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Generalized fuzzy sets. (English) Zbl 0676.06017
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:
 06D20 Heyting algebras 03E99 Set theory (logic) 03E72 Fuzzy set theory
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##### References:
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