SUPREMUM

Etymology

Noun

supremum (plural suprema)

(set theory) (real analysis): Given a subset X of R, the smallest real number that is ≥ every element of X; (order theory): given a subset X of a partially ordered set P (with partial order ≤), the least element y of P such that every element of X is ≤ y.

Usage notes

• Commonly denoted sup(X).

• The supremum of X may not exist, and, if it does, may not be an element of X.

• (order theory)

Formally: Let $\backslash (S\; =\backslash \{\; t\; :\; t\; \backslash in\; P\; :\; \backslash forall\; x\backslash in\; X,\; x\; \backslash le\; t\; \backslash \}\backslash )$ be the set of upper bounds of X. Then sup(X), if it exists, is the element $\backslash (s\backslash in\; S:\; \backslash forall\; y\backslash in\; S,\; s\; \backslash le\; y\backslash )$.

The concept of supremum is closely related to the function ∨ (called join). The supremum of two elements, denoted $\backslash (\backslash sup\backslash \{x,y\backslash \}\backslash )$ can also be written $\backslash (x\backslash lor\; y\backslash )$. The supremum of a set may be denoted $\backslash (\backslash sup(X)\backslash )$ or $\backslash (\backslash bigvee\; X\backslash )$.

Synonyms

• (element of a set greater than or equal to all members of a given subset): least upper bound, LUB, sup

Coordinate terms

• infimum

Anagrams

• supermum

Source: Wiktionary

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