Let $\mathfrak{A}$ be a lattice with join denoted $\cup$, meet denoted $\cap$, and least element $\bot$.
Consider a set $S\in\mathscr{P}\mathfrak{A}$ such that the following property holds (for every $a,b\in\mathfrak{A}$):
$a\in S\land b\in S\land a\cap b\ne\bot \Rightarrow a\cup b\in S$.
Is there a "standard" name for this property?