Boolean theory of coteries

A coterie under a ground set U consists of a set of subsets (quorums) of U such that any pair of quorums intersect each other. 'Nondominated' coteries are of particular interest, since they are 'optimal' in some sense. By assigning a Boolean variable to each element in U, we represent a coterie by a Boolean function of these variables. The authors characterize the nondominated coteries as exactly those which can be represented by positive, self-dual functions. They take advantage of Boolean decomposition theorems to investigate 'decomposition' (or 'composition') of coteries, and prove that any function representing a nondominated coteries can be composed from copies of the 3-majority function. They also introduce a 'decomposition tree' to represent any nondominated coterie. A number of other new results are also obtained, demonstrating the usefulness of the Boolean approach.<
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