Pyramids and weak hierarchies in the ordinal model for clustering

There are several well known bijections between classes of dissimilarity coefficients and structures such as indexed or weakly indexed pyramids, as well as indexed closed weak hierarchies. Our goal will be to approach these results from the viewpoint developed by Jardine and Sibson (Mathematical Taxonomy, Wiley, New York, 1971). Properties of dissimilarity coefficients will be related to properties of the maximal linked subsets defined by the family of relations associated with the underlying dissimilarity coefficient. Our approach also involves a close study of the inclusion and diameter conditions introduced by Diatta and Fichet (in: E. Diday et al. (Eds.), New Approaches in Classification and Data Analysis, Springer, Berlin, 1994, p. 111). Typical results include showing that the diameter condition is equivalent to a weakening of the Bandelt four-point characterization that appears in Bandelt (Mathematisches Seminar, Universitat Hamburg, Germany, 1992) as well as Bandelt and Dress (Discrete Math. 136 (1994) 21), and this in turn is equivalent to the maximal linked subsets being closed under nonempty intersections; the inclusion condition is equivalent to the 2-balls coinciding with the weak clusters; the Bandelt four-point characterization is equivalent to the maximal linked subsets coinciding with the weak clusters; and a Robinsonian dissimilarity coefficient is strongly Robinsonian (in the sense of Fichet (in: Y.A. Prohorov, V.V. Sazonov (Eds.), Proceedings of the First World Congress of the BERNOULLI SOCIETY, Tachkent, 1986, V.N.U. Science Press, Vol. 2, 1987, p. 123)) if and only if it satisfies the inclusion condition.