On isomorphisms between the lattice of tolerance relations and lattices of clusterings

Abstract

By "mathematical foundations of cluster analysis" we understand the study of (one-to-one) correspondences between "similarity relations" on a given universe U and "clusterings" an U. From classical set theory such correspondences are well-known as bijections and lattice isomorphisms between the set and the lattice of all equivalence relations on a universe U and the set and the lattice of all partitions of U, respectively. In this paper we show that the lattice (even the complete atomistic boolean algebra) of all (crisp!) tolerance relations on U is isomorphic to the lattice (even the complete atomistic boolean algebra) of all subset closed strongly model-compact coverings of U, on the one hand, and to the lattice (even the complete atomistic boolean algebra) of all tolerance coverings, on the other hand.