Set Systems and Dissimilarities

Different one–one correspondences exist between classes of indexed clustering structures (such as indexed hierarchies) and classes of dissimilarities (such as ultrametrics). Following the line developed in previous works (i.e., Johnson (1967), Diday (1984, 1986), Bertrand and Diday (1991), Batbedat (1988), Bandelt and Dress (1994)), we associate each pseudo-dissimilarityδ defined on a finite set E with the set system of all maximal linked subsets of E. This provides a one–one correspondence that maps the set of pseudo-dissimilarities onto the collection of all the valued set systems (S, f) that satisfy two conditions. One of these conditions was introduced by Batbedat (1988) and is related to the characterization of 2-conformity in hypergraphs (see also Bandelt and Dress (1994)). We introduce the other condition which requires the index f to be a weak index. Our approach includes a characterization of the valued set systems (S, f) such that S contains (respectively is contained in) the set of all finite nonempty intersections of maximal linked subsets with respect to the pseudo-dissimilarity induced by the pair (S, f).