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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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