Cluster structures and collections of Galois closed entity subsets

We discuss relations between cluster structures and so-called cluster prestructures. On the other hand, we place ourselves in the framework of a context where entity descriptions belong to a complete meet-semilattice. Such a context induces a Galois correspondence which, in turn, induces a closure operator on the powerset of the entity set. We give a necessary and sufficient condition for a particular collection of fixed points of this closure operator to be hierarchical. Moreover, we specify the collection of all entity subsets which are both fixed points of this closure operator and strong clusters associated with a given pairwise dissimilarity function, as well as that of all entity subsets which are both fixed points of this closure operator and weak clusters associated with a given kk-way dissimilarity function.