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Fuzzy measures and the entropy of fuzzy partitions. (English) Zbl 0782.28012
The author defines an entropy of a fuzzy partition and proves some basic properties of it. His aim is to get such properties which are very similar to analogous properties for the entropy of probability measures. Therefore, the consideration must be restricted to bounded-sum-union and bounded-product-intersection of fuzzy sets (only in this case a fuzzy partition consists of disjoint fuzzy sets) and to the Butnariu-fuzzy- measure [which is additive for disjoint fuzzy sets, see D. Butnariu, J. Math. Anal. Appl. 93, 436-452 (1983; Zbl 0516.28006)]. A somewhat artificial definition of a product between fuzzy sets (the used product of membership functions is nothing more than another intersection operator generated by the algebraic-product-\(t\)-norm) allows the introduction of a conditional Butnariu-fuzzy-measure and a conditional entropy. Refinements of partitions lead to monotonicity properties of the entropy.

28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.
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