maximal element
Let $\le $ be an ordering^{} on a set $S$, and let $A\subseteq S$. Then, with respect to the ordering $\le $,

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$a\in A$ is the least element of $A$ if $a\le x$, for all $x\in A$.

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$a\in A$ is a minimal^{} element of $A$ if there exists no $x\in A$ such that $x\le a$ and $x\ne a$.

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$a\in A$ is the greatest element of $A$ if $x\le a$ for all $x\in A$.

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$a\in A$ is a maximal element of $A$ if there exists no $x\in A$ such that $a\le x$ and $x\ne a$.
Examples.

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The natural numbers^{} $\mathbb{N}$ ordered by divisibility ($\mid $) have a least element, $1$. The natural numbers greater than 1 ($\mathbb{N}\setminus \{1\}$) have no least element, but infinitely many minimal elements (the primes.) In neither case is there a greatest or maximal element.

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

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The natural numbers $\mathbb{N}$ ordered by the standard $\le $ have a least element, $1$, which is also a minimal element. They have no greatest or maximal element.

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

Canonical name  MaximalElement 
Date of creation  20130322 12:30:44 
Last modified on  20130322 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 