You are here
Homemaximal element
Primary tabs
maximal element
Let $\leq$ be an ordering on a set $S$, and let $A\subseteq S$. Then, with respect to the ordering $\leq$,

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

$a\in A$ is a minimal element of $A$ if there exists no $x\in A$ such that $x\leq a$ and $x\neq a$.

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

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

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.

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

The natural numbers $\mathbb{N}$ ordered by the standard $\leq$ 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 $\leq$ have no least element or minimal element, and no maximal or greatest element.
Mathematics Subject Classification
03E04 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections