# Supremum Element

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A Supremum Element is the set element of a Partially Ordered Set that is GreaterThanOrEqualTo all Set Elements of some Subset of the set.

## References

### 2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Infimum_and_supremum Retrieved:2015-6-13.
• In mathematics, the infimum (abbreviated inf ; plural infima) 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 (abbreviated as GLB) is also commonly used.

The supremum (abbreviated sup ; plural suprema) 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 (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.

If the supremum exists, it is unique for the subset. If S contains a greatest element, then that element is the supremum; otherwise, the supremum does not belong to S (or does not exist). Likewise, 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 (or does not exist).

The concepts of supremum and infimum 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 positive real numbers do not have a minimum, because any given positive real number could simply be divided in half resulting in a smaller number. 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 number which could be used as a lower bound. Note that 0 is not part of the subset, since 0 is not a positive real number.