Skip to main content
SpringerLink
Log in

Homogeneous Effect Algebras and Observables vs Spectral Resolutions

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

A block of an effect algebra is a maximal sub-effect algebra with some kind of compatibility property. Jenča (Australian Math. Soc. 64, 81–98, 2001), shows that every block of a homogeneous effect algebra satisfies the Riesz Decomposition Property (RDP). We establish that in a monotone \(\sigma\)-complete effect algebra, every block is a monotone \(\sigma\)-complete sub-effect algebra with (RDP). This result is used to show a one-to-one relationship between observables and spectral resolutions, as well as for n-dimensional observables and n-dimensional spectral resolutions. This result extends the class of effect algebras where this relationship holds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    Article  MathSciNet  Google Scholar 

  2. Catlin, D.: Spectral theory in quantum logics. Inter. J. Theor. Phys. 1, 285–297 (1968)

    Article  MathSciNet  Google Scholar 

  3. Di Nola, A., Dvurečenskij, A., Lenzi, G.: Observables on perfect MV-algebras. Fuzzy Sets and Systems 369, 57–81 (2019). https://doi.org/10.1016/j.fss.2018.11.018

    Article  MathSciNet  MATH  Google Scholar 

  4. Dvurečenskij, A.: Quantum observables and effect algebras. Inter. J. Theor. Phys. 57, 637–651 (2018). https://doi.org/10.1007/s10773-017-3594-1

    Article  MathSciNet  MATH  Google Scholar 

  5. Dvurečenskij, A.: Perfect effect algebras and spectral resolutions of observables. Found. Phys. 49, 607–628 (2019). https://doi.org/10.1007/s10701-019-00238-2

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Dvurečenskij, A.: Sum of n-dimensional observables on MV-effect algebras. Soft Comput. 25, 8073–8084 (2021). https://doi.org/10.1007/s00500-021-05911-1

    Article  Google Scholar 

  7. Dvurečenskij, A., Kuková, M.: Observables on quantum structures. Inf. Sci. 262, 215–222 (2014). https://doi.org/10.1016/j.ins.2013.09.014

    Article  MathSciNet  MATH  Google Scholar 

  8. Dvurečenskij, A., Lachman, D.: Two-dimensional observables and spectral resolutions. Rep. Math. Phys. 85, 163–191 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  9. Dvurečenskij, A., Lachman, D.: Lifting, \(n\)-dimensional spectral resolutions, and \(n\)-dimensional observables. Algebra Universalis 34, Art. Num. 34 (2020). https://doi.org/10.1007/s00012-020-00664-8

  10. Dvurečenskij, A., Lachman, D.: \(n\)-dimensional observables on \(k\)-perfect MV-algebras and effect algebras. I. Characteristic points. Fuzzy Sets and Systems 442, 1–16 (2022). https://doi.org/10.1016/j.fss.2021.05.005

  11. Dvurečenskij, A., Lachman, D.: Spectral resolutions and quantum observables. Inter. J. Theor. Phys. 59, 2362–2383 (2020). https://doi.org/10.1007/s10773-020-04507-z

    Article  MathSciNet  MATH  Google Scholar 

  12. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publ., Dordrecht, Ister Science, Bratislava, 541 + xvi pp (2000)

  13. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  14. Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs No. 20, Amer. Math. Soc., Providence, Rhode Island (1986)

  15. Halmos, P.R.: Measure Theory. Springer, Berlin (1974)

    MATH  Google Scholar 

  16. Jenča, G.: Blocks of homogeneous effect algebras. Bull. Australian Math. Soc. 64, 81–98 (2001). https://doi.org/10.1017/S0004972700019705

    Article  MathSciNet  MATH  Google Scholar 

  17. Jenča, G.: Sharp and meager elements in orthocomlete homogeneous effect algebras. Order 27, 41–61 (2010)

    Article  MathSciNet  Google Scholar 

  18. Ravindran, K.: On a Structure Theory of Effect Algebras. Ph.D. Thesis, Kansas State University, Manhattan (1996)

  19. Riečanová, Z.: Generalization of blocks for D-lattices and lattice ordered effect algebras. Inter. J. Theor. Phys. 39, 231–237 (2000). https://doi.org/10.1023/A:1003619806024

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The paper acknowledges the support by the grant of the Slovak Research and Development Agency under contract APVV-20-0069 and the grant VEGA No. 2/0142/20 SAV, A.D., and the Austrian Science Fund (FWF), project I 4579-N and the Czech Science Foundation (GAČR), project 20-09869L, D.L.

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

  2. Department Algebra Geometry, Palacký University, 17. listopadu 12, CZ-771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij & Dominik Lachman

Authors
  1. Anatolij Dvurečenskij
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Lachman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatolij Dvurečenskij.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dvurečenskij, A., Lachman, D. Homogeneous Effect Algebras and Observables vs Spectral Resolutions. Int J Theor Phys 61, 214 (2022). https://doi.org/10.1007/s10773-022-05185-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-022-05185-9

Keywords

Search

Navigation