International Journal of Theoretical Physics

, Volume 52, Issue 6, pp 2171–2180 | Cite as

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

  • Jiří Janda
  • Zdenka Riečanová


In (Riečanová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set \(\mathcal{M}\) of states can by embedded into a Hilbert space effect algebra \(\mathcal{E}(l_{2}(\mathcal{M}))\). We consider the problem when its effect algebraic MacNeille completion \(\hat{E}\) can be also embedded into the same Hilbert space effect algebra \(\mathcal {E}(l_{2}(\mathcal{M}))\). That is when the ordering set \(\mathcal{M}\) of states on E can be extended to an ordering set of states on \(\hat{E}\). We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.


Effect algebra MV-effect algebra MacNeille completion Positive linear operators in Hilbert space Hilbert space effect-representation 



Jiří Janda kindly acknowledges the support by Masaryk University, grant 0964/2009 and ESF Project CZ.1.07 /2.3.00/20.0051 Algebraic Methods in Quantum Logic of the Masaryk University.

Zdenka Riečanová kindly acknowledges the support by the Science and Technology Assistance Agency under the contract APVV-0178-11 Bratislava SR, and VEGA-grant of MŠ SR No. 1/0297/11.


  1. 1.
    Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics, 2nd edn. Springer, Berlin (2008) MATHGoogle Scholar
  2. 2.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Acad./Ister Science, Dordrecht/Bratislava (2000) MATHGoogle Scholar
  3. 3.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  4. 4.
    Kôpka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21 (1994) MathSciNetMATHGoogle Scholar
  5. 5.
    Kôpka, F.: D-posets of fuzzy sets. Tatra Mt. Math. Publ. 1, 83–87 (1992) MathSciNetMATHGoogle Scholar
  6. 6.
    Janda, J., Riečanová, Z.: Intervals in generalized effect algebra. Preprint Google Scholar
  7. 7.
    Paseka, J., Riečanová, Z.: Inherited properties of effect algebras preserved by isomorphism. Acta Polytech. (2013) to appear Google Scholar
  8. 8.
    Riečanová, Z.: MacNeille completions of d-posets and effect algebras. Int. J. Theor. Phys. 39(3), 859–869 (2000) MATHCrossRefGoogle Scholar
  9. 9.
    Riečanová, Z.: Archimedean and block-finite lattice effect algebra. Demonstr. Math. 33(3), 443–452 (2000) MATHGoogle Scholar
  10. 10.
    Riečanová, Z.: Generalization of blocks for d-lattices and lattice-ordered effect algebras. Int. J. Theor. Phys. 39(2), 231–237 (2000) MATHCrossRefGoogle Scholar
  11. 11.
    Riečanová, Z.: Distributive atomic effect algebras. Demonstr. Math. 36(2), 247–259 (2003) MATHGoogle Scholar
  12. 12.
    Riečanová, Z.: Sub-effect algebras and Boolean sub-effect algebras. Soft Comput. 5, 400–403 (2001) MATHCrossRefGoogle Scholar
  13. 13.
    Riečanová, Z., Marinová, I.: Generalized homogenoeus, prelattice and MV-effect algebras. Kybernetika 41(2), 129–142 (2005) MathSciNetMATHGoogle Scholar
  14. 14.
    Paseka, J., Riečanová, Z.: The inheritance of BDE-property in sharply dominating lattice effect algebras and (o)-continuous states. Soft Comput. 15, 543–555 (2011) MATHCrossRefGoogle Scholar
  15. 15.
    Riečanová, Z., Zajac, M.: Hilbert space effect-representations of effect algebras. Rep. Math. Phys. 70(2), 283–290 (2012) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Riečanová, Z., Zajac, M.: Intervals in generalized effect algebras and their sub-generalized effect algebras. Acta Polytech. (2013) to appear Google Scholar
  17. 17.
    Schmidt, J.: Zur Kennzeichnung der Dedekind-MacNeilleschen Hulle einer Geordneten Menge. Arch. Math. 7, 241–249 (1956) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Mathematics, Faculty of Electrical Engineering and Information TechnologySlovak University of TechnologyBratislavaSlovak Republic

Personalised recommendations