Fuzzy sets and fuzzy logic: the foundations of application—from a mathematical point of view
Fuzzy sets and fuzzy logic: the foundations of application—from a mathematical point of view
Fuzzy set theory—and its applications (3rd ed.)
Fuzzy set theory—and its applications (3rd ed.)
Pattern Recognition with Fuzzy Objective Function Algorithms
Pattern Recognition with Fuzzy Objective Function Algorithms
On Lattice-Isomorphism Between Fuzzy Equivalence Relations and Fuzzy Partitions
ISMVL '95 Proceedings of the 25th International Symposium on Multiple-Valued Logic
Similarity relations and fuzzy orderings
Information Sciences: an International Journal
Fuzzy Sets and Systems - Theme: Basic notions
On algebraic foundations of information granulation
Technologies for constructing intelligent systems
Information Sciences: an International Journal
On Lattice-Isomorphism Between Fuzzy Equivalence Relations and Fuzzy Partitions
ISMVL '95 Proceedings of the 25th International Symposium on Multiple-Valued Logic
Fuzzy equivalence relations and their equivalence classes
Fuzzy Sets and Systems
Uniform fuzzy relations and fuzzy functions
Fuzzy Sets and Systems
On local entropy of fuzzy partitions
Fuzzy Sets and Systems
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Abstract: 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 /spl Uscr/ and the class of partitions of /spl Uscr/. 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 /spl Uscr/. 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.