Not to be confused with maximum and minimum!
Let $X$ be a preordered set and $a \in X$.
$a$ is a maximal element of $X$ if
\[\forall x \in X :\enspace a \leq x \enspace\rimp\enspace x \leq a.\]$a$ is a minimal element of $X$ if
\[\forall x \in X :\enspace x \leq a \enspace\rimp\enspace a \leq x.\]
Let $X$ be a partially ordered set and $a \in X$.
$a$ is a maximal element of $X$ if and only if there is no other element $x\in X$ such that $a\leq x$, i.e.
\[\forall x\in X :\enspace a\leq x \enspace\rimp\enspace x=a.\]$a$ is a minimal element of $X$ if and only if there is no other element $x\in X$ such that $x\leq a$, i.e.
\[\forall x\in X :\enspace x\leq a \enspace\rimp\enspace x=a.\]
Proof. By antisymmetry of $\leq$.