Galois closed entity sets and k-balls of quasi-ultrametric multi-way dissimilarities

Quasi-ultrametric multi-way dissimilarities and their respective sets of k-balls extend the fundamental bijection in classification between ultrametric pairwise dissimilarities and indexed hierarchies. We show that nonempty Galois closed subsets of a finite entity set coincide with k-balls of some quasi-ultrametric multi-way dissimilarity. This result relates the order theoretic Galois connection based clustering approach to the dissimilarity based one. Moreover, it provides an effective way to specify easy-to-interpret cluster systems, from complex data sets, as well as to derive informative attribute implications.