Definition. Let $P=(S,\leq)$ be a partially ordered set. If it is true that $$(\exists s_0\in S)\ (\forall s\in S)\ \ s\leq s_0$$ then $s_0$ is said to be a greatest element of $S$.
Of course, if a greatest element exists, it is the unique greatest element, and coincides with the unique maximal element, $\max S$. For this reason, the usefulness of such a notation is debatable but the question nevertheless stands. If anything, one could simply write $\operatorname{ges} S =\max S $, or $\operatorname{ges S}$ does not exist etc, for example $\operatorname{ges} S$ does not exist etc.[1]
- Is there some common notation for the least/greatest element of a set?
(Used simply for the sake of an example, not a suggestion: ges – greatest element of set.)
[1] Example: $P=(\{1,3,7\},|)$, $\max_1 S= 3$, $\max_2 S= 7$, $\operatorname{ges} S$ does not exist. Therefore, any notation used for maximal elements is ill-suited for the greatest element in this case.
I’ve commonly seen the following:
$0, \bot$- minimal element in a partial ordered set
$1, \top$- maximal element