Correspondences between Pre-pyramids, Pyramids and Robinsonian Dissimilarities

7 pages; International audience; We consider cluster structures in a general setting where they do not necessarily contain all singletons of the ground set. Then we provide a direct proof of the bijection between semi-proper robinsonian dissimilarities and indexed pre-pyramids. This result generalizes its analogue proven by Batbedat in the particular case of definite cluster structures. Moreover, the proposed proof shows that the clusters of the indexed pre-pyramid corresponding to a semi-proper robinsonian dissimilarity are particular 2-balls of the considered dissimilarity.