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Algebraic properties of fuzzy sets. (English) Zbl 0211.31002
Summary: Some new algebraic properties of the class of the “fuzzy sets” \(\mathcal F(I)\) are stressed; in particular it is pointed out that the class of the generalized characteristic functions furnished with the lattice operations proposed by Zadeh is a Brouwerian lattice. The possibility of inducing other different lattice operations to the whole class \(\mathcal F(I)\) or to a suitable subclass of it is considered. The problem of the relationship between “fuzzy sets” and classical set theory is finally remarked. A qualitative comparison with similar situations appearing in the axiomatic formulation of quantum mechanics and in the classical theory of probability is made.

03E72 Theory of fuzzy sets, etc.
94D05 Fuzzy sets and logic (in connection with information, communication, or circuits theory)
Full Text: DOI
[1] Zadeh, L. A.: Fuzzy sets. Information and control 8, 338 (1965) · Zbl 0139.24606
[2] Goguen, J. A.: L-fuzzy sets. J. math. Anal. appl. 18, 145 (1967) · Zbl 0145.24404
[3] Birkhoff, G.; Von Neumann, J.: The logic of quantum mechanics. Ann. of math. 37, 823 (1936) · Zbl 0015.14603
[4] Piron, C.: Axiomatique quantique. Helv. phys. Acta 37, 439 (1964) · Zbl 0141.23204
[5] Watanabe, S.: Modified concepts of logic, probability and information based on generalized continuous characteristic functions. Information and control 15, 1 (1969) · Zbl 0205.00801
[6] De Luca, A.; Termini, S.: Algorithmic aspects in the analysis of complex systems. Scientia 106, 659 (1972) · Zbl 0239.94028
[7] De Luca, A.; Termini, S.: A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Information and control (1972) · Zbl 0239.94028
[8] Varadarajan, V. S.: Probability in physics and a theorem on simultaneous observability. Comm. pure appl. Math. 15, 189 (1962) · Zbl 0109.44705
[9] Finch, P. D.: On the structure of quantum logic. J. symbolic logic 34, 275 (1969) · Zbl 0205.00803
[10] Finch, P. D.: Quantum mechanical physical quantities as random variables. Preprint monash university (1969) · Zbl 0349.02025
[11] Mckinsey, J. C. C; Tarski, A.: On closed elements in closure algebras. Ann. of math. 47, 122 (1946) · Zbl 0060.06207
[12] Birkhoff, G.: Lattice theory. American mathematical society colloquium publications (1967) · Zbl 0126.03801
[13] Jauch, J. M.; Piron, C.: On the structure of quantal proposition systems. Helv. phys. Acta 42, 842 (1969) · Zbl 0181.27603
[14] Heyting, A.: Intuitionism. (1966) · Zbl 0229.02008
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