Tensor product of difference posets

Dvurečenskij, Anatolij

doi:10.1090/S0002-9947-1995-1249874-8

Tensor product of difference posets
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Trans. Amer. Math. Soc. 347 (1995), 1043-1057 Request permission

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.

References

Dirk Aerts and Ingrid Daubechies, Physical justification for using the tensor product to describe two quantum systems as one joint system, Helv. Phys. Acta 51 (1978), no. 5-6, 661–675 (1979). MR 542798
Paul Busch, Pekka J. Lahti, and Peter Mittelstaedt, The quantum theory of measurement, Lecture Notes in Physics. New Series m: Monographs, vol. 2, Springer-Verlag, Berlin, 1991. MR 1176754, DOI 10.1007/978-3-662-13844-1
Gianpiero Cattaneo and Giuseppe Nisticò, Brouwer-Zadeh posets and three-valued Łukasiewicz posets, Fuzzy Sets and Systems 33 (1989), no. 2, 165–190. MR 1024221, DOI 10.1016/0165-0114(89)90239-X
Anatolij Dvurečenskij and Sylvia Pulmannová, Difference posets, effects, and quantum measurements, Internat. J. Theoret. Phys. 33 (1994), no. 4, 819–850. MR 1286161, DOI 10.1007/BF00672820

Decomposition of measures on orthoalgebras and difference posets

33

David J. Foulis, Coupled physical systems, Found. Phys. 19 (1989), no. 7, 905–922. MR 1013911, DOI 10.1007/BF01889305
D. J. Foulis and M. K. Bennett, Tensor products of orthoalgebras, Order 10 (1993), no. 3, 271–282. MR 1267193, DOI 10.1007/BF01110548
D. J. Foulis, R. J. Greechie, and G. T. Rüttimann, Filters and supports in orthoalgebras, Internat. J. Theoret. Phys. 31 (1992), no. 5, 789–807. MR 1162623, DOI 10.1007/BF00678545

On the tensor product of a Boolean algebra and an orthoalgebra

D. J. Foulis and C. H. Randall, Empirical logic and tensor products, Interpretations and foundations of quantum theory (Marburg, 1979) Grundlagen Exakt. Naturwiss., vol. 5, Bibliographisches Inst., Mannheim, 1981, pp. 9–20. MR 683888
Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496
M. Kläy, C. Randall, and D. Foulis, Tensor products and probability weights, Internat. J. Theoret. Phys. 26 (1987), no. 3, 199–219. MR 892411, DOI 10.1007/BF00668911
František Kôpka, $D$-posets of fuzzy sets, Tatra Mt. Math. Publ. 1 (1992), 83–87. Fuzzy sets (Liptovský Mikuláš, 1992). MR 1230466
František Kôpka and Ferdinand Chovanec, $D$-posets, Math. Slovaca 44 (1994), no. 1, 21–34. MR 1290269
Robin H. Lock, The tensor product of generalized sample spaces which admit a unital set of dispersion-free weights, Found. Phys. 20 (1990), no. 5, 477–498. MR 1060619, DOI 10.1007/BF01883236
Tamás Matolcsi, Tensor product of Hilbert lattices and free orthodistributive product of orthomodular lattices, Acta Sci. Math. (Szeged) 37 (1975), no. 3-4, 263–272. MR 388122

Difference posets and orthoalgebras

Orthomodular structures as quantum logics

Sylvia Pulmannová, Tensor product of quantum logics, J. Math. Phys. 26 (1985), no. 1, 1–5. MR 776118, DOI 10.1063/1.526784
C. H. Randall and D. J. Foulis, Operational statistics and tensor products, Interpretations and foundations of quantum theory (Marburg, 1979) Grundlagen Exakt. Naturwiss., vol. 5, Bibliographisches Inst., Mannheim, 1981, pp. 21–28. MR 683889

Boolean algebras

Alexander Wilce, Tensor product of frame manuals, Internat. J. Theoret. Phys. 29 (1990), no. 8, 805–814. MR 1070453, DOI 10.1007/BF00675098
Antonio Zecca, On the coupling of logics, J. Mathematical Phys. 19 (1978), no. 6, 1482–1485. MR 503035, DOI 10.1063/1.523816