Supremum

In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). The supremum may, or may not, belong to the set S. If S contains a greatest element, then that element is the supremum; and if not, then the supremum does not belong to the set.

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Suprema are often considered for subsets of real numbers, rational numbers, or any other well-known mathematical structures for which it is immediately clear what it means for an element to be "greater-or-equal" than another element. Nonetheless, the definition generalizes easily to the more abstract setting of order theory where one considers arbitrary partially ordered sets.

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In any case, suprema must not be confused with minimal upper bounds, or with maximal or greatest elements. Some notes on these issues follow below.

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Ordered set: An ordered set or ordered collection may be any of the following....

Least element: REDIRECT Greatest element...

Greatest element: In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. A bounded poset is a poset that has both a greatest element and a l...