maximal element

Let be an orderingMathworldPlanetmath on a set S, and let AS. Then, with respect to the ordering ,

  • aA is the least element of A if ax, for all xA.

  • aA is a minimalPlanetmathPlanetmath element of A if there exists no xA such that xa and xa.

  • aA is the greatest element of A if xa for all xA.

  • aA is a maximal element of A if there exists no xA such that ax and xa.


  • The natural numbersMathworldPlanetmath ordered by divisibility () have a least element, 1. The natural numbers greater than 1 ({1}) have no least element, but infinitely many minimal elements (the primes.) In neither case is there a greatest or maximal element.

  • The negative integers ordered by the standard definition of have a maximal element which is also the greatest element, -1. They have no minimal or least element.

  • The natural numbers ordered by the standard have a least element, 1, which is also a minimal element. They have no greatest or maximal element.

  • The rationals greater than zero with the standard ordering have no least element or minimal element, and no maximal or greatest element.

Title maximal element
Canonical name MaximalElement
Date of creation 2013-03-22 12:30:44
Last modified on 2013-03-22 12:30:44
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 9
Author akrowne (2)
Entry type Definition
Classification msc 03E04
Defines greatest element
Defines least element
Defines minimal element