Several Remarks on Dissimilarities and Ultrametrics

We investigate the relationships between tolerance relations, equivalence relations, and ultrametrics. The set of spheres associated to an ultrametric space has a tree structure that reects a hierarchy on the set of equivalences associated to that space. We show that every ultrametric dened on a nite space is a linear combination of binary ultrametric and we introduce the notion of ultrametricity for dissimilarities, which has applications in many data mining problems.