A characterization of Boolean collections of set-valued functions

We consider functions on subsets of a finite element set r = {0,…, r-1}. A small fraction of these functions are Boolean functions, i.e., functions that can be constructed from constants and variables, using union, intersection, and complementation. We introduce the notion of Boolean collection of sets, and explore several combinatorial aspects of these collections. A collection C is a set consisting of several n-tuples of subsets of r. We solve the following problem. Given n ⩾ 1, characterize collections C for which there exists a Boolean function F : P ( r ) n → P ( r ) such that F( X 1,…, X n ) = ⊘ if and only if ( X 1,…, X n ) ε C, where X 1 , ... , X n ∈ P ( r ) and P ( r ) is the set of subsets of r.