Generalized fuzzy sets.

*(English)*Zbl 0676.06017Summary: 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. |

##### Keywords:

ring of generalized fuzzy sets; complete Heyting algebra; extension lattice; \({\mathbb{L}}\)-fuzzy sets
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\textit{N. Nakajima}, Fuzzy Sets Syst. 32, No. 3, 307--314 (1989; Zbl 0676.06017)

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##### References:

[1] | Birkhoff, G, Lattice theory, (1967), AMS Colloquim Publications Providence, RI · Zbl 0126.03801 |

[2] | Goguen, J.A, L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404 |

[3] | Iwamura, T, Sokorun, (1966), Kyoritsu Shuppan Tokyo |

[4] | McGoveran, D, Fuzzy logic and non-distributive truth valuations, (), 49-57 · Zbl 0593.03013 |

[5] | Nakajima, N, On determination of membership functions, (), 154-159, Preprints of |

[6] | Norwich, A.M; Turksen, I.B, A model for the measurement of membership and the consequences of its empirical implication, Fuzzy sets and systems, 12, 1-25, (1984) · Zbl 0538.94026 |

[7] | Takeuti, G, Senkeidaisu to ryosirikigaku, (1981), Shokabo Tokyo |

[8] | Ryumae, S; Morita, Y; Oka, Y, A measurement model for membership functional representation, (), 254-257, Preprints of |

[9] | Watanabe, S, Modified concepts of logic, probability, and information based on generalized continuous characteristic function, Inform. and control, 15, 1-21, (1969) · Zbl 0205.00801 |

[10] | Zadeh, L.A, Fuzzy sets, Inform. and control., 8, 338-353, (1965) · Zbl 0139.24606 |

[11] | Zadeh, L.A, Calculus of fuzzy restrictions, (), 1-39 · Zbl 0327.02018 |

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