, Volume 10, Issue 3, pp 271–282 | Cite as

Tensor products of orthoalgebras

  • D. J. Foulis
  • M. K. Bennett


We define a tensor product via a universal mapping property on the class oforthoalgebras, which are both partial algebras and orthocomplemented posets. We show how to construct such a tensor product forunital orthoalgebras, and use the Fano plane to show that tensor products do not always exist.

Mathematics Subject Classifications (1991)

81B10 06C15 03G12 06E25 08A55 

Key words

Orthoalgebra tensor product test space orthocomplemented poset probability measure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beltrametti, E. and Cassinelli, G. (1981) The Logic of Quantum Mechanics, in Gian-Carlo Rota (ed.),Encyclopedia of Mathematics and its Applications 15, Addison-Wesley, Reading, MA.Google Scholar
  2. 2.
    Bennett, M. K. and Foulis, D. J. (1990) Superposition in quantum and classical mechanics,Foundations of Physics 20(6), 733–744.Google Scholar
  3. 3.
    Beren, L. (1984)Orthomodular Lattices, D. Reidel, Dordrecht.Google Scholar
  4. 4.
    Birkhoff, G. and von Neumann, J. (1936) The logic of quantum mechanics,Ann. of Math. 37, 823–843.Google Scholar
  5. 5.
    Foulis, D. J. and Randall, C. H. (1981) Empirical logic and tensor products, in H. Neumann (ed.),Interpretations and Foundations of Quantum Theory, Wissenschaftsverlag, Bibliographisches Institut 5, Mannheim/Wien/Zürich, pp. 9–20.Google Scholar
  6. 6.
    Foulis, D. J. (1989) Coupled physical systems,Foundations of Physics 7, 905–922.Google Scholar
  7. 7.
    Foulis, D. J., Greechie, R. J. and Rüttimann, G. T. (1992) Filters and supports in orthoalgebras,Int. J. of Theoretical Physics 31(5), 789–807.Google Scholar
  8. 8.
    Foulis, D. J. (1992) Tensor products of quantum logics, to appear in the Proceedings of SymposiumThe Interpretation of Quantum Theory: Where Do We Stand?, April, 1992, sponsored by the Instituto Della Enciclopedia Italiana.Google Scholar
  9. 9.
    Greechie, R. J. (1971) Orthomodular lattices admitting no states,J. Combinatorial Theory 10, 119–132.Google Scholar
  10. 10.
    Greechie, R. J. and Gudder, S. P. (1975) Quantum logics, in C. A. Hooker (ed.),The Logico-Algebraic Approach to Quantum Mechanics, Volume I:Historical Evolution, D. Reidel, Dordrecht.Google Scholar
  11. 11.
    Halmos, P. R. (1963)Lectures on Boolean Algebras, Van Nostrand, Princeton.Google Scholar
  12. 12.
    Hardegree, G. and Frazer, P. (1981) Charting the labyrinth of quantum logics, in E. Beltrametti and B. van Fraassen (eds),Current Issues in Quantum Logic, Ettore Majorana International Science Series,8, Plenum Press, N.Y.Google Scholar
  13. 13.
    Kalmbach, G. (1983)Orthomodular Lattices, Academic Press, N.Y.Google Scholar
  14. 14.
    Kläy, M. P., Randall, C. H. and Foulis, D. J. (1987) Tensor products and probability weights,Int. J. of Theoretical Physics 26(3), 199–219.Google Scholar
  15. 15.
    Lock, P. F. (1981) Categories of Manuals, Ph.D. Dissertation, Univ. of Mass., Amherst, MA.Google Scholar
  16. 16.
    Lock, R. H. (1990) The tensor product of generalized sample spaces which admit a unital set of dispersion-free weights,Foundations of Physics 20(5), 477–498.Google Scholar
  17. 17.
    Randall, C. H. and Foulis, D. J. (1979)New Definitions and Results, Univ. of Mass. Mimeographed Notes, Sept. 1979.Google Scholar
  18. 18.
    Randall, C. H. and Foulis, D. J. (1981) Operational statistics and tensor products, in H. Neumann (ed.),Interpretations and Foundations of Quantum Theory, Band 5, Wissenschaftsverlag, Bibliographisches Institut, Mannheim/Wien/Zürich, pp. 21–28.Google Scholar
  19. 19.
    Sikorski, R. (1960)Boolean Algebras, Springer-Verlag, Berlin/Göttingen/Heidelberg.Google Scholar
  20. 20.
    Stone, M. H. (1936) The theory of representations for a Boolean algebra,Trans. Amer. Math. Soc. 40, 37–111.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • D. J. Foulis
    • 1
  • M. K. Bennett
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstU.S.A.

Personalised recommendations