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Tensor product of difference posets


Author: Anatolij Dvurečenskij
Journal: Trans. Amer. Math. Soc. 347 (1995), 1043-1057
MSC: Primary 03G12; Secondary 06C15, 81P10
DOI: https://doi.org/10.1090/S0002-9947-1995-1249874-8
MathSciNet review: 1249874
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Abstract: A tensor product of difference posets, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. In particular, we show that a tensor product for difference posets with a sufficient system of probability measures exists, as well as a tensor product of any difference poset and any Boolean algebra, which is isomorphic to a bounded Boolean power.


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  • [1] D. Aerts and I. Daubechies, Physical justification for using tensor product to describe quantum systems as one joint system, Helv. Phys. Acta 51 (1978), 661-675. MR 542798 (81e:81008)
  • [2] P. Busch, P.J. Lahti, and P. Mittelstaedt, The quantum theory of measurement, Lecture Notes in Phys.,, Springer-Verlag, Berlin, Heidelberg, New York, London, and Budapest, 1991. MR 1176754 (93m:81014)
  • [3] G. Cattaneo and G. Nisticò, Brower--Zadeh posets and three-valued Lukasiewicz posets, Fuzzy Sets and Systems 33 (1989), 165-190. MR 1024221 (91b:03105)
  • [4] A. Dvurečenskij and S. Pulmannová, Difference posets, effects, and quantum measurements, Interat. J. Theoret. Phys. 33 (1994), 819-850. MR 1286161 (95k:81008)
  • [5] A. Dvurečenskij and B. Riečan, Decomposition of measures on orthoalgebras and difference posets, Internat. J. Theoret. Phys. 33 (1994), 1403-1418.
  • [6] D. Foulis, Coupled physical systems, Found. Phys. 19 (1989), 905-922. MR 1013911 (90i:81021)
  • [7] D.J. Foulis and M.K. Bennett, Tensor products of orthoalgebras, Order 10 (1993), 271-282. MR 1267193 (95a:81020)
  • [8] D.J. Foulis, R.J. Greechie, and G.T. Rüttimann, Filters and supports in orthoalgebras, Internat. J. Theoret. Phys. 31 (1992), 787-807. MR 1162623 (93c:06014)
  • [9] D. Foulis and P. Pták, On the tensor product of a Boolean algebra and an orthoalgebra, preprint 1993.
  • [10] D. Foulis and C. Randall, Empirical logic and tensor products, Interpretations and Foundations of Quantum Theories (A. Neumann, ed.), Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1981, pp. 9-20. MR 683888
  • [11] G. Kalmbach, Orthomodular lattices, Academic Press, London and New York, 1983. MR 716496 (85f:06012)
  • [12] M. Kläy, C. Randall and D. Foulis, Tensor products and probability weights, Internat. J. Theoret. Phys. 26 (1987), 199-219. MR 892411 (88h:81008)
  • [13] F. Kôpka, $ D$-posets of fuzzy sets, Tatra Mountains Math. Publ. 1 (1992), 83-87. MR 1230466 (94e:04008)
  • [14] F. Kôpka and F. Chovanec, $ D$-posets, Math. Slovaca 44 (1994), 21-34. MR 1290269 (95i:03134)
  • [15] R. Lock, The tensor product of generalized sample spaces which admit a unital set of dispersion--free weights, Found. Phys. 20 (1990), 477-498. MR 1060619 (91d:81008)
  • [16] T. Matolcsi, Tensor product of Hilbert lattices and free orthodistributive product of orthomodular lattices, Acta Sci. Math. 37 (1975), 263-272. MR 0388122 (52:8959)
  • [17] M. Navara and P. Pták, Difference posets and orthoalgebras, submitted.
  • [18] P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, Kluwer, Dordrecht, Boston and London, 1991.
  • [19] S. Pulmannová, Tensor product of quantum logics, J. Math. Phys. 26 (1985), 1-5. MR 776118 (86h:03111)
  • [20] C. Randall and D. Foulis, Empirical statistics and tensor products, Interpretations and Foundations of Quantum Theory (H. Neumann, ed.), Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1981, pp. 21-28. MR 683889
  • [21] R. Sikorski, Boolean algebras, Springer-Verlag, Berlin, Heidelberg and New York, 1964.
  • [22] A. Wilce, Tensor product of frame manuals, Internat. J. Theoret. Phys. 29 (1990), 805-814. MR 1070453 (91g:81006)
  • [23] A. Zecca, On the coupling of quantum logics, J. Math. Phys. 19 (1978), 1482-1485. MR 0503035 (58:19896)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1249874-8
Keywords: Difference poset, orthomodular poset, orthoalgebra, tensor product, bimorphism, probability measure, Boolean power, effects
Article copyright: © Copyright 1995 American Mathematical Society

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