Let $(X,\preceq)$ be a partially ordered set.
If $X$ has a greatest element, then it is unique and is called the greatest element / maximum of $X$, denoted by $\max X$.
If $X$ has a least element, then it is unique and is called the least element / minimum of $X$, denoted by $\min X$.
The maximum and minimum are indeed unique:
Proof.If both $a$ and $a’$ are greatest elements of $X$, $a’\preceq a$ and $a\preceq a’$, so $a=a’$ by antisymmetry. The uniqueness of the minimum follows analogously.