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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 $\text{GF}\left(X\right)$ of $X$, a complete Heyting algebra (cHa) which contains the power set $P\left(X\right)$ of $X$, an extension lattice $\overline{B\left(L\right)}$ where $B=P\left(X\right)$, and the set of $𝐋$-fuzzy sets where $𝐋=\left\{Lx\mid x\in X\right\}$ 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 Fuzzy set theory 06D20 Heyting algebras