DEF-MAX.
Let $(X,\leq)$ be a partially ordered set.
If $X$ has a top, then it is unique and is called the greatest element / maximum of $X$, denoted by $\max X$.
If $X$ has a bottom, then it is unique and is called the least element / minimum of $X$, denoted by $\min X$.
Proof.Let $a$ and $b$ both be tops of $X$. By definition, we have $b\leq a$ and $a\leq b$. By partial order’s antisymmetry, we have $a=b$. Analogously, if $a$ and $b$ are both bottoms of $X$, we also have $a=b$.$\square$