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Upper and lower sets

In mathematics, an upper set S {\displaystyle S} of a partially ordered set X {\displaystyle X} is a subset such that if s is in S and if x in X is larger than s, then x is in S. A lower set is defined similarly as being a subset S of X with the property that any element x of X that precedes an element of S is necessarily also an element of S.

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Examples

Upper closure and lower closure

Scott topology

Birkhoff's theorem

Related notions

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Map overview Semantic statistics

Upper and lower sets

Nodes50
Edges49
Triples1
Avg. degree1.96
Density0.04
Components1

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Important terminology

displaystyle set upper lower sets also subset leq ordered filter closure topology ideal called inclusion isbn directed every theory poset

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
that a least upper bound always existsinstance ofthe above construction can be used to replace a given ordering by set inclusion and also yields advantages0.80text

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