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On the Word Problem for Orthocomplemented Modular Lattices

Published online by Cambridge University Press:  20 November 2018

Michael S. Roddy
Michael S. Roddy*
Affiliation:
Brandon University, Brandon, Manitoba
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In [16] Freese showed that the word problem for the free modular lattice on 5 generators is unsolvable. His proof makes essential use of Mclntyre's construction of a finitely presented field with unsolvable word problem [30]. (We follow Cohn [7] in calling what is commonly called a division ring a field, and what is commonly called a field a commutative field.) In this paper we will use similar ideas to obtain unsolvability results for varieties of modular ortholattices. The material in this paper is fairly wide ranging, the following are recommended as reference texts.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 6 , 01 December 1989 , pp. 961 - 1004
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Berberian, S. K.. Baer “rings, Grundl. math. Wiss. 195 (Springer, 1972).Google Scholar
2. Bergman, G. M.. Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1–32.Google Scholar
3. Birkhoff, G., Lattices in applied mathematics, in Lattice theory, Proceedings of symp. pure math. A.M.S., Providence (1961), 155–184.Google Scholar
4. Birkhoff, G.and von Neumann, J., The logic of quantum mechanics, Annals of Math. 37 (1936), 823–843.Google Scholar
5. Bruns, G., Free ortholattices, Can. J. Math. 28 (1976), 977–985.Google Scholar
6. Burris, S. and Sankappanavar, H. P., A course in universal algebra, Graduate Texts in Mathematics 78 (Springer-Verlag, 1981).Google Scholar
7. Cohn, P. M., Skew field constructions, London Math. Soc. Lecture Note Series 27 (Cambridge University Press, 1977).Google Scholar
8. Cohn, P. M., Algebra, vols. 1 and 2, 2nd ed. (Wiley, 1982).Google Scholar
9. Cohn, P. M., Free rings and their relations, L.M.S. Monographs 19 (Academic Press, 1985).Google Scholar
10. Collins, D. J., A simple presentation of a group with unsolvable word problem, Illinois J. of Math. 30 (1986), 230–234.Google Scholar
11. Cornish, W. H., Antimorphic action, Research and Exposition in Mathematics 12 (Heldermann Verlag, Berlin, 1986).Google Scholar
12. Crawley, P. and Dilworth, R. P., Algebraic theory of lattices (Prentice Hall, 1973).Google Scholar
13. Day, R. A., Geometrical applications in modular lattices, in Universal algebra and lattice theory, Proceedings, Puebla (1972), 111–141.Google Scholar
14. Evans, T., Word problems, Bull. Amer. Math. Soc. 84 (1978), 789–802.Google Scholar
15. Foulis, D. J., Baer “semigroups, Proc. Amer. Math. Soc. 11 (1963), 889–894.Google Scholar
16. Freese, R., Free modular lattices, Trans. Amer. Math. Soc. 261 (1980), 81–91.Google Scholar
17. Grätzer, G., Lattice theory (Freeman, 1971).Google Scholar
18. Greechie, R. J. and Godowski, R., Some equations related to states on orthomodular lattices, Demonstratio Math. 17 (1984), 241–250.Google Scholar
19. Herrmann, C., On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983), 513–527.Google Scholar
20. Herrmann, C., Rahmen un erzeugende quadrupel in modularen verbanden, Algebra Universalis 14 (1982), 357–387.Google Scholar
21. Herstein, I., Rings with involution (Chicago University Press, 1976).Google Scholar
22. Holland, S. S. Jr.., The current interest in orthomodular lattices,in Trends in lattice theory (van Nostrand, 1970), 41–126.Google Scholar
23. Hughes, I., Division rings of fractions for group rings, Communications in Pure and Applied Mathematics 13 (1970), 181–188.Google Scholar
24. Husimi, K., Studies on the foundations of quantum mechanics 1, Proc. of the Physicomath. Soc. of Japan 19 (1937), 766–789.Google Scholar
25. Hutchinson, G., Recursively unsolvable word problems for modular lattices and diagram chasing, J. Algebra 26 (1973), 385–399.Google Scholar
26. Kalmbach, G., Orthomodular lattices (Academic Press, 1983).Google Scholar
27. Kotas, J., An axiom system for the modular logic, Studia Logica 21 (1967), 17–38.Google Scholar
28. Lewin, J., Fields of fractions for group algebras of free groups, Trans. Amer. Math. Soc. 192 (1974), 339–346.Google Scholar
29. Lipshitz, L., The undecidability of the word problem for projective geometries and modular lattices, Trans. Amer. Math. Soc. 193 (1974), 171–180.Google Scholar
30. Maclntyre, A., The word problem for division rings, J. Symb. Logic 38 (1973), 428–36.Google Scholar
31. Maeda, F., Representations of orthocomplemented modular lattices, J. Sci. Hiroshima Univ. 14 (1950), 93–96.Google Scholar
32. Mayet, R., Equational bases for some varieties of orthomodular lattices related to states, Algebra Universalis, preprint.Google Scholar
33. Mayet, R., Varieties of orthomodular lattices related to states, Algebra Univesalis 20 (1985), 368- 396.Google Scholar
34. Mayet, R.and Roddy, M., N-distributivity in ortholattices, in Contributions to general algebra 5, Proceedings of the Salzburg conference, Mai 29 - June 1, 1986 (1987), 285–294.Google Scholar
35. Meckler, A., Nelson, E.and Shelah, S., A variety with solvable, but not uniformly solvable, word problem, preprint (1987), 1–62.Google Scholar
36. Mittlestaedt, P., Quantum logic (Reidel, Dordrecht, 1978).Google Scholar
37. Neumann, B. H., On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202—252.Google Scholar
38. Novikov, P. S., On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44 (1955).Google Scholar
39. Post, E., Recursive unsolvability of a problem of thue, J. Symb. Logic 12 (1947), 1–11.Google Scholar
40. Roddy, M., Varieties of modular ortholattices, Order 3 (1987), 405–426.Google Scholar
41. Rotman, J. J., Theory of groups, 3rd ed. (Wm. Brown, C., 1988).Google Scholar
42. Skornyakov, L. A., Complemented modular lattices and regular rings, (Oliver and Boyd, 1964).Google Scholar
43. von Neumann, J., Continuous geometry (Princeton University Press, 1960).Google Scholar
44. Whitman, P. M., Free lattices I, Ann. of Math. 42 (1941), 325–330.Google Scholar