A set-theoretical approach to the problem of hierarchical clustering

A set-theoretical formalization is developed for the problem of generating hierarchically organized collections of subsets, or to use a phrase common in the applied substantive literature, for the problem of hierarchical clustering. A number of terms are introduced to characterize those clustering methods that attempt to limit the size of the overlap between each pair of subsets constructed at a specific “compactness” level. Several examples, motivated primarily by graph theory, are discussed briefly to illustrate the various set-theoretical concepts presented.