Fuzzy set representations of some quantum structures

Fuzzy Sets and Systems

Volumn 101, Issue 1, 1999, Pages 67-78

  Dvurečenskij, Anatolij ^a  

^a INSTITUTE OF CHEMISTRY   (Slovakia)

Author keywords

Effect algebra; Fuzzy set; Lukasiewicz t norm; MV algebras; Order determining system of states; Orthoalgebra; Orthomodular poset; State

Indexed keywords

EID: 0002257843     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0165-0114(97)00050-X     Document Type: Article

References (23)

1. L.P. Belluce, Semisimple algebras of infinite valued logic and Bold fuzzy set theory, Canad. J. Math. 38 (1986) 1356-1379.
2. G. Birkhoff, J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 (1936) 823-834.
3. C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958) 467-490.
4. D.W. Cohen, An Introduction to Hubert Space and Quantum Logic, Springer, New York, Berlin, Heidelberg, London, Paris, Tokyo, 1989.
5. A. Dvurečenskij, Gleason's Theorem and its Applications, Kluwer Academic Publ., Dordrecht, Boston, London, Ister Science Press, Bratislava, 1993.
6. D.J. Foulis, M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 31 (1994) 21-34.
7. D.J. Foulis, C.H. Randall, Operational statistics. I. Basic concepts, J. Math. Phys. 13 (1972) 1667-1675.
8. R. Giles, Lukasiewicz logic and fuzzy set theory, Internat. J. Man-Machine Stud. 8 (1976) 313-327.
9. F. Kôpka, D-posets of fuzzy sets, Tatra Mountains Math. Publ. 1 (1992) 83-97.
10. F. Kôpka, F. Chovanec, D-posets, Math. Slovaca 44 (1994) 21-34.
11. G.W. Mackey, The Mathematical Foundations of Quantum Mechanics, W.A. Benjamin Inc., Reading, 1963.
12. M.J. Ma̧czyński, On a functional representation of lattice of projections on a Hilbert space, Studia Math. 47 (1973) 253-259.
13. M.J. Ma̧czyński, Orthogonality postulate in axiomatic quantum mechanics, Internat. J. Theor. Phys. 8 (1973) 353-360.
14. M.J. Ma̧czyński, Functional properties of quantum logics, Internat. J. Theor. Phys. 11 (1974) 149-156.
15. M.J. Ma̧czyński, T. Traczyk, A characterization of orthomodular partially ordered sets admitting a full set of states, Bull. Acad. Polon. Sci. Sèr, Sci. Math. Astr. Phys. 21 (1973) 3-8.
16. R. Mesiar, h-fuzzy quantum logics, Internat. J. Theor. Phys. 33 (1994) 1417-1425.
17. R. Mesiar, Do fuzzy quantum structures exist?, Internat. J. Theor. Phys. 34 (1995) 1609-1614.
18. D. Mundici, Interpretation of AF C*-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986) 15-63.
19. P. Pták, S. Pulmannová, Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht, Boston, London, 1991.
20. J. Pykacz, Fuzzy set ideas in quantum logics, Internat. J. Theor. Phys. 31 (1992) 1767-1783.
21. J. Pykacz, Fuzzy quantum logics and infinite-valued Lukasiewicz logic, Internat. J. Theor. Phys. 33 (1994) 1403-1416.
22. E. Trillas, Sobre functiones de negacion en la teoria de conjuntos difusos, Stochastica 3 (1979) 47-60.
23. V.S. Varadarajan, Geometry of Quantum Theory, vol. 1, van Nostrand, Princeton, NJ, 1968.