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Measure in generalized fuzzy sets. (English) Zbl 1120.03312
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 [{\it E. P. Klement} and {\it W. Schwyhla}, Fuzzy Sets Syst. 7, 57--70 (1982; Zbl 0478.28006)].

03E72Fuzzy set theory
28E10Fuzzy measure theory
06D20Heyting algebras