, Volume 32, Issue 2, pp 189–204 | Cite as

Unitizing a Generalized Pseudo Effect Algebra

  • David J. Foulis
  • Sylvia Pulmannová


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.


Pseudo effect algebra Generalized pseudo effect algebra Unitization Unitizing automorphism Two-valued state 


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  1. 1.
    Bennett, M. K., Foulis, D. J.: Interval and scale effect algebras. Adv. in Appl. Math. 19(2), 200–215 (1997)MATHMathSciNetCrossRefGoogle Scholar
  2. 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)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Beran, L.: Orthomodular Lattices, An Algebraic Approach, Mathematics and its Applications, vol. 18. D. Reidel Publishing Company, Dordrecht (1985)Google Scholar
  4. 4.
    Dvurečenskij, A.: Kite pseudo effect algebras. Found. Phys. 43, 1314–1338 (2013). arXiv:1306.0304.v1 MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer, Dordrecht (2000)MATHCrossRefGoogle Scholar
  6. 6.
    Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras I. Basic properties. Internat. J. Theor. Phys. 40, 685–701 (2001)MATHCrossRefGoogle Scholar
  7. 7.
    Dvurečenskij, A., Vetterlein, T.: Pseudoeffect algebras II. Group representation. Internat. J. Theor. Phys. 40, 703–726 (2001)MATHCrossRefGoogle Scholar
  8. 8.
    Dvurečenskij, A. , Vetterlein, T.: Generalized pseudo-effect algebras. Lectures on Soft Computing and Fuzzy Logic. Adv. Soft. Comput. Physica. Heidelberg, 89–111 (2001)Google Scholar
  9. 9.
    Dvurečenskij, A., Vetterlein, T.: Algebras in the positive cone of po-groups. Order 19, 127–146 (2002). doi: 10.1023A:1016551707-476 MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dvurečenskij, A., Xie, Y., Yang: Discrete (n+1)-valued states and n-perfect pseudo-effect algebras. Soft. Comput. 17, 1537–1552 (2013)CrossRefGoogle Scholar
  11. 11.
    Foulis, D.J.: MV and Heyting effect algebras. Found. Phys. 30(10), 1687–1706 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1331–1352 (1994)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Foulis, D.J., Pulmannová, S: The center of a generalized effect algebra. Demonstratio Math. 47, 1–21 (2014)MATHCrossRefGoogle Scholar
  14. 14.
    Foulis, D.J., Pulmannová, S: The exocenter of a generalized effect algebra. Rep. Math. Phys. 68(3), 347–371 (2011)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Foulis, D.J., Greechie, R.J., Rüttimann, G.T.: Filters and supports in orthoalgebras. Int. J. Theor. Phys. 31(5), 789–807 (1992)MATHCrossRefGoogle Scholar
  16. 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)CrossRefGoogle Scholar
  17. 17.
    Harding, J.: Regularity in quantum logic. Int. J. Theor. Phys. 37(4), 1173–1212 (1998)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Hedlíková, J., Pulmannová, S.: Generalized difference posets and orthoalgebras. Acta. Math. Univ. Comenianae 45, 247–279 (1996)Google Scholar
  19. 19.
    Janowitz, M.F.: A note on generalized orthomodular lattices. J. Natur. Sci. and Math. 8, 89–94 (1968)MATHMathSciNetGoogle Scholar
  20. 20.
    Jenča, G., Pulmannová, S.: Quotients of partial abelian monoids and the Riesz decomposition property. Algebra. Univ. 47, 443–477 (2002)MATHCrossRefGoogle Scholar
  21. 21.
    Kalmbach, G.: Orthomodular Lattices. Academic Press Inc., London/New York (1983)MATHGoogle Scholar
  22. 22.
    Kalmbach, G., Riečanová, Z.: An axiomatization for abelian relative inverses. Demonstratio Math. 27, 535–537 (1994)Google Scholar
  23. 23.
    Mundici, D.: Interpretation of AF C -algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Mayet-Ippolito, A.: Generalized orthomodular posets. Demonstratio Math. 24, 263–274 (1991)MATHMathSciNetGoogle Scholar
  25. 25.
    Pulmannová, S., Vinceková, E.: Riesz ideals in generalized effect algebras and in their unitizations. Algebra Univ. 57, 393–417 (2007)MATHCrossRefGoogle Scholar
  26. 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 MATHCrossRefGoogle Scholar
  27. 27.
    Stone, M.H.: Postulates for Boolean algebras and generalized Boolean algebras. Amer. J. Math. 57, 703–732 (1935)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Riečanová, Z.: Effect algebraic extensions of generalized effect algebras and two-valued states. Fuzzy Sets and Systems 159, 1116–1122 (2008)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Wilce, A.: Perspectivity and congruence in partial abelian semigroups. Math. Slovaca 48, 117–135 (1998)MATHMathSciNetGoogle Scholar
  30. 30.
    Li, H-Y., Li, S-G.: Congruences and ideals in pseudo-effect algebras. Soft. Comput. 12, 487–492 (2008)MATHCrossRefGoogle Scholar
  31. 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 MATHCrossRefGoogle Scholar
  32. 32.
    Xie, Y., Li, Y., Guo. J., Ren F., Li, D.: Weak commutative pseudo-effect algebras. Int. J. Theor. Phys 50, 1186–1197 (2011)MATHMathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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