On the Word Problem for Orthocomplemented Modular Lattices

Roddy, Michael S.

Canadian journal of mathematics, Tome 41 (1989) no. 6, pp. 961-1004

Résumé

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.

Détail

DOI : 10.4153/CJM-1989-044-8

Bibliographie

[1] 1. Berberian, S. K.. Baer “rings, Grundl. math. Wiss. 195 (Springer, 1972). Google Scholar

[2] 2. Bergman, G. M.. Modules over coproducts of rings, Trans. Amer. Math. Soc. 200 (1974), 1–32. Google Scholar

[3] 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] 4. Birkhoff, G.and von Neumann, J., The logic of quantum mechanics, Annals of Math. 37 (1936), 823–843. Google Scholar

[5] 5. Bruns, G., Free ortholattices, Can. J. Math. 28 (1976), 977–985. Google Scholar

[6] 6. Burris, S. and Sankappanavar, H. P., A course in universal algebra, Graduate Texts in Mathematics 78 (Springer-Verlag, 1981). Google Scholar

[7] 7. Cohn, P. M., Skew field constructions, London Math. Soc. Lecture Note Series 27 (Cambridge University Press, 1977). Google Scholar

[8] 8. Cohn, P. M., Algebra, vols. 1 and 2, 2nd ed. (Wiley, 1982). Google Scholar

[9] 9. Cohn, P. M., Free rings and their relations, L.M.S. Monographs 19 (Academic Press, 1985). Google Scholar

[10] 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] 11. Cornish, W. H., Antimorphic action, Research and Exposition in Mathematics 12 (Heldermann Verlag, Berlin, 1986). Google Scholar

[12] 12. Crawley, P. and Dilworth, R. P., Algebraic theory of lattices (Prentice Hall, 1973). Google Scholar

[13] 13. Day, R. A., Geometrical applications in modular lattices, in Universal algebra and lattice theory, Proceedings, Puebla (1972), 111–141. Google Scholar

[14] 14. Evans, T., Word problems, Bull. Amer. Math. Soc. 84 (1978), 789–802. Google Scholar

[15] 15. Foulis, D. J., Baer “semigroups, Proc. Amer. Math. Soc. 11 (1963), 889–894. Google Scholar

[16] 16. Freese, R., Free modular lattices, Trans. Amer. Math. Soc. 261 (1980), 81–91. Google Scholar

[17] 17. Grätzer, G., Lattice theory (Freeman, 1971). Google Scholar

[18] 18. Greechie, R. J. and Godowski, R., Some equations related to states on orthomodular lattices, Demonstratio Math. 17 (1984), 241–250. Google Scholar

[19] 19. Herrmann, C., On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983), 513–527. Google Scholar

[20] 20. Herrmann, C., Rahmen un erzeugende quadrupel in modularen verbanden, Algebra Universalis 14 (1982), 357–387. Google Scholar

[21] 21. Herstein, I., Rings with involution (Chicago University Press, 1976). Google Scholar

[22] 22. Holland, S. S. Jr.., The current interest in orthomodular lattices,in Trends in lattice theory (van Nostrand, 1970), 41–126. Google Scholar

[23] 23. Hughes, I., Division rings of fractions for group rings, Communications in Pure and Applied Mathematics 13 (1970), 181–188. Google Scholar

[24] 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] 25. Hutchinson, G., Recursively unsolvable word problems for modular lattices and diagram chasing, J. Algebra 26 (1973), 385–399. Google Scholar

[26] 26. Kalmbach, G., Orthomodular lattices (Academic Press, 1983). Google Scholar

[27] 27. Kotas, J., An axiom system for the modular logic, Studia Logica 21 (1967), 17–38. Google Scholar

[28] 28. Lewin, J., Fields of fractions for group algebras of free groups, Trans. Amer. Math. Soc. 192 (1974), 339–346. Google Scholar

[29] 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] 30. Maclntyre, A., The word problem for division rings, J. Symb. Logic 38 (1973), 428–36. Google Scholar

[31] 31. Maeda, F., Representations of orthocomplemented modular lattices, J. Sci. Hiroshima Univ. 14 (1950), 93–96. Google Scholar

[32] 32. Mayet, R., Equational bases for some varieties of orthomodular lattices related to states, Algebra Universalis, preprint. Google Scholar

[33] 33. Mayet, R., Varieties of orthomodular lattices related to states, Algebra Univesalis 20 (1985), 368- 396. Google Scholar

[34] 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] 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] 36. Mittlestaedt, P., Quantum logic (Reidel, Dordrecht, 1978). Google Scholar

[37] 37. Neumann, B. H., On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202—252. Google Scholar

[38] 38. Novikov, P. S., On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44 (1955). Google Scholar

[39] 39. Post, E., Recursive unsolvability of a problem of thue, J. Symb. Logic 12 (1947), 1–11. Google Scholar

[40] 40. Roddy, M., Varieties of modular ortholattices, Order 3 (1987), 405–426. Google Scholar

[41] 41. Rotman, J. J., Theory of groups, 3rd ed. (Wm. Brown, C., 1988). Google Scholar

[42] 42. Skornyakov, L. A., Complemented modular lattices and regular rings, (Oliver and Boyd, 1964). Google Scholar

[43] 43. von Neumann, J., Continuous geometry (Princeton University Press, 1960). Google Scholar

[44] 44. Whitman, P. M., Free lattices I, Ann. of Math. 42 (1941), 325–330. Google Scholar