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Generalized fuzzy sets. (English) Zbl 1092.03028
Summary: Our aim is to generalize Lemma 2 and Lemma 3 of N. Nakajima’s article “Generalized fyzzy sets” [Fuzzy Sets Syst. 32, 307–314 (1989; Zbl 0676.06017)]. In this article, a construction of fuzzy sets without depending on a membership function, algebraic properties of a family of fuzzy sets, a ring of generalized fuzzy sets $$\text{GF}(X)$$ of $$X$$, a complete Heyting algebra (cHa) which contains the power set $$P(X)$$ of $$X$$, an extension lattice $$\overline {B(L)}$$ where $$B=P(X)$$, and the set of $${\mathbf L}$$-fuzzy sets where $${\mathbf L}=\{Lx\mid x\in X\}$$ were proposed and shown that they are equivalent depending on Lemma 2 and Lemma 3. In his article Lemma 2, Lemma 3 were not shown in their generalized version. In our article we show that these lemmas can be generalized.

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