Maximal Element (original) (raw)

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Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

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• Foundations of Mathematics
• Set Theory
• Partial Orders
• MathWorld Contributors
• Nakahara

Let be a partially ordered set. Then an element is said to be maximal if, for all , . Alternatively, an element is maximal such that if for any , then .

Note that the definition for a maximal element above is true for any two elements of a partially ordered set that are comparable. However, it may be the case that two elements of a given partial ordering are not comparable.

See also

Comparable Elements, Maximal Set

This entry contributed by Jay S. Nakahara

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References

Jech, T. Set Theory. Berlin: Springer-Verlag, 2003.

Referenced on Wolfram|Alpha

Maximal Element

Cite this as:

Nakahara, Jay S. "Maximal Element." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MaximalElement.html

Subject classifications

• Foundations of Mathematics
• Set Theory
• Partial Orders
• MathWorld Contributors
• Nakahara