Supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset



S


{\displaystyle S}
of a partially ordered set



T


{\displaystyle T}
is the greatest element in



T


{\displaystyle T}
that is less than or equal to all elements of



S
,


{\displaystyle 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


{\displaystyle S}
of a partially ordered set



T


{\displaystyle T}
is the least element in



T


{\displaystyle T}
that is greater than or equal to all elements of



S
,


{\displaystyle 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 no minimum or maximum. For instance, the set of positive real numbers





R


+




{\displaystyle \mathbb {R} ^{+}}
(not including 0) does not have a minimum, because any given element of





R


+




{\displaystyle \mathbb {R} ^{+}}
could simply be divided in half resulting in a smaller number that is still in





R


+




{\displaystyle \mathbb {R} ^{+}}
. 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.

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