On the mutual definability of fuzzy tolerance relations and fuzzy tolerance coverings

Studies the mathematical foundations of cluster analysis, i.e. the correspondences between binary relations (“similarity relations”) and systems of sets (“clusters”) with respect to a fixed universe. For a long time, from crisp set theory and the classical (crisp) theory of universal algebras, such correspondences have been well-known as bijections and lattice isomorphisms between the class of equivalence relations on a universe 𝒰 and the class of partitions of 𝒰. In the middle of the 1960s, there began the study of tolerance relations, i.e. binary relations where, in contrast to equivalence relations, only reflexivity and symmetry are assumed. It was proved that there exists a bijection between the class of tolerance relations on a universe U and a class of special coverings of 𝒰. Schmechel (1995) generalized the classical result about crisp equivalence relations and crisp partitions to the “fuzzy case”, i.e. to several classes of fuzzy equivalence relations and corresponding classes of fuzzy partitions. This paper contains a generalization of the result on crisp tolerance relations and crisp coverings to fuzzy tolerance relations and special sets of fuzzy clusters