Sorry, I couldn't find anything and without proper names it's even harder. It sounds like something reasonably simple to consider and name though.
Given a set of sets $\bf S$, we define some kind of upward closure: $\delta({\bf S})=\{T\subseteq\bigcup {\bf S}\mid T \textit{ maximal such that for any }a,b\in T\textit{ there is }S\in{\bf S}\textit{ with }a,b\in S\}$
A few motivational words: For any $T\in\delta({\bf S})$ there will be some $S\in {\bf S}$ with $T\subseteq S$, if $a,b\in\bigcup{\bf S}$ are never member of the same set $S\in{\bf S}$ they won't be for any $T\in\delta({\bf S})$ either, in a way $\delta({\bf S})$ is an upward closure of $\bf S$ for these properties. And in some way, at least to me, set theoretic topology has very similar operators for filters and filter bases.
I came up with a more elegant definition: $\delta({\bf S})=\{T\subseteq\bigcup {\bf S}\mid b\in\bigcup{\bf S}\setminus T\textit{ iff there is }a\in T\textit{ with no }S\in{\bf S}\textit{ such that }a,b\in S\}$.
Upon consideration I think I'm going to call it a boundary operator, combined with an interior operator $Int$ on $\bf S$ such that $Int({\bf S})=\{T\subseteq 2^{\bigcup{\bf S}}\mid\textit{there is }S\in{\bf S}\textit{ with }T\subseteq S\}$ and a closure operator $cl({\bf S})=Int(\delta({\bf S}))$.
The intuition here is that the boundary are the maximally justified sets considering $\bf S$, the interior are all smaller or equal justified sets considering $\bf S$, the closure is a combination.
