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.
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References
Beran, L.: Orthomodular Lattices. Algebraic Approach. Academia, Prague (1984)
Bruns G.: Free ortholattices. Canad. J. Math. 28, 977–985 (1976)
Chevalier G., Pulmannová S.: Compositions of Sasaki projections. Internat. J. Theoret. Phys. 31, 1599–1614 (1992)
D’Hooghe, B., Pykacz, J.: On some new operations on orthomodular lattices. Internat. J. Theoret. Phys. 39, no. 3, 641–652 (2000)
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)
Foulis D.: A note on orthomodular lattices. Port. Math. 21, 65–72 (1962)
Gabriëls J.J.M., Navara M.: Associativity of operations on orthomodular lattices. Math. Slovaca 62, 1069–1078 (2012)
Gabriëls J., Navara M.: Computer proof of monotonicity of operations on orthomodular lattices. Information Sci. 236, 205–217 (2013)
Gleason A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)
Greechie R.J.: On generating distributive sublattices of orthomodular lattices. Proc. Amer. Math. Soc. 67, 17–22 (1977)
Greechie R.J.: An addendum to “On generating distributive sublattices of orthomodular lattices”. Proc. Amer. Math. Soc. 76, 216–218 (1979)
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)
Holland S.S. Jr.: A Radon-Nikodym theorem in dimension lattices. Trans. Amer. Math. Soc. 108, 66–87 (1963)
Hyčko M.: Implications and equivalences in orthomodular lattices. Demonstr. Math. 38, 777–792 (2005)
Hyčko, M.: Computations in OML. http://www.mat.savba.sk/~hycko/oml
Hyčko, M., Navara, M.: Decidability in orthomodular lattices. Internat. J. Theoret. Phys. 44, no. 12, 2239–2248 (2005)
Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)
Mayet R.: Varieties of orthomodular lattices related to states. Algebra Universalis 20, 368–396 (1985)
Megill N.D., Pavičić M.: Orthomodular lattices and a quantum algebra. Internat. J. Theoret. Phys. 40, 1387–1410 (2001)
Navara M.: On generating finite orthomodular sublattices. Tatra Mt. Math. Publ. 10, 109–117 (1997)
Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991)
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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.
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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
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DOI: https://doi.org/10.1007/s00012-015-0332-5