HansJürgen Bandelt and
Andreas W. M. Dress. An order theoretic framework for overlapping clustering. In DM, Vol. 136(13):2137, 1994.
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"Cluster analysis deals with procedures which  given a finite collection X of objects together with some kind of local dissimilarity information  identify those subcollections C of objects from X, called clusters, which exhibit a comparatively low degree of internal dissimilarity. In this note we study arbitrary mappings φ which assign to each subcollection A ⊆ X of objects its internal degree of dissimilarity φ (A), subject only to the natural condition that A ⊆ B ⊆ X implies φ (A) ̌ φ (B), and we analyse on a rather abstract, purely order theoretic level how assumptions concerning the way such a mapping φ might be constructed from local data (that is, data involving only a few objects at a time) influence the degree of overlapping observed within the resulting family of clusters,  and vice versa. Hence, unlike previous order theoretic approaches to cluster analysis, we do not restrict our attention to nonoverlapping, hierarchical clustering. Instead, we regard a dissimilarity function φ as an arbitrary isotone mapping from a finite partially ordered set I  e.g. the set P(X) of all subsets A of a finite set X  into a (partially) ordered set R  e.g. the nonnegative real numbers  and we study the correspondence between the two subsets C(φ) and D(φ) of I, formed by the elements whose images are inaccessible from above and from below, respectively. While D(φ) constitutes the local data structure from which φ can be built up, C(φ) embodies the family of clusters associated with φ. Our results imply that in case I: = P(X) and R: = R≥0 one has # D ̌ n for all Dε{lunate}D(φ) and some fixed nε{lunate}N if and only if{A figure is presented} for all C0,..., Cnε{lunate}C(φ) if and only if this holds for all subsets C0,..., Cn ⊆ X, generalizing a wellknown criterion for nconformity of hypergraphs as well as corresponding results due to Batbedat, dealing with the case n = 2. © 1994."
