Algebra universalis

, Volume 68, Issue 3–4, pp 197–220 | Cite as

Triple Representation Theorem for orthocomplete homogeneous effect algebras

Article
First Online:
Received:
Accepted:

Abstract

The aim of our paper is twofold. First, we thoroughly study the set of meager elements M(E), the set of sharp elements S(E), and the center C(E) in the setting of meager-orthocomplete homogeneous effect algebras E. Second, we prove the Triple Representation Theorem for sharply dominating meager-orthocomplete homogeneous effect algebras, in particular orthocomplete homogeneous effect algebras.

2010 Mathematics Subject Classification

Primary: 03G12 Secondary: 06D35 06F25 81P10 

Key words and phrases

homogeneous effect algebra orthocomplete effect algebra meager-orthocomplete effect algebra lattice effect algebra center atom sharp element meager element hypermeager element 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cattaneo G.: A unified framework for the algebra of unsharp quantum mechanics. Internat. J. Theoret. Phys. 36, 3085–3117 (1997)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chajda I., Halaš R., Kühr J.: Implication in MV-algebras. Algebra Universalis 52, 377–382 (2004)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chovanec, F., Kôpka, F.: D-posets. In: Engesser, K., Gabbay, D.M., Lehmann, D. (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Structures, pp. 367–428. Elsevier, Amsterdam (2007)Google Scholar
  4. 4.
    Cīrulis J.: On implication in MV-algebras. Algebra Universalis 56, 237–239 (2007)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dvurečcenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht, Ister Science, Bratislava (2000)Google Scholar
  6. 6.
    Foulis D.J.: The universal group of a Heyting effect algebra. Studia Logica 84, 407–424 (2006)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Foulis D.J., Bennett M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Foulis D.J., Greechie R.J., Rütimann G.: Filters and supports in orthoalgebras. Internat. J. Theoret. Phys. 35, 789–802 (1995)Google Scholar
  9. 9.
    Greechie R.J., Foulis D.J., Pulmannová S.: The center of an effect algebra. Order 12, 91–106 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gudder S.P.: Sharply dominating effect algebras. Tatra Mt. Math. Publ. 15, 23–30 (1998)MathSciNetMATHGoogle Scholar
  11. 11.
    Gudder S.P.: S-dominating effect algebras. Internat. J. Theoret. Phys. 37, 915–923 (1998)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Jenča G.: Blocks of homogeneous effect algebras. Bull. Austral. Math. Soc. 64, 81–98 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Jenča G.: Sharp and meager elements in orthocomplete homogeneous effect algebras. Order 27, 41–61 (2010)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Jenča G., Pulmannová S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra Universalis 47, 443–477 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Jenča G., Pulmannová S.: Orthocomplete effect algebras. Proc. Amer. Math. Soc. 131, 2663–2672 (2003)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Kôpka F.: Compatibility in D-posets. Internat. J. Theoret. Phys. 34, 1525–1531 (1995)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Niederle J., Paseka J.: More about sharp and meager elements in Archimedean atomic lattice effect algebras. Soft Computing 16, 109–119 (2012)CrossRefGoogle Scholar
  18. 18.
    Niederle J., Paseka J.: Homogeneous orthocomplete effect algebras are covered by MV-algebras. Fuzzy Sets and Systems 210, 89–101 (2013)MATHCrossRefGoogle Scholar
  19. 19.
    Pulmannová S.: Blocks in homogeneous effect algebras and MV-algebras. Math. Slovaca 53, 525–539 (2003)MathSciNetMATHGoogle Scholar
  20. 20.
    Riečanová Z.: Proper effect algebras admitting no states. Internat. J. Theoret. Phys. 40, 1683–1691 (2001)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Riečanová Z., Wu J.D.: States on sharply dominating effect algebras. Sci. China Ser. A 51, 907–914 (2008)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic

Personalised recommendations