Abstract
Generalized Boolean quasirings (GBQRs) are extensions of partial algebras thatare in one-to-one correspondence to bounded lattices with an involutoryantiautomorphism. This correspondence generalizes the bijection betweenBoolean rings and Boolean algebras and provides for a large variety of presumptivequantum logics (including logics which can be defined by means of Mackey'sprobability function). It is shown how properties of the corresponding latticesare reflected in GBQRs and what the implications are of the associativity of the+-operation of GBQRs, which can be interpreted as some kind of an “exclusiveor”-operation. We prove that under very weak conditions, which, however, seemto be essential for experimental verifications, the associativity of + implies theclassicality of the considered quantum mechanical system.
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REFERENCES
L. Beran, Orthomodular Lattices, Reidel, Dordrecht, 1985.
I. Chajda, Pseudosemirings induced by ortholattices, Czechosl. Math. J. 46 (1996), 405–411.
G. Dorfer, Kongruenzen und symmetrische Differenzen auf orthomodularen Verbänden, Diss. TU Wien, 1998.
D. Dorninger, Sublogics of ring-like quantum logics, Tatra Mt. Math. Publ. 15 (1998), 75–83.
D. Dorninger, H. Länger, and M. Maczyński, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstr. Math. 30 (1997), 215–232.
D. Dorninger, H. Länger, and M. Maczyński, On ring-like structures occurring in axiomatic quantum mechanics, Österr. Akad. Wiss. Sitzungsber. Math.-Naturw. Kl. Abt. II 206 (1997), 279–289.
D. Dorninger, H. Länger, and M. Maczyński, On ring-like structures induced by Mackey's probability function, Rep. Math. Phys. 43 (1999), 499–515.
C. Luo and X. Ma, On Stone theorem for de Morgan algebra, J. Fuzzy Math. 5 (1997), 543–551.
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Dorninger, D., Länger, H. & Maczyński, M. Lattice Properties of Ring-like Quantum Logics. International Journal of Theoretical Physics 39, 1015–1026 (2000). https://doi.org/10.1023/A:1003646323230
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DOI: https://doi.org/10.1023/A:1003646323230