Skip to main content
SpringerLink
Log in

Weaker forms of associativity in orthomodular lattices

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

Computations in orthomodular lattices are rather difficult because there is no known analogue of the standard disjunctive or conjunctive forms from boolean algebras. Tools admitting simplifications of formulas are rather limited. In previous work, we investigated the possibility of finding associative operations in orthomodular lattices that would admit new techniques of theorem proving. It was found that there are only 6 operations in orthomodular lattices which are (already known to be) associative. In this paper, we replaced associativity by more general properties, in particular by the identities that hold in alternative algebras. We have found that there are 8 nonassociative operations in orthomodular lattices satisfying the identities of alternative algebras. For these operations, we clarified the validity of other similar identities, including the Moufang identities. We hope that this effort can contribute to clarifying to what extent the validity of identities in orthomodular lattices can be algorithmized.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beran, L.: Orthomodular Lattices. Algebraic Approach. Academia, Prague (1984)

  2. Bruns G.: Free ortholattices. Canad. J. Math. 28, 977–985 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chevalier G., Pulmannová S.: Compositions of Sasaki projections. Internat. J. Theoret. Phys. 31, 1599–1614 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. D’Hooghe, B., Pykacz, J.: On some new operations on orthomodular lattices. Internat. J. Theoret. Phys. 39, no. 3, 641–652 (2000)

  5. Egly, U., Tompits, H.: Gentzen-like methods in quantum logic. Proceedings of the Eight International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX ’99), Institute for Programming and Logics, University at Albany - SUNY, (1999)

  6. Foulis D.: A note on orthomodular lattices. Port. Math. 21, 65–72 (1962)

    MATH  MathSciNet  Google Scholar 

  7. Gabriëls J.J.M., Navara M.: Associativity of operations on orthomodular lattices. Math. Slovaca 62, 1069–1078 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gabriëls J., Navara M.: Computer proof of monotonicity of operations on orthomodular lattices. Information Sci. 236, 205–217 (2013)

    Article  MATH  Google Scholar 

  9. Gleason A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)

    MATH  MathSciNet  Google Scholar 

  10. Greechie R.J.: On generating distributive sublattices of orthomodular lattices. Proc. Amer. Math. Soc. 67, 17–22 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  11. Greechie R.J.: An addendum to “On generating distributive sublattices of orthomodular lattices”. Proc. Amer. Math. Soc. 76, 216–218 (1979)

    MATH  MathSciNet  Google Scholar 

  12. Harding J.: The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. Algebra Universalis 48, 171–182 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Holland S.S. Jr.: A Radon-Nikodym theorem in dimension lattices. Trans. Amer. Math. Soc. 108, 66–87 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hyčko M.: Implications and equivalences in orthomodular lattices. Demonstr. Math. 38, 777–792 (2005)

    MATH  Google Scholar 

  15. Hyčko, M.: Computations in OML. http://www.mat.savba.sk/~hycko/oml

  16. Hyčko, M., Navara, M.: Decidability in orthomodular lattices. Internat. J. Theoret. Phys. 44, no. 12, 2239–2248 (2005)

  17. Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)

  18. Mayet R.: Varieties of orthomodular lattices related to states. Algebra Universalis 20, 368–396 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  19. Megill N.D., Pavičić M.: Orthomodular lattices and a quantum algebra. Internat. J. Theoret. Phys. 40, 1387–1410 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Navara M.: On generating finite orthomodular sublattices. Tatra Mt. Math. Publ. 10, 109–117 (1997)

    MATH  MathSciNet  Google Scholar 

  21. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, 2050, Johannesburg, South Africa

    Stephen M. Gagola III

  2. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27, Prague 6, Czech Republic

    Jeannine J. M. Gabriëls & Mirko Navara

Authors
  1. Stephen M. Gagola III
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeannine J. M. Gabriëls
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mirko Navara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen M. Gagola III.

Additional information

Presented by S. Pulmannova.

The second and third authors were supported by the Czech Technical University in Prague under project SGS12/187/OHK3/3T/13.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gagola, S.M., Gabriëls, J.J.M. & Navara, M. Weaker forms of associativity in orthomodular lattices. Algebra Univers. 73, 249–266 (2015). https://doi.org/10.1007/s00012-015-0332-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-015-0332-5

2010 Mathematics Subject Classification

Keywords and phrases

Search

Navigation