A step of an algorithm I’ve designed requires computing the minimal closure under intersection of a set of sets of arbitrary size. By the "minimal closure (of a set $S$) under intersection", I mean:
Given a set $S$ containing sets $s_1, \cdots , s_k$, the smallest set $𝑆′$ such that $S\subseteq S'$ and $x\cap y \in S'$ for any two sets $x, y \in S'$.
While I can come up with a pretty straightforward naive approach (loop over the sets and store all the intersections in a new set, then update the set as the union of previous step’s set and the new set, and repeat this process until the new and old set are the same), I am looking for a faster method, but haven’t been able to find any papers working on a similar problem. Does anybody know if there are faster existing algorithms to solve this problem before I attempt to potentially reinvent the — or a worse version of — the wheel? Or if not exact solutions, maybe fast approximation algorithms on other data structures (e.g. sets of strings) that translate to this problem setup?