Maximal and Minimal Elements

Definition 1. 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

  \[\forall x \in X :\enspace x \leq a \enspace\rimp\enspace a \leq x.\]

Proposition 1. 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.

  \[\forall x \in X :\enspace x \leq a \enspace\rimp\enspace x = a.\]