# 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.

Title maximal element MaximalElement 2013-03-22 12:30:44 2013-03-22 12:30:44 akrowne (2) akrowne (2) 9 akrowne (2) Definition msc 03E04 greatest element least element minimal element