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Zbl 0676.06017
Nakajima, Nobuyuki
Generalized fuzzy sets. (English)
[J] Fuzzy Sets Syst. 32, No.3, 307-314 (1989). ISSN 0165-0114

Summary: This paper is concerned with a construction of fuzzy sets not depending on a membership function. Algebraic properties of a family of fuzzy sets are investigated. In this paper, three notions are proposed: (1) a ring of generalized fuzzy sets ${\cal G}{\cal F}(X)$ of X, a complete Heyting algebra (cHa) which contains the power set ${\cal P}(X)$ of X; (2) an extension lattice $\overline{{\cal B}(L)}$, where ${\cal B}={\cal P}(X)$; and (3) the set of ${\bbfL}$-fuzzy sets, where ${\bbfL}=[L\sb x\vert$ $x\in X]$. It is shown that these three notions are equivalent. The mathematical structure of ${\cal G}{\cal F}(X)$ is studied, and a ring of generalized fuzzy sets of type 2 is introduced.

MSC 2000:: *06D20 Heyting algebras
03E99 Set theory (logic)
03E72 Fuzzy sets (logic)

Keywords: ring of generalized fuzzy sets; complete Heyting algebra; extension lattice; ${\bbfL}$-fuzzy sets

Cited in: Zbl 1092.03028

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