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Characterization of Galois closed sets using multiway dissimilarities

We place ourselves in a so-called meet-closed description context; that is a context consisting of a finite nonempty entity set E whose elements are described in a complete meet-semilattice D, by means of a descriptor δ. Then we consider multiway quasi-ultrametric dissimilarities on E, a class of multiway dissimilarities that, with their relative k-balls, extend the fundamental in classification bijection between ultrametric dissimilarities and indexed hierarchies. We also consider multiway dissimilarities agreeing with entity descriptions in a quite natural sense called δ-meet compatibility. It turns out that there exists an integer k such that any strictly δ-meet compatible k-way dissimilarity is quasi-ultrametric. On the other hand, the descriptor δ induces a Galois connection between the powerset P(E) and D, which, in turn, induces a closure operator, say φδ, on P(E). then it is proved that nonempty φδ-closed subsets of E coincide with k-balls relative to some strictly δ-meet compatible multiway dissimilarities on E.