maximal element

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

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$a\in A$ is the least element of $A$ if $a\leq x$ , 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 $x\leq a$ and $x\neq a$ .
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$a\in A$ is the greatest element of $A$ if $x\leq a$ 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 $a\leq x$ and $x\neq a$ .

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 $\leq$ 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 $\leq$ 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 $\leq$ have no least element or minimal element, and no maximal or greatest element.