Definition 4

Let E be an ordered set. An element $a \in E$ is said to be the least (resp. greatest) element of E if for all $x \in E$ we have $a \leqslant x$ (resp. $x \leqslant a$).

An ordered set need not have a greatest element nor a least element. If E has a least element $a$, then $a$ is the greatest element with respect to the opposite ordering.

¶ If E has a least element $a$, then $a$ is the unique minimal element of E; for if $x \in E$ is distinct from $a$, we have $a < x$.

Examples