A colleague of mine recently asked me if this set family had a name (see definition of *this* below) . I didn't know the answer, so I thought I would consult the MO oracle.

Let $\mathcal{S}:=\{ S_1, \dots, S_k \}$ be a family of subsets of $[n]$. Consider the family $\mathcal{F}_{\mathcal{S}}$ formed by taking all sets of the form

$ S_1' \cap \dots \cap S_k' $

where each $S_i'$ is either $S_i$ or the complement of $S_i$. Note that we are forced to intersect exactly $k$ such sets.

Do set families arising in this way have a well-established name?