Skip to main content
SpringerLink
Log in

Unitizing a Generalized Pseudo Effect Algebra

  • Published:
Order Aims and scope Submit manuscript

Abstract

As is well-known, every generalized effect algebra can be embedded as a maximal proper ideal in an effect algebra called its unitization. We show that a necessary and sufficient condition that a generalized pseudo effect algebra can similarly be embedded as a maximal proper ideal in a pseudo effect algebra is that it admits a so-called unitizing automorphism. On the other hand, we show that a pseudo effect algebra is a unitization of a generalized pseudo effect algebra if and only if it admits a two-valued state.

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

References

  1. Bennett, M. K., Foulis, D. J.: Interval and scale effect algebras. Adv. in Appl. Math. 19(2), 200–215 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bennett, M. K., Foulis, D. J., Greechie, R. J.: Quotients of interval effect algebras. Quantum physics at the Einstein Meets Magritte Conference (Brussels, 1995). Internat. J. Theoret. Phys. 35(11), 2321–2338 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beran, L.: Orthomodular Lattices, An Algebraic Approach, Mathematics and its Applications, vol. 18. D. Reidel Publishing Company, Dordrecht (1985)

    Google Scholar 

  4. Dvurečenskij, A.: Kite pseudo effect algebras. Found. Phys. 43, 1314–1338 (2013). arXiv:1306.0304.v1

    Article  MATH  MathSciNet  Google Scholar 

  5. Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  6. Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras I. Basic properties. Internat. J. Theor. Phys. 40, 685–701 (2001)

    Article  MATH  Google Scholar 

  7. Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras II. Group representation. Internat. J. Theor. Phys. 40, 703–726 (2001)

    Article  MATH  Google Scholar 

  8. Dvurečenskij, A. , Vetterlein, T.: Generalized pseudo-effect algebras. Lectures on Soft Computing and Fuzzy Logic. Adv. Soft. Comput. Physica. Heidelberg, 89–111 (2001)

  9. Dvurečenskij, A., Vetterlein, T.: Algebras in the positive cone of po-groups. Order 19, 127–146 (2002). doi:10.1023A:1016551707-476

    Article  MATH  MathSciNet  Google Scholar 

  10. Dvurečenskij, A., Xie, Y., Yang: Discrete (n+1)-valued states and n-perfect pseudo-effect algebras. Soft. Comput. 17, 1537–1552 (2013)

    Article  Google Scholar 

  11. Foulis, D.J.: MV and Heyting effect algebras. Found. Phys. 30(10), 1687–1706 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Foulis, D.J., Pulmannová, S: The center of a generalized effect algebra. Demonstratio Math. 47, 1–21 (2014)

    Article  MATH  Google Scholar 

  14. Foulis, D.J., Pulmannová, S: The exocenter of a generalized effect algebra. Rep. Math. Phys. 68(3), 347–371 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  16. Foulis, D.J., Pulmannová, S., Vinceková, E.: The exocenter and type decomposition of a generalized pseudo effect algebra. Discuss. Math. Gen. Algebra Appl. 33, 13–47 (2014)

    Article  Google Scholar 

  17. Harding, J.: Regularity in quantum logic. Int. J. Theor. Phys. 37(4), 1173–1212 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hedlíková, J., Pulmannová, S.: Generalized difference posets and orthoalgebras. Acta. Math. Univ. Comenianae 45, 247–279 (1996)

    Google Scholar 

  19. Janowitz, M.F.: A note on generalized orthomodular lattices. J. Natur. Sci. and Math. 8, 89–94 (1968)

    MATH  MathSciNet  Google Scholar 

  20. Jenča, G., Pulmannová, S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra. Univ. 47, 443–477 (2002)

    Article  MATH  Google Scholar 

  21. Kalmbach, G.: Orthomodular Lattices. Academic Press Inc., London/New York (1983)

    MATH  Google Scholar 

  22. Kalmbach, G., Riečanová, Z.: An axiomatization for abelian relative inverses. Demonstratio Math. 27, 535–537 (1994)

    Google Scholar 

  23. Mundici, D.: Interpretation of AF C ∗-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mayet-Ippolito, A.: Generalized orthomodular posets. Demonstratio Math. 24, 263–274 (1991)

    MATH  MathSciNet  Google Scholar 

  25. Pulmannová, S., Vinceková, E.: Riesz ideals in generalized effect algebras and in their unitizations. Algebra Univ. 57, 393–417 (2007)

    Article  MATH  Google Scholar 

  26. Riečanová, Z.: Subalgebras, intervals, and central elements of generalized effect algebras. Int. J. Theor. Phys. 38, 3209–3220 (1999). doi:10.1023/A:1026682215765

    Article  MATH  Google Scholar 

  27. Stone, M.H.: Postulates for Boolean algebras and generalized Boolean algebras. Amer. J. Math. 57, 703–732 (1935)

    Article  MathSciNet  Google Scholar 

  28. Riečanová, Z.: Effect algebraic extensions of generalized effect algebras and two-valued states. Fuzzy Sets and Systems 159, 1116–1122 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wilce, A.: Perspectivity and congruence in partial abelian semigroups. Math. Slovaca 48, 117–135 (1998)

    MATH  MathSciNet  Google Scholar 

  30. Li, H-Y., Li, S-G.: Congruences and ideals in pseudo-effect algebras. Soft. Comput. 12, 487–492 (2008)

    Article  MATH  Google Scholar 

  31. Xie, Y., Li, Y.: Riesz ideals in generalized pseudo effect algebras and in their unitizations. Soft Comput. 14, 387–398 (2010). doi:10.1007/s00500-009-0412-6

    Article  MATH  Google Scholar 

  32. Xie, Y., Li, Y., Guo. J., Ren F., Li, D.: Weak commutative pseudo-effect algebras. Int. J. Theor. Phys 50, 1186–1197 (2011)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, 01002, 1 Sutton Court, Amherst, MA, USA

    David J. Foulis

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

    Sylvia Pulmannová

Authors
  1. David J. Foulis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sylvia Pulmannová
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvia Pulmannová.

Additional information

The second author was supported by Research and Development Support Agency under the contract No. APVV-0178-11 and grant VEGA 2/0059/12.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Foulis, D.J., Pulmannová, S. Unitizing a Generalized Pseudo Effect Algebra. Order 32, 189–204 (2015). https://doi.org/10.1007/s11083-014-9325-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-014-9325-9

Keywords

Search

Navigation