Infimum and supremum
In mathematics, the
infimum of a subset
S of a partially ordered set
T is the greatest element of
T that is less than or equal to all elements of
S. Consequently the term
greatest lower bound is also commonly used. Infima of real numbers are a common special case that is especially important in analysis. However, the general definition remains valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The
supremum of a subset
S of a totally or partially ordered set
T is the least element of
T that is
greater than or equal to all elements of
S. Consequently, the supremum is also referred to as the
least upper bound. If the supremum exists, it is unique, meaning that there will be only one supremum. If
S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to
S. 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.