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Maximal Element

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

$a \in A$ is the least element of if $a \le x$ , for all $x \in A$ .
$a \in A$ is a minimal element of if there exists no $x \in A$ such that $x \le a$ and $x \ne a$ .
$a \in A$ is the greatest element of if $x \le a$ for all $x \in A$ .
$a \in A$ is a maximal element of if there exists no $x \in A$ such that $a \le x$ and $x \ne a$ .

The natural numbers $\mathbb{N}$ ordered by divisibility ( $\mid$ ) have a least element, . 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 $\le$ have a maximal element which is also the greatest element, . They have no minimal or least element.
The natural numbers $\mathbb{N}$ ordered by the standard $\le$ have a least element, , which is also a minimal element. They have no greatest or maximal element.
The rationals greater than zero with the standard ordering $\le$ have no least element or minimal element, and no maximal or greatest element.