Maximal and Minimal Element • Lutalli

DEF-MA. Maximal and Minimal Element.

Let $(X,\preceq)$ be a preordered set and $a\in X$.

$a$ is a maximal element of $X$ if
\[\forall x\in X :\enspace a\preceq x \enspace\rimp\enspace x\preceq a.\]

$a$ is a minimal element of $X$ if
\[\forall x\in X :\enspace x\preceq a \enspace\rimp\enspace a\preceq x.\]

PROP-MA-PO.

Let $(X,\preceq)$ be a partially ordered set and $a\in X$. Because $\preceq$ is antisymmetric, we have:

$a$ is a maximal element of $X$ if and only if there is no other element $x\in X$ such that $a\preceq x$, i.e.
\[\forall x\in X :\enspace a\preceq x \enspace\rimp\enspace x=a.\]

$a$ is a minimal element of $X$ if and only if there is no other element $x\in X$ such that $x\preceq a$, i.e.
\[\forall x\in X :\enspace x\preceq a \enspace\rimp\enspace x=a.\]