# Does this set family have a name?

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?

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I am not sure how standard this is, but It makes sense to call this family the atoms of the corresponding lattice of sets obtained from $\mathcal S$. They can be illustrated as the different regions in a Venn diagram.