Consider a partially ordered set $(A,\le)$, an element $a\in A$, and a set of elements $B\subseteq A$ with $b\ge a$ for all $b\in B$, such that any maximal chain in $(A,\le)$ that includes $a$ must contain at least an element from $B$. In other words, $B$ is a set of "upper bounds" of $a$.
For example, in the partially ordered set of subsets of $\{1,2,3\}$, $(\mathcal P(\{1,2,3\}), \subseteq)$, $\{2\}$ would be bounded by $\{\{1,2\},\{2,3\}\}$.
Does this set of elements have a standard name? Does this concept have some interesting known properties?
{ x : a <= x } is called the up set or upper set of {a}.
It is notated by a small up pointing arrow in front of a.
{ x : some a in A with a <= x } is called the up set or upper
set of A, written by small up pointing arrow in front of A.
Simular, in the down direction, are the down ward sets.