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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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up vote 3 down vote accepted

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.

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Because of complements, I would not say lattice of sets, but (Boolean) algebra of sets. – Gerald Edgar Nov 16 '10 at 14:39
I agree with both, although I would only call the inclusion-minimal sets atoms. – Andrew D. King Nov 16 '10 at 18:45
Thanks everyone for taking the time to answer/comment. Said information has been passed along. – Tony Huynh Nov 17 '10 at 11:04

In my class of probability and measure theory I am calling it the partition generated by the family.

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