Infimum and supremum

In mathematics, the infimum (abbreviated inf; pl.

Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.

is less than or equal to b. Consequently, the supremum is also referred to as the least upper bound (or LUB).

[1] 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 no minimum or maximum.

could simply be divided in half resulting in a smaller number that is still in

is called an infimum (or greatest lower bound, or meet) of

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 nonempty finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all 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.

For example, consider the set of negative real numbers (excluding zero).

On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set.

In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique.

In a totally ordered set, like the real numbers, the concepts are the same.

nor is the converse true: both sets are minimal upper bounds but none is a supremum.

is the least upper bound, a contradiction is immediately deduced because between any two reals

Another example is the hyperreals; there is no least upper bound of the set of positive infinitesimals.

For instance, the negative real numbers do not have a greatest element, and their supremum is

[1] The completeness of the real numbers implies (and is equivalent to) that any bounded nonempty subset

is any non-empty set of real numbers then there always exists a non-decreasing sequence

Expressing the infimum and supremum as a limit of a such a sequence allows theorems from various branches of mathematics to be applied.

) can also be used to help prove many of the formula given below, since addition and multiplication of real numbers are continuous operations.

The following formulas depend on a notation that conveniently generalizes arithmetic operations on sets.

consisting of all possible arithmetic sums of pairs of numbers, one from each set.

of real numbers is defined similarly to their Minkowski sum:

For subsets of the real numbers, another kind of duality holds:

denotes "divides", is the lowest common multiple of the elements of

is the union of the subsets when considering the partially ordered set