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 $\min X$.
- Synonym of maximum: the greatest element
- Synonym of minimum: the least element
The maximum and minimum are indeed unique:
Proof.If both $a$ and $a’$ are greatest elements of $X$, $a’\leq a$ and $a\leq a’$, so $a=a’$ by antisymmetry. The uniqueness of the minimum follows analogously.