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A New View of Effects in a Hilbert Space

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Abstract

We investigate certain Brouwer-Zadeh lattices that serve as abstract counterparts of lattices of effects in Hilbert spaces under the spectral ordering. These algebras, called PBZ*-lattices, can also be seen as generalisations of orthomodular lattices and are remarkable for the collapse of three notions of “sharpness” that are distinct in general Brouwer-Zadeh lattices. We investigate the structure theory of PBZ*-lattices and their reducts; in particular, we prove some embedding results for PBZ*-lattices and provide an initial description of the lattice of PBZ*-varieties.

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References

  1. Blok W. J., Raftery J. G.: Assertionally equivalent quasivarieties. International Journal of Algebra and Computation 18(4), 589–681 (2008)

    Article  Google Scholar 

  2. Blyth T. S.: Lattices and Ordered Algebraic Structures. Springer, Berlin (2005)

    Google Scholar 

  3. Blyth T. S., Janowitz M. F.: Residuation Theory. Pergamon Press, Oxford (1972)

    Google Scholar 

  4. Bruns G.: Varieties of modular ortholattices. Houston Journal of Mathematics 9, 1–7 (1983)

    Google Scholar 

  5. Bruns G., Harding J. et al.: Algebraic aspects of orthomodular lattices. In: Coecke, B. (eds) Current Research in Operational Quantum Logic., pp. 37–65. Springer, Berlin (2000)

    Chapter  Google Scholar 

  6. Bruns G., Kalmbach G.: Varieties of orthomodular lattices II. Canadian Journal of Mathematics 24(2), 328–337 (1972)

    Article  Google Scholar 

  7. Cattaneo G., Dalla Chiara M. L., Giuntini R.: Some algebraic structures for many–valued logics. Tatra Mountains Mathematical Publications 15, 173–196 (1998)

    Google Scholar 

  8. Cattaneo G., Giuntini R.: Some results on BZ structures from Hilbertian unsharp quantum mechanics. Foundations of Physics 25, 1147–1183 (1995)

    Article  Google Scholar 

  9. Cattaneo G., Giuntini R., Pilla R.: BZMVdM and Stonian MV algebras (applications to fuzzy sets and rough approximations). Fuzzy Sets and Systems 108, 201–222 (1999)

    Article  Google Scholar 

  10. Cattaneo G., Gudder S.: Algebraic structures arising in axiomatic unsharp quantum physics. Foundations of Physics 29, 1607–1637 (1999)

    Article  Google Scholar 

  11. Cattaneo G., Hamhalter J.: De Morgan property for effect algebras of von Neumann algebras. Letters in Mathematical Physics 59(3), 243–252 (2002)

    Article  Google Scholar 

  12. Cattaneo, G., and G. Marino, Brouwer-Zadeh posets and fuzzy set theory, in A. Di Nola, A. G. S. Ventre (eds.), Proceedings of Mathematics of Fuzzy Systems, Napoli, 1984.

  13. Chajda I., Gil Férez J., Giuntini R., Kolarik M., Ledda A., Paoli F.: On some properties of directoids. Soft Computing 9(4), 955–964 (2015)

    Article  Google Scholar 

  14. Chajda I., Länger H.: Horizontal sums of bounded lattices. Mathematica Pannonica 20(1), 1–5 (2009)

    Google Scholar 

  15. Chajda, I., and H. Länger, H., Directoids. An Algebraic Approach to Ordered Sets, Heldermann Verlag, Lemgo, 2011.

  16. Chajda, I., and H. Länger, Bounded lattices with an antitone involution the complemented elements of which form a sublattice, draft.

  17. Dalla Chiara M. L., Giuntini R., Greechie R. J.: Reasoning in Quantum Theory. Kluwer, Dordrecht (2004)

    Book  Google Scholar 

  18. Dixmier J.: Von Neumann Algebras. North Holland, Amsterdam (1981)

    Google Scholar 

  19. Dvurečenskij A.: Gleason’s Theorem and Its Applications. Kluwer, Dordrecht-Boston-London (1993)

    Book  Google Scholar 

  20. Font J. M.: An abstract algebraic logic view of some multiple-valued logics. In: Fitting, M., Orlowska, E. (eds) Beyond Two: Theory and Applications of Multiple-Valued Logic, pp. 25–58. Physica-Verlag, Heidelberg (2003)

    Chapter  Google Scholar 

  21. Font J.M.: Taking degrees of truth seriously. Studia Logica 91(3), 383–406 (2009)

    Article  Google Scholar 

  22. Font J.M., Jansana R., Pigozzi D.: A survey of abstract algebraic logic. Studia Logica 74, 13–97 (2003)

    Article  Google Scholar 

  23. Foulis D.J., Bennett M.K.: Effect algebras and unsharp quantum logics. Foundations of Physics 24, 1325–1346 (1994)

    Article  Google Scholar 

  24. Giuntini R.: Quantum MV algebras. Studia Logica 56, 393–417 (1996)

    Article  Google Scholar 

  25. Gleason A.M.: Measures on the closed subspaces of a Hilbert space. Journal of Mathematics and Mechanics 6, 885–893 (1957)

    Google Scholar 

  26. Greechie R.J.: On the structure of orthomodular lattices satisfying the chain condition. Journal of Combinatorial Theory 4(3), 210–218 (1968)

    Article  Google Scholar 

  27. Greechie, R. J., A non-standard quantum logic with a strong set of states, in E. Beltrametti and B. van Fraassen (eds.), Current Issues in Quantum Logic, Plenum, New York, 1981, pp. 375–380.

  28. de Groote, H. F., On a canonical lattice structure on the effect algebra of a von Neumann algebra, 2005. arXiv:math/0410018v2.

  29. Gumm H. P., Ursini A.: Ideals in universal algebra. Algebra Universalis 19, 45–54 (1984)

    Article  Google Scholar 

  30. Hamhalter J.: Spectral order of operators and range projections. Journal of Mathematical Analysis and Applications 331, 1122–1134 (2007)

    Article  Google Scholar 

  31. Harzheim E.: Ordered Sets. Springer, Berlin (2005)

    Google Scholar 

  32. Jansana R.: Selfextensional logics with a conjunction. Studia Logica 84, 63–104 (2006)

    Article  Google Scholar 

  33. Kalman J.A.: Lattices with involution. Transactions of the American Mathematical Society 87, 485–491 (1958)

    Article  Google Scholar 

  34. Kalmbach G.: Orthomodular Lattices. Academic Press, New York (1983)

    Google Scholar 

  35. Kozen, D., On Kleene algebras and closed semirings, in Mathematical Foundations of Computer Science 1990, Lecture Notes in Computer Science, vol. 452, Springer, Berlin, 1990, pp. 26–47.

  36. Kreyszig E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)

    Google Scholar 

  37. Manzonetto, G., and A. Salibra, From λ-calculus to universal algebra and back, in MFCS’08, vol. 5162 of LNCS, 2008, pp. 479–490.

  38. Olson M.P.: The self-adjoint operators of a von Neumann algebra form a conditionally complete lattice. Proceedings of the American Mathematical Society 28, 537–544 (1971)

    Article  Google Scholar 

  39. Paoli F., Spinks M., Veroff R.: Abelian logic and the logics of pointed lattice-ordered varieties. Logica Universalis 2(2), 209–233 (2008)

    Article  Google Scholar 

  40. Roddy M.: Varieties of modular ortholattices. Order 3(4), 405–426 (1987)

    Article  Google Scholar 

  41. Salibra A., Ledda A., Paoli F., Kowalski T.: Boolean-like algebras. Algebra Universalis 69(2), 113–138 (2013)

    Article  Google Scholar 

  42. Stroock D.W.: A Concise Introduction to the Theory of Integration 3rd ed. Birkhäuser, Basel (1998)

    Google Scholar 

  43. Ursini A.: On subtractive varieties I. Algebra Universalis 31, 204–222 (1994)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Pedagogy, Psychology, Philosophy, University of Cagliari, Cagliari, Italy

    Roberto Giuntini, Antonio Ledda & Francesco Paoli

Authors
  1. Roberto Giuntini
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  2. Antonio Ledda
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  3. Francesco Paoli
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Correspondence to Francesco Paoli.

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Giuntini, R., Ledda, A. & Paoli, F. A New View of Effects in a Hilbert Space. Stud Logica 104, 1145–1177 (2016). https://doi.org/10.1007/s11225-016-9670-3

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  • DOI: https://doi.org/10.1007/s11225-016-9670-3

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