maximal element

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

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

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

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

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

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