In
mathematics, the infimum (abbreviated inf; plural infima) of a
subset of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
is a
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 that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
in
that is less than or equal to each element of
if such an element exists.
Consequently, the term ''greatest lower bound'' (abbreviated as ) is also commonly used.
The supremum (abbreviated sup; plural suprema) of a subset
of a partially ordered set
is the
least 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 that is greater than every other element of S. The term least element is defined dually, that is, it is an elem ...
in
that is greater than or equal to each element of
if such an element exists.
Consequently, the supremum is also referred to as the ''least upper bound'' (or ).
The infimum is in a precise sense
dual to the concept of a supremum. Infima and suprema of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s are common special cases that are important in
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, and especially in
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
. However, the general definitions remain valid in the more abstract setting of
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article int ...
where arbitrary partially ordered sets are considered.
The concepts of infimum and supremum are close to
minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
and
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
, but are more useful in analysis because they better characterize special sets which may have . For instance, the set of
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
(not including
) does not have a minimum, because any given element of
could simply be divided in half resulting in a smaller number that is still in
There is, however, exactly one infimum of the positive real numbers relative to the real numbers:
which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part.
Formal definition
A of a subset
of a
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
is an element
of
such that
*
for all
A lower bound
of
is called an (or , or
) of
if
* for all lower bounds
of
in
(
is larger than or equal to any other lower bound).
Similarly, an of a subset
of a partially ordered set
is an element
of
such that
*
for all
An upper bound
of
is called a (or , or
) of
if
* for all upper bounds
of
in
(
is less than or equal to any other upper bound).
Existence and uniqueness
Infima and suprema do not necessarily exist. Existence of an infimum of a subset
of
can fail if
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
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
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
exists, it is unique. If
contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to
(or does not exist). Likewise, if the infimum exists, it is unique. If
contains a least element, then that element is the infimum; otherwise, the infimum does not belong to
(or does not exist).
Relation to maximum and minimum elements
The infimum of a subset
of a partially ordered set
assuming it exists, does not necessarily belong to
If it does, it is a
minimum or least element of
Similarly, if the supremum of
belongs to
it is a
maximum or greatest element of
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
there is another negative real number
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,
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
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is def ...
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
be the set of all finite subsets of natural numbers and consider the partially ordered set obtained by taking all sets from
together with the set of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s
and the set of positive real numbers
ordered by subset inclusion as above. Then clearly both
and
are greater than all finite sets of natural numbers. Yet, neither is
smaller than
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
has the property that every nonempty subset of
having an upper bound also has a least upper bound, then
is said to have the least-upper-bound property. As noted above, the set
of all real numbers has the least-upper-bound property. Similarly, the set
of integers has the least-upper-bound property; if
is a nonempty subset of
and there is some number
such that every element
of
is less than or equal to
then there is a least upper bound
for
an integer that is an upper bound for
and is less than or equal to every other upper bound for
A
well-order
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
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 the least-upper-bound property is
the set of rational numbers. Let
be the set of all rational numbers
such that
Then
has an upper bound (
for example, or
) but no least upper bound in
: If we suppose
is the least upper bound, a contradiction is immediately deduced because between any two reals
and
(including
and
) there exists some rational
which itself would have to be the least upper bound (if
) or a member of
greater than
(if
). Another example is the
hyperreals
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbe ...
; 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
every bounded subset has a supremum, this applies also, for any set
in the function space containing all functions from
to
where
if and only if
for all
For example, it applies for real functions, and, since these can be considered special cases of functions, for real
-tuples and sequences of real numbers.
The
least-upper-bound property
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if ev ...
is an indicator of the suprema.
Infima and suprema of real numbers
In
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, infima and suprema of subsets
of the
real numbers are particularly important. For instance, the negative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s do not have a greatest element, and their supremum is
(which is not a negative real number).
The
completeness of the real numbers
Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number li ...
implies (and is equivalent to) that any bounded nonempty subset
of the real numbers has an infimum and a supremum. If
is not bounded below, one often formally writes
If
is
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
, one writes
Properties
The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets: Let the sets
and scalar
Define
*
; the scalar product of a set is just the scalar multiplied by every element in the set. The case
is denoted by
*
; called the
Minkowski sum
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set
: A + B = \.
Analogously, the Minkowski ...
, it is the arithmetic sum of two sets is the sum of all possible pairs of numbers, one from each set.
*
; 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
and
exist, the following identities hold:
*
if and only if
and otherwise
* If
then there exists a sequence
in
such that
Similarly, there will exist a (possibly different) sequence
in
such that
Consequently, if the limit
is a real number and if
is a continuous function, then
is necessarily an
adherent point
In mathematics, an adherent point (also closure point or point of closure or contact point) Steen, p. 5; Lipschutz, p. 69; Adamson, p. 15. of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (or equivalen ...
of
*
if and only if
is a lower bound and for every
there is an
with
*
if and only if
is an upper bound and if for every
there is an
with
* If
and then
and
* If
then
and
* If
then
and
In particular,
and
*
and
* If
and
are nonempty sets of positive real numbers then
and similarly for suprema
* If
is non-empty and if
then
where this equation also holds when
if the definition
is used.
[The definition is commonly used with the ]extended real number
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
s; in fact, with this definition the equality will also hold for any non-empty subset However, the notation is usually left undefined, which is why the equality is given only for when This equality may alternatively be written as
Moreover,
if and only if
where if
then
Duality
If one denotes by
the partially-ordered set
with the
Converse relation, opposite order relation; that is, for all
declare:
then infimum of a subset
in
equals the supremum of
in
and vice versa.
For subsets of the real numbers, another kind of duality holds:
where
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
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of the set.
*
*
*
*
* 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.
*
*
*
*
In the last example, the supremum of a set of
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
is
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
, which means that the rationals are
incomplete.
One basic property of the supremum is
for any
functionals and
The supremum of a subset
of
where
denotes "
divides
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
", is the
lowest common multiple of the elements of
The supremum of a set
containing subsets of some set
is the
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of the subsets when considering the partially ordered set
, where
is the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
and
is
subset.
See also
*
*
*
* (infimum limit)
*
Notes
References
*
External links
*
* {{MathWorld, Supremum, author=Breitenbach, Jerome R., author2=Weisstein, Eric W., name-list-style=amp
Order theory