Grouping fuzzy sets by similarity

Authors:
Radim Belohlavek;Michal Krupka
Affiliations:
State University of New York at Binghamton, Department of Systems Science and Industrial Engineering, Binghamton, NY 13902, USA and Palacký University, Tomkova 40, CZ-779 00 Olomouc, Czech Re ...;Palacký University, Tomkova 40, CZ-779 00 Olomouc, Czech Republic
Venue:
Information Sciences: an International Journal
Year:
2009

Citing 4
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Abstract

The paper presents results on factorization of systems of fuzzy sets. The factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a. An obstacle to such factorization, well known in fuzzy set theory, is the fact that ''being similar at least to degree a'' is not an equivalence relation because, in general, it is not transitive. As a result, ordinary factorization using equivalence classes cannot be used. This obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a. We show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice. This lattice plays the role of a factor structure for the original system of fuzzy sets. Particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices.