Abstract
It is known that orthomodular lattices admit 96 binary operations, out of which 16 are commutative. We clarify which of them are associative.
[1] BERAN, L.: Orthomodular Lattices. Algebraic Approach, Academia, Praha, 1984. 10.1007/978-94-009-5215-7Search in Google Scholar
[2] BIRKHOFF, G.— VON NEUMANN, J.: The logic of quantum mechanics, Ann. of Math. (2) 37 (1936), 823–843. http://dx.doi.org/10.2307/196862110.2307/1968621Search in Google Scholar
[3] D’HOOGHE, B.— PYKACZ, J.: On some new operations on orthomodular lattices, Internat. J. Theoret. Phys. 39 (2000), 641–652. http://dx.doi.org/10.1023/A:100363780463210.1023/A:1003637804632Search in Google Scholar
[4] FOULIS, D.: A note on orthomodular lattices, Portugal. Math. 21 (1962), 65–72. Search in Google Scholar
[5] GREECHIE, R. J.: On generating distributive sublattices of orthomodular lattices, Proc. Amer. Math. Soc. 67 (1977), 17–22. http://dx.doi.org/10.1090/S0002-9939-1977-0450157-910.1090/S0002-9939-1977-0450157-9Search in Google Scholar
[6] GREECHIE, R. J.: An addendum to “On generating distributive sublattices of orthomodular lattices”, Proc. Amer. Math. Soc. 76 (1979), 216–218. Search in Google Scholar
[7] GUDDER, S. P.: Quantum Probability, Academic Press, New York, 1988. 10.1007/BF00670748Search in Google Scholar
[8] HERRMANN, C.: The free orthomodular word problem is solved (review of the paper by G. Kalmbach), Zent.bl. MATH, Zbl 0585.06004; MR, MR 87k:06023. Search in Google Scholar
[9] HOLLAND Jr., S. S.: A Radon-Nikodym theorem in dimension lattices, Trans. Amer. Math. Soc. 108 (1963), 66–87. http://dx.doi.org/10.1090/S0002-9947-1963-0151407-310.1090/S0002-9947-1963-0151407-3Search in Google Scholar
[10] HYČKO, M.: Implications and equivalences in orthomodular lattices, Demonstratio Math. 38 (2005), 777–792. Search in Google Scholar
[11] HYČ KO, M.— NAVARA, M.: Decidability in orthomodular lattices, Internat. J. Theoret. Phys. 44 (2005), 2239–2248. http://dx.doi.org/10.1007/s10773-005-8019-x10.1007/s10773-005-8019-xSearch in Google Scholar
[12] KALMBACH, G.: Orthomodular Lattices, Academic Press, London, 1983. Search in Google Scholar
[13] KALMBACH, G.: The free orthomodular word problem is solved, Bull. Austral. Math. Soc. 34 (1986), 219–223. http://dx.doi.org/10.1017/S000497270001008X10.1017/S000497270001008XSearch in Google Scholar
[14] KRÖGER, H.: Zwerch-Assoziativität und verbandsähnliche Algebren, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. 3 (1973), 23–48. Search in Google Scholar
[15] MEGILL, N. D.— PAVIČIĆ, M.: Orthomodular lattices and a quantum algebra, Internat. J. Theoret. Phys. 40 (2001), 1387–1410. http://dx.doi.org/10.1023/A:101756782644810.1023/A:1017567826448Search in Google Scholar
[16] MEGILL, N. D.— PAVIČIĆ, M.: Equivalencies, identities, symmetric differences, and congruences in orthomodular lattices, Internat. J. Theoret. Phys. 42 (2003), 2797–2805. http://dx.doi.org/10.1023/B:IJTP.0000006006.18494.1c10.1023/B:IJTP.0000006006.18494.1cSearch in Google Scholar
[17] MEGILL, N. D.— PAVIČIĆ, M.: Quantum implication algebras, Internat. J. Theoret. Phys. 42 (2003), 2807–2822. http://dx.doi.org/10.1023/B:IJTP.0000006007.58191.da10.1023/B:IJTP.0000006007.58191.daSearch in Google Scholar
[18] NAVARA, M.: On generating finite orthomodular sublattices, Tatra Mt. Math. Publ. 10 (1997), 109–117. Search in Google Scholar
© 2012 Mathematical Institute, Slovak Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
