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?
> 
> 
> 



Does this set family have a name?
I am not sure how standard this is, but It makes sense to call this family the [atoms](http://en.wikipedia.org/wiki/Atom_(order_theory)) of the corresponding lattice of sets obtained from $\mathcal S$. They can be illustrated as the different regions in a Venn diagram.


