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A characterization of arbitrary Ruspini partitions by fuzzy similarity relations

One of the fundamental problems in clustering theory is the mutual definability of clusterings and binary relations establishing bijections between certain classes of clusterings and certain classes of binary relations. Whereas a fuzzy clustering can be generated by a binary fuzzy relation in a standard procedure, the definition of a suitable binary fuzzy relation from a given clustering holds some problems. After describing some different approaches for solving this problem we introduce a new concept in order to construct a binary fuzzy relation starting with a given clustering. This concept, called relation generating function, can be derived from the concepts of a-disjointness and c-covering for clusterings and gives the possibility to define bijections between arbitrary Ruspini (1969) partitions and certain binary fuzzy relations.