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Abstract
Let B be a collection of fuzzy sets. What are the fuzzy sets which are sufficiently similar to every fuzzy set from B, i.e. ‘central’ fuzzy sets for B? Such a question naturally arises if B is large and one wishes to replace B by a single fuzzy set – the representative of B. In this paper, we develop a framework which enables us to answer this question and related ones. We use complete residuated lattices as the structures of truth degrees, covering thus the real unit interval with left-continuous t-norm and its residuum as an important but particular case. We present results describing central fuzzy sets and optimal central fuzzy sets, provided similarity of fuzzy sets is assessed by Leibniz rule.
Acknowledgements
R. Belohlavek acknowledges support by project MSM 6198959214. This paper is based on ‘Factor Structures and Central Points by Similarity’, by R. Belohlavek and M. Krupka, which appeared in the Proceedings of the 4th International IEEE Conference on Intelligent Systems IS'08, pp. 15-8–15-11.