In mathematics, the infimum (abbreviated inf; : infima) of a subset of a partially ordered set is the greatest element if such an element exists. exists, it is unique, and if b is a lower bound, then b. Consequently, the term greatest lower bound (abbreviated as ) is also commonly used. The supremum (abbreviated sup; : suprema is the least element if such an element exists. exists, it is unique, and if b is an upper bound is less than or equal to b. 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 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 close 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 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 partially ordered set such that

is called an (or , or meet if

• is larger than any other lower bound).

Similarly, an such that

is called a (or , or join if

• is less than any other upper bound).

We can also define suprema and infima without restricting to sets. For example, there is no set containing all cardinal numbers (and there is no greatest cardinal number), but the axiom of choice implies that every set of cardinal numbers has a least upper bound among cardinal numbers. The axiom of choice is equivalent to the statement that every nonempty set of cardinal numbers has a minimum element (which is also the infimum of the set). The empty set of cardinal numbers has many lower bounds but no greatest lower bound among cardinal numbers.

Existence and uniqueness

.)

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.

(or does not exist).

Relation to maximum and minimum elements

If it does, it is a minimum or least element it is a maximum or greatest element

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.

together with the set of integers 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 .

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 (including \sqrt{2}). 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.

-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 of the real numbers are particularly important. For instance, the negative real numbers (which is not a negative real number). The completeness of the real numbers is empty

Properties

Set inclusion

Image under functions

Identifying infima and suprema

Relation to limits of sequences

In particular, the infimum and supremum of a set belong to its closure

Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied. Consider for example the well-known fact from topology is a continuous function which (for instance) guarantees is an adherent point is also an increasing or non-decreasing function is a real (or complex whose sup norm

be non-increasing rather than non-decreasing). Other norms include the L^{p,w} space), the norm on Lebesgue space and operator norms) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.

Arithmetic operations on sets

are sets of real numbers.

Sum of sets

The Minkowski sum

Product of sets

Scalar product of a set

Multiplicative inverse of a set

is used. where if

Duality

with the opposite order relation and vice versa.

Examples

Infima

• 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.

Suprema

• 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 is irrational, which means that the rationals are incomplete.

for any functionals

denotes "divides", is the lowest common multiple

is the union is the power set is subset.

See also

• (infimum limit)

Notes

References

External links