Skip to main content
SpringerLink
Log in

Notes on Sharp and Principal Elements in Effect Algebras

  • Published:
Order Aims and scope Submit manuscript

Abstract

In this paper, we characterize the sharp elements and principal elements in effect algebras. Furthermore, the sufficient and necessary conditions for ES (the set of sharp elements in an effect algebra E) to be an orthoalgebra, and to be an orthomodular poset are given. We also answer the open problems raised in 1996 by Gudder. Moreover, we show that in effect algebras, every pre-lattice ideal is an effect algebra ideal if and only if the effect algebra is an orthomodular poset.

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

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Beltrametti, E.G., Cassinelli, G.: The logic of quantum mechanics. Addison-Wesley (1981)

  2. Bennett, M.K., Foulis, D.J.: Phi-symmetric effect algebras. Found. Phys 25(12), 1699–1722 (1995)

    Article  MathSciNet  Google Scholar 

  3. Bińczak, G., Kaleta, J.: Sharp and principal elements in effect algebras. J. Math. Phys. 56(10), 101102–1–7 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dietzel, C., Rump, W.: The structure group of a non-degenerate effect algebra. Algebra Univers. 81(27), 1–30 (2020)

    MathSciNet  MATH  Google Scholar 

  5. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava (2000)

    Book  MATH  Google Scholar 

  6. Foulis, D.J.: A note on orthomodular lattices. Port. Math. 21(1), 65–72 (1962)

    MathSciNet  MATH  Google Scholar 

  7. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24(10), 1331–1352 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Foulis, D.J., Greechie, R.J., Rüttimann, G.T.: Filters and supports in orthoalgebras. Int. J. Theor. Phys. 31(5), 789–807 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Greechie, R.J., Foulis, D.J., Pulmannová, S.: The center of an effect algebra. Order 12(1), 91–106 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gudder, S.: Examples, problems, and results in effect algebras. Int. J. Theor. Phys. 35(11), 2365–2376 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gudder, S.: S-Dominating Effect algebras. Int. J. Theor. Phys. 37 (3), 915–923 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jenča, G., Rievcanovǎ, Z.: On sharp elements in lattice ordered effect algebras. BUSEFAL 80, 24–49 (1999)

    Google Scholar 

  13. Ma, Z.H.: Note on ideals of effect algebras. Inform. Sci. 179(5), 505–507 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moreland, T., Gudder, S.: Infima of Hilbert space effects. Linear Algebra Appl. 286(1-3), 1–17 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics Fundamental Theories of Physics, vol. 44. Kluwer Academic Publishers Group, Dordrecht (1991)

    MATH  Google Scholar 

  16. Riečanová, Z.: Sharp elements in effect algebras. Int. J. Theor. Phys. 40(5), 913–920 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Varadarajan, V.S.: Geometry of Quantum Theory. Springer, New York (1985)

    MATH  Google Scholar 

Download references

Acknowledgments

The first author is supported by Hebei Youth Natural Science Fund (Grant: A2019403031). We thank the anonymous referees for their helpful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Hebei GEO University, 050031, Shijiazhuang, China

    Yali Wu & Xia Li

  2. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), 250353, Jinan, China

    Jing Wang

Authors
  1. Yali Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xia Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing Wang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) 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

Wu, Y., Li, X. & Wang, J. Notes on Sharp and Principal Elements in Effect Algebras. Order 40, 525–536 (2023). https://doi.org/10.1007/s11083-022-09619-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-022-09619-1

Keywords

Search

Navigation