Maximal and Minimal Element

MA#DEF. Maximal and Minimal Element.

Let $X$ be a preordered set and $a\in X$.

• $a$ is a maximal element of $X$ if for all $x\in X$,

  \[a\leq x \enspace\rimp\enspace x\leq a.\]

• $a$ is a minimal element of $X$ if for all $x\in X$,

  \[x\leq a \enspace\rimp\enspace a\leq x.\]

MA#PROP-PO.

Let $X$ be a partially ordered set and $a\in X$. Because $\leq$ 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 $a\leq x$, 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.

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