Algebra universalis

, 80:49 | Cite as

Observables on lexicographic effect algebras

  • Anatolij DvurečenskijEmail author
  • Dominik Lachman


We study lexicographic effect algebras which are intervals in lexicographic products \(H\,\overrightarrow{\times }\,G\), where (Hu) is a unital po-group and G is a monotone \(\sigma \)-complete po-group with interpolation. We prove that there is a one-to-one correspondence between observables, which are a special kind of \(\sigma \)-homomorphisms and analogues of measurable functions, and spectral resolutions which are systems \(\{x_t : t \in {\mathbb {R}}\}\) of elements of a lexicographic effect algebra that are monotone, “left continuous”, and going to 0 if \(t\rightarrow -\infty \) and to 1 if \(t\rightarrow +\infty \). We show that this correspondence in lexicographic effect algebras holds only for spectral resolutions with the finiteness property. Otherwise, they do not determine any observable. Whence, the information involved in a spectral resolution with the finiteness property completely describes information about an observable.


Effect algebra Lexicographic effect algebra Monotone \(\sigma \)-complete po-group Observable Spectral resolution Finiteness property 

Mathematics Subject Classification

03G12 03B50 06C15 81P15 



The authors are very indebted to anonymous referees for their suggestions and remarks that improve the readability of the paper.


  1. 1.
    Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Loomis-Sikorski representation of monotone \(\sigma \)-complete effect algebras. Fuzzy Sets Syst. 157, 683–690 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Catlin, D.: Spectral theory in quantum logics. Int. J. Theor. Phys. 1, 285–297 (1968)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Di Nola, A., Dvurečenskij, A., Lenzi, G.: Observables on perfect MV-algebras. Fuzzy Sets Syst. 369, 57–81 (2019)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Di Nola, A., Grigolia, R.: Gödel spaces and perfect MV-algebras. J. Appl. Log. 13, 270–284 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dvurečenskij, A.: Perfect effect algebras are categorically equivalent with Abelian interpolation po-groups. J. Aust. Math. Soc. 82, 183–207 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dvurečenskij, A.: Representable effect algebras and observables. Int. J. Theor. Phys. 53, 2855–2866 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dvurečenskij, A.: Lexicographic effect algebras. Algebra Universalis 75, 451–480 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dvurečenskij, A.: Perfect effect algebras and spectral resolutions of observables. Found. Phys. 49, 607–628 (2019)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dvurečenskij, A., Kuková, M.: Observables on quantum structures. Inf. Sci. 262, 215–222 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dvurečenskij, A., Lachman, D.: Observables in \(n\)-perfect MV-algebras (Submitted) Google Scholar
  11. 11.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publ., Dordrecht, Ister Science, Bratislava (2000)CrossRefGoogle Scholar
  12. 12.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford (1963)zbMATHGoogle Scholar
  14. 14.
    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation. Math. Surveys and Monographs No. 20, American Mathematical Society, Providence (1986)Google Scholar
  15. 15.
    Halmos, P.R.: Measure Theory. Springer, Berlin (1974)zbMATHGoogle Scholar
  16. 16.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland Publisher, Amsterdam (1982)zbMATHGoogle Scholar
  17. 17.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)zbMATHGoogle Scholar
  18. 18.
    Ravindran, K.: On a structure theory of effect algebras. Ph.D. Thesis, Kansas State University, Manhattan (1996)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia
  2. 2.Faculty of SciencesPalacký University OlomoucOlomoucCzech Republic

Personalised recommendations