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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 $${\mathcal G}{\mathcal F}(X)$$ of X, a complete Heyting algebra (cHa) which contains the power set $${\mathcal P}(X)$$ of X; (2) an extension lattice $$\overline{{\mathcal B}(L)}$$, where $${\mathcal B}={\mathcal P}(X)$$; and (3) the set of $${\mathbb{L}}$$-fuzzy sets, where $${\mathbb{L}}=[L_ x|$$ $$x\in X]$$. It is shown that these three notions are equivalent. The mathematical structure of $${\mathcal G}{\mathcal F}(X)$$ is studied, and a ring of generalized fuzzy sets of type 2 is introduced.

##### MSC:
 06D20 Heyting algebras (lattice-theoretic aspects) 03E99 Set theory 03E72 Theory of fuzzy sets, etc.
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##### References:
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