Maximum and Minimum

Not to be confused with maximal and minimal element!

Definition 1. Let $X$ be a partially ordered set.

If $X$ has a greatest element, then it is unique and is called the maximum of $X$, denoted by
\[\max X.\]
If $X$ has a least element, then it is unique and is called the minimum of $X$, denoted by
\[\min X.\]

Proof (uniqueness). If both $a$ and $a’$ are greatest elements of $X$, $a’ \leq a$ and $a \leq a’$, so $a=a’$ by antisymmetry. Similarly for the minimum.