Constructing Hierarchical Set Systems

In this note, it is shown that by applying ranking procedures to data that allow, for any three objects a1; a2; b in a collection X of objects of interest, to make consistent decisions about which of the two objects a1 or a2 is more similar to b, a family of cluster systems A (k) (k = 0; 1;:::) can be constructed that start with the associated Apresjan Hierarchy and keep growing until, for k = #X 1, the full set P (X) of all subsets of X is reached. Various ideas regarding canonical modications of the similarity values so that these cluster systems contain as many clusters as possible for small values of k (and in particular for k := 0) and/or are rooted at a specic element in X, possible applications, e.g. concerning (i) the comparison of distinct dissimilarity data dened on the same set X or (ii) diversity optimization, and new tasks arising in ranking statistics are also discussed.