Orthomodular lattices that are \(Z_2\)-rich

  • Milan Matoušek
  • Pavel Pták


We study the orthomodular lattices (OMLs) that have an abundance of \(Z_2\)-valued states. We call these OMLs \(Z_2\)-rich. The motivation for the investigation comes from a natural algebraic curiosity that reflects the state of the (orthomodular) art, the consideration also has a certain bearing on the foundation of quantum theories (OMLs are often identified with “quantum logics”) and mathematical logic (\(Z_2\)-states are fundamental in mathematical logic). Before we launch on the subject proper, we observe, for a potential application elsewhere, that there can be a more economic introduction of \(Z_2\)-richness - the \(Z_2\)-richness in the orthocomplemented setup is sufficient to imply orthomodularity. In the further part we review basic examples of OMLs that are \(Z_2\)-rich and that are not. Then we show, as a main result, that the \(Z_2\)-rich OMLs form a large and algebraicly “friendly” class—they form a variety. In the appendix we note that the OMLs that allow for a natural introduction of a symmetric difference provide a source of another type of examples of \(Z_2\)-rich OMLs. We also formulate open questions related to the matter studied.


Orthocomplemented poset Quantum logic Projection logic Group-valued state Symmetric difference Boolean algebra Variety of algebras 

Mathematics Subject Classification

06A11 03G12 28E99 81P10 08B 


  1. 1.
    Bunce, L., Navara, M., Pták, P., Wright, J.: Quantum logics with Jauch–Piron states. Q. J. Math. (Oxford) 36, 261–271 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)CrossRefzbMATHGoogle Scholar
  3. 3.
    de Lucia, P., Pták, P.: Quantum logics with classically determined states. Colloquium Mathematicae 80(1), 147–154 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Simone, A., Navara, M., Pták, P.: States on systems of sets that are closed under symmetric difference. Mathematische Nachrichten 288(17–18), 1995–2000 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Simone, A., Pták, P.: Measures on circle coarse-grained systems of sets. Positivity 14, 247–256 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dvurečenskij, A.: Gleason’s Theorem and Its Applications. Kluwer Academic Publishers, Dordrecht-Boston-London (1992)zbMATHGoogle Scholar
  7. 7.
    Einstein, A., Podolski, B., Rosen, N.: Can quantum mechanical description of reality be considered complete? Phys. Rev. 47, 777–780 (1935)CrossRefzbMATHGoogle Scholar
  8. 8.
    Greechie, R.J.: Orthomodular lattices admitting no states. J. Comb. Theory 10, 119–132 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gudder, S.P.: Stochastic Methods in Quantum Mechanics. Elsevier, North-Holland, Amsterdam (1979)zbMATHGoogle Scholar
  10. 10.
    Hamhalter, J.: Quantum Measure Theory. Kluwer Academic Publishers, Dordrecht, Boston, London (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Engesser, K., Gabbay, D.M., Lehmann, D.: Handbook of Quantum Logic and Quantum Structures. Elsevier, Amsterdam (2007)zbMATHGoogle Scholar
  12. 12.
    Harding, J., Jager, E., Smith, D.: Group-valued measures on the lattice of closed subspaces of a hilbert space. Int. J. Theor. Phys. 44, 539–548 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hroch, M., Pták, P.: States on orthocomplemented difference posets (extensions). Lett. Math. Phys. 106(8), 1131–1137 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)zbMATHGoogle Scholar
  15. 15.
    Matoušek, M.: Orthocomplemented lattices with a symmetric difference. Algebra Universalis 60, 185–215 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Matoušek, M., Pták, P.: Symmetric difference on orthomodular lattices and \(Z_2\)-valued states. Comment. Math. Univ. Carolin. 50(4), 535–547 (2009)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Matoušek, M., Pták, P.: On identities in orthocomplemented difference lattices. Mathematica Slovaca 60(5), 583–590 (2010)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mayet, R.: Varieties of orthomodular lattices related to states. Algebra Universalis 20, 368–386 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Navara, M.: An orthomodular lattice admitting no group-valued measure. Proc. Am. Math. Soc. 122, 7–12 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Navara, M., Pták, P.: For \(n \ge 5\) there is no nontrivial \(Z_2\)-measure on \(L(R^n)\). Int. J. Theor. Phys. 43, 1595–1598 (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Navara, M., Pták, P., Rogalewicz, V.: Enlargements of quantum logics. Pac. J. Math. 135, 361–369 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht, Boston, London (1991)zbMATHGoogle Scholar
  23. 23.
    Pták, P., Weber, H.: Relatively additive states on quantum logics. Comment. Math. Univ. Carolin. 46(2), 327–338 (2005)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Pták, P., Weber, H.: Lattice properties of subspace families in an inner product space. Proc. Am. Math. Soc. 129(7), 2111–2117 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Weber, H.: There are orthomodular lattices without non-trivial group-valued states: a computer-based construction. J. Math. Anal. Appl. 183, 89–94 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Prague 10Czech Republic
  2. 2.Department of Mathematics, Faculty of Electrical EngineeringCzech Technical UniversityPrague 6Czech Republic

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