; y$ in $P,$
then infimum of a subset $S$ in $P$ equals the supremum of $S$ in $P^$ and vice versa.
For subsets of the real numbers, another kind of duality holds: inf S = −sup(−''S''), where $-S\; =\; \backslash .$

** Examples **

** Infima **

* The infimum of the set of numbers is . The number is a lower bound, but not the greatest lower bound, and hence not the infimum.
* More generally, if a set has a smallest element, then the smallest element is the infimum for the set. In this case, it is also called the minimum of the set.
* $\backslash inf\; \backslash \; =\; 1.$
* $\backslash inf\; \backslash \; =\; 0.$
* $\backslash inf\; \backslash left\backslash \; =\; \backslash sqrt$
* $\backslash inf\; \backslash left\backslash \; =\; -1.$
* If is a decreasing sequence with limit , then .

** Suprema **

*The supremum of the set of numbers is . The number is an upper bound, but it is not the least upper bound, and hence is not the supremum.
* $\backslash sup\; \backslash \; =\; \backslash sup\; \backslash \; =\; 1.$
* $\backslash sup\; \backslash left\backslash \; =\; 1.$
* $\backslash sup\; \backslash \; =\; \backslash sup\; A\; +\; \backslash sup\; B.$
* $\backslash sup\; \backslash \; =\; \backslash sqrt.$
In the last example, the supremum of a set of rationals is irrational, which means that the rationals are incomplete.
One basic property of the supremum is
:$\backslash sup\; \backslash \; \backslash leq\; \backslash sup\; \backslash \; +\; \backslash sup\; \backslash $
for any functionals $f$ and $g.$
The supremum of a subset $S$ of (ℕ,|) where | denotes "divides", is the lowest common multiple of the elements of $S.$
The supremum of a subset $S$ of (''P'',⊆), where $P$ is the power set of some set, is the supremum with respect to ⊆ (subset) of a subset $S$ of $P$ is the union of the elements of $S.$

** See also **

* Essential supremum and essential infimum
* Greatest element and least element
* Maximal and minimal elements
* Limit superior and limit inferior (infimum limit)
* Upper and lower bounds

** References **

*

** External links **

*
* {{MathWorld|Supremum|author=Breitenbach, Jerome R.|author2=Weisstein, Eric W.|name-list-style=amp
Category:Order theory