DEF-MA. Maximal and Minimal Element.
Let $(X,\leq)$ be a preordered set and let $a\in X$.
$a$ is a maximal element of $X$, if
\[\forall x\in X :\enspace a\leq x \enspace\Rightarrow\enspace x\leq a.\]$a$ is a minimal element of $X$, if
\[\forall x\in X :\enspace x\leq a \enspace\Rightarrow\enspace a\leq x.\]
PROP-MA-PO.
Let $(X,\leq)$ be a partially ordered set and let $a\in X$. Because of partial order’s antisymmetry, we have:
$a$ 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.
\[\forall x\in X :\enspace a\leq x \enspace\Rightarrow\enspace x=a.\]$a$ 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.
\[\forall x\in X :\enspace x\leq a \enspace\Rightarrow\enspace x=a.\]