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Sasaki projections

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Abstract

We collect, correct, and extend results on the properties of the Sasaki projection in orthomodular lattices. We bring arguments as to why this operation can extend tools for simplification of formulas and automated computing.

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References

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

  2. Birkhoff G., von Neumann J.: The logic of quantum mechanics. Ann. of Math. 37, 823–843 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chevalier G.: Relations binaires et congruences dans un treillis orthomodulaires. C. R. Math. Acad. Sci. Paris 296, 785–788 (1983)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Finch P.D.: Sasaki projections on orthocomplemented posets. Bull. Austral. Math. Soc. 1, 319–324 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  6. Finch P.D.: A transposition principle in orthomodular lattices. Bull. London Math. Soc 2, 49–52 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  7. Foulis D.: Baer *-semigroups. Proc. Amer. Math. Soc. 11, 648–654 (1960)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Foulis D.: Relative inverse in Baer *-semigroups. Michigan Math. J. 10, 65–84 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foulis D., Pulmannová S.: Projections in a synaptic algebra. Order 27, 235–257 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Gagola S. III, Gabriëls J., Navara M.: Weaker forms of associativity in orthomodular lattices. Algebra Universalis 73, 249–266 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Godowski R., Greechie R.J.: Some equations related to states on orthomodular lattices. Demonstratio Math. 17, 241–250 (1984)

    MathSciNet  MATH  Google Scholar 

  17. Halmos P.R.: Algebraic logic I. Monadic Boolean algebras. Composition Math. 12, 217–249 (1955)

    Google Scholar 

  18. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  21. Hyčko, M.: Computations in OML. http://www.mat.savba.sk/~hycko/oml (Ver. June 7, 2011)

  22. Hyčko M., Navara M.: Decidability in orthomodular lattices. Internat. J. Theoret. Phys. 44, 2239–2248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

  24. Maeda, F., Maeda, S.: Theory of Symmetric Lattices. Springer (1970)

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Mittelstaedt, P.: Quantenlogische Interpretation orthokomplementärer quasimodularer Verbände. Zeitschrift für Naturforschung 25a, 1773–1778 (1970)

  27. Mittelstaedt, P.: On the interpretation of the lattice of subspaces of the Hilbert space as a propositional calculus. Zeitschrift für Naturforschung 27a, 1358–1362 (1972)

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

    MathSciNet  MATH  Google Scholar 

  29. Navara, M.: Constructions of quantum structures. In: D. Gabbay, D. Lehmann, K. Engesser (eds.) Handbook of Quantum Logic, vol. 1, pp. 335–366. Elsevier (2007)

  30. Pykacz, J.: Quantum Physics, Fuzzy Sets and Logic. Steps Towards a Many-Valued Interpretation of Quantum Mechanics. Springer Briefs in Physics. Springer, Cham (2015)

  31. Sasaki U.: Orthocomplemented lattices satisfying the exchange axiom. J. Sci. Hiroshima Univ. A, 17, 293–302 (1954)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. 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

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

    Stephen M. Gagola III

Authors
  1. Jeannine J. M. Gabriëls
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  2. Stephen M. Gagola III
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  3. Mirko Navara
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Correspondence to Mirko Navara.

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Presented by S. Pulmannova.

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Gabriëls, J.J.M., Gagola, S.M. & Navara, M. Sasaki projections. Algebra Univers. 77, 305–320 (2017). https://doi.org/10.1007/s00012-017-0428-1

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