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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.

06D20Heyting algebras
03E99Set theory (logic)
03E72Fuzzy set theory
Full Text: DOI
[1] Birkhoff, G.: Lattice theory. (1967) · Zbl 0153.02501
[2] Goguen, J. A.: L-fuzzy sets. J. math. Anal. appl. 18, 145-174 (1967) · Zbl 0145.24404
[3] Iwamura, T.: Sokorun. (1966)
[4] Mcgoveran, D.: Fuzzy logic and non-distributive truth valuations. Fuzzy sets, 49-57 (1980) · Zbl 0593.03013
[5] Nakajima, N.: On determination of membership functions. Second fuzzy system symposium, 154-159 (1986)
[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)
[8] Ryumae, S.; Morita, Y.; Oka, Y.: A measurement model for membership functional representation. Second IFSA congress, 254-257 (1987)
[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. Fuzzy sets and their applications to cognitive and decision processes, 1-39 (1975)