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Measure in generalized fuzzy sets. (English) Zbl 1120.03312
Far East J. Math. Sci. (FJMS) 24, No. 2, 217-226 (2007).
Summary: In this article, using a construction of fuzzy sets without depending on a membership function, algebraic properties of a family of fuzzy sets, three notions, including rings of generalized fuzzy sets $$GF(X)$$ of $$X$$, complete Heyting algebras (cHa) which contain the power set $$P(X)$$ of $$X$$, extension lattices $$\overline{B(L)}$$, where $$B=P(X)$$, and sets of $$L$$-fuzzy sets, where $$L=\{L_x\mid x\in X\}$$, definitions of fuzzy $$\sigma$$-algebra and fuzzy measure are generalized. We obtain some results using these definitions, which include a generalization of Proposition 2 in [E. P. Klement and W. Schwyhla, Fuzzy Sets Syst. 7, 57–70 (1982; Zbl 0478.28006)].

##### MSC:
 03E72 Theory of fuzzy sets, etc. 28E10 Fuzzy measure theory 06D20 Heyting algebras (lattice-theoretic aspects)