On some chains of fuzzy sets |
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Authors: | Antonio Di Nola Aldo GS Ventre |
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Institution: | Istituto Matematico, Facoltà di Architettura. Università di Napoli. Via Monteoliveto, 3 80134 Naples, Italy |
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Abstract: | A partial order relation σ is defined in the set (X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on (X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval 0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure. |
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Keywords: | Measure space Entropy Sharpened version Chain Valuation on a lattice |
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