Fuzzy partitions and relations; an axiomatic basis for clustering

In this paper some connections between fuzzy partitions and similarity relations are explored. A new definition of transitivity for fuzzy relations yields a relation-theoretic characterization of the class of all psuedo-metrics on a fixed (finite) data set into the closed unit interval. This notion of transitivity also links the triangle inequality to convex decompositions of fuzzy similarity relations in a manner which may generate new techniques for fuzzy clustering. Finally, we show that every fuzzy c-partition of a finite data set induces a psuedo-metric of the type described above on the data.