Maximal Element

Let (A,<=) be a partially ordered set. Then an element m in A is said to be maximal if, for all a in A, m!<=a. Alternatively, an element m in A is maximal such that if m<=a for any a in A, then m=a.

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

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Maximal Element

Cite this as:

Nakahara, Jay S. "Maximal Element." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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