Minimal generator for a set of sets

Is this a known problem?

Given a set of sets $$S$$ find a set of sets $$B$$ s.t. each set in $$S$$ can be obtained through unions of some sets in $$B$$. The set $$S$$ is already a solution but the objective is to find the smallest $$B$$.

For instance $$\{\{a,b\}, \{c\}, \{a,b,c\}\}$$ can be obtained from $$\{\{a,b\}, \{c\}\}$$.

Since it's somewhat similar to set-cover, I suspect it's NP-hard.

Additional remark: $$B$$ does not have to be a subset of $$S$$.

A decision variant, without the minimality condition, asking whether there is a set $$B$$ of size $$n$$ is called the set basis problem [SP7] in Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson. The NP-completeness of it was proved by Stockmeyer.