On isomorphisms between the De Morgan algebra of fuzzy tolerance relations and De Morgan algebras of fuzzy clusterings

摘要

By "mathematical foundations of cluster analysis" we understand the investigation of (one-to-one) correspondences between "similarity relations" on a universe U and "clusterings" on U. The paper presented is a continuation of a series of the author's publications concerning this field. In particular, in this paper we have shown that a bijection existing between the set of all fuzzy tolerance relations on U and the set of all clusterings and proper clusterings on U, respectively, is an isomorphism between certain De Morgan algebras constructed on the sets mentioned above.