Min-indistinguishability Operators and Hierarchical Trees

Abstract

min -indistinguishability operators are widely used in Taxonomy because they are closely related to hierarchical trees. Indeed, given a min -indistinguishability operator on a set X and α ∈ [0,1], the α-cuts of E are partitions of X and if α ≥ β, then the α-cut is a refinement of the partition corresponding to the β-cut. Therefore, E generates an indexed hierarchical tree. Reciprocally, from an indexed hierarchical tree a min -indistinguishability operator can be generated. These results follow from the fact that 1–E is a pseudo ultrametric. Pseudo ultrametrics are pseudodistances where, in the triangular inequality, the addition is replaced by the more restrictive max operation. The topologies generated by ultrametrics are very peculiar, since if two balls are non disjoint, then one of them is included in the other one.