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Algebra universalis

, Volume 73, Issue 2, pp 157–178 | Cite as

Small orthomodular partial algebras

  • Richard Holzer
  • Peter Burmeister
Article
  • 84 Downloads

Abstract

Orthomodular partial algebras (OMAs) can be seen as the algebraic representation of orthomodular posets. We use Greechie diagrams for the graphical representation of OMAs and investigate characterizations for the strong embeddability of a given OMA into a Boolean OMA. We present a complete list of the Greechie diagrams of OMAs up to 24 elements, and we show that there exists an infinite OMA that is generated by 4 elements.

Key words and phrases

orthomodular partial algebra orthomodular poset Greechie diagram 

2010 Mathematics Subject Classification

Primary: 08A55 Secondary: 06F99 

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References

  1. 1.
    Berge, C.: Graphs and Hypergraphs. Graphs and Hypergraphs. North-Holland (1976)Google Scholar
  2. 2.
    Burmeister, P.: A model theoretic oriented approach to partial algebras. Introduction to Theory and Application of Partial Algebras - Part I, Mathematical Research, vol. 32. Akademie (1986)Google Scholar
  3. 3.
    Burmeister P., Ma̧czyński M.: Orthomodular (partial) algebras and their representations. Demonstratio Math. 27, 701–722 (1994)MATHMathSciNetGoogle Scholar
  4. 4.
    Burmeister, P., Ma̧czyński, M.: Quasi-rings and congruences in the theory of orthomodular algebras. Tech. Rep. 2014, Department of Mathematics of the Darmstadt University of Technology (1998)Google Scholar
  5. 5.
    Dichtl M.: Astroids and pastings. Algebra Universalis 18, 380–385 (1984)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Erné, M., Heitzig, J., Reinhold, J.: On the number of distributive lattices. Electron. J. Combin. 9, Research Paper 24, 2381–2406 (2002)Google Scholar
  7. 7.
    Godowski R.: Varieties of orthomodular lattices with a strongly full set of states. Demonstr. Math. 14, 725–733 (1981)MATHMathSciNetGoogle Scholar
  8. 8.
    Godowski R.: States on orthomodular lattices. Demonstr. Math. 15, 817–822 (1982)MATHMathSciNetGoogle Scholar
  9. 9.
    Greechie R.J.: Orthomodular lattices admitting no states. J. Combin. Theory Ser. A 10, 119–132 (1971)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Greechie R.J.: Some results from the combinatorial approach to quantum logic. Synthese 29, 113–127 (1974)CrossRefMATHGoogle Scholar
  11. 11.
    Gudder, S.: Stochastic methods in quantum mechanics. Developments in Toxicology and Environmental Science. North-Holland (1979)Google Scholar
  12. 12.
    Harding J.: Remarks on concrete orthomodular lattices. Internat. J. Theoret. Phys. 43, 2149–2168 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Heitzig J., Reinhold J.: Counting finite lattices. Algebra Universalis 48, 43–53 (2002)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Holzer R.: On subdirectly irreducible omas. Studia Logica 78, 261–277 (2004)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Holzer R.: Greechie diagrams of orthomodular partial algebras. Algebra Universalis 57, 419–453 (2007)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Kalmbach, G.: Orthomodular lattices. L.M.S. monographs. Academic Press (1983)Google Scholar
  17. 17.
    McKay B.D., Megill N.D., Pavičić M.: Algorithms for greechie diagrams. International Journal of Theoretical Physics 39, 2381–2406 (2000)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Navara, M.: Measure Theory, chap. Kernel logics, pp. 27–30. Tatra Mountain Mathematical Publishers (1993)Google Scholar
  19. 19.
    Ovchinnikov P.G., Sultanbekov F.F.: Finite concrete logics: Their structure and measures on them. Proceedings of the International Quantum Structures Association (Berlin). Internat. J. Theoret. Phys. 37, 147–153 (1998)MATHMathSciNetGoogle Scholar
  20. 20.
    Pták, P.: Concrete quantum logics. Quantum Structures '98. Internat. J. Theoret. Phys. 39, 827–837 (2000).Google Scholar
  21. 21.
    Pták, P., Pulmannová, S.: Orthomodular structures as quantum logics. Translated from the 1989 Slovak original by the authors. Fundamental Theories of Physics, vol. 44, Kluwer (1991)Google Scholar
  22. 22.
    Pulmannová S.: A remark on orthomodular partial algebras. Demonstr. Math. 27, 687–699 (1994)MATHGoogle Scholar
  23. 23.
    Rogalewicz V.: Any orthomodular poset is a pasting of boolean algebras. Comment. Math. Univ. Carolin. 29, 557–558 (1988)MATHMathSciNetGoogle Scholar
  24. 24.
    Rogalewicz V.: On generating and concreteness in quantum logics. Math. Slovaca 41, 431–435 (1991)MATHMathSciNetGoogle Scholar
  25. 25.
    Shultz, F.W.: Axioms for quantum mechanics: a generalized probability theory. Ph.D. thesis, University of Wisconsin, Madison (1972)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.OFFIS, R&D Division TransportationOldenburgGermany
  2. 2.Department of Mathematics, AG 1Darmstadt University of TechnologyDarmstadtGermany

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