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Transactions of the American Mathematical Society

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Free modular lattices


Author: Ralph Freese
Journal: Trans. Amer. Math. Soc. 261 (1980), 81-91
MSC: Primary 06C05; Secondary 03D40
DOI: https://doi.org/10.1090/S0002-9947-1980-0576864-X
MathSciNet review: 576864
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Abstract: It is shown that the word problem for the free modular lattice on five generators is recursively unsolvable.


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  • [1] B. Artmann, On coordinates in modular lattices with a homogeneous basis, Illinois J. Math. 12 (1968), 626-648. MR 0236075 (38:4373)
  • [2] G. Birkhoff, Lattice theory, Amer. Math. Soc. Colloq. Publ., vol. 25, rev. ed., Amer. Math. Soc., New York, 1948. MR 0029876 (10:673a)
  • [3] P. M. Cohn, Skew field constructions, London Math. Soc. Lecture Note Series, no. 27, Cambridge Univ. Press, Cambridge, 1977. MR 0463237 (57:3190)
  • [4] P. Crawley and R. P. Dilworth, Algebraic theory of lattices, Prentice-Hall, Englewood Cliffs, N. J., 1973.
  • [5] M. Davis, Computability and unsolvability, McGraw-Hill, New York, 1958. MR 0124208 (23:A1525)
  • [6] A. Day, C. Herrman and R. Wille, On modular lattices with four generators. Algebra Universalis 2 (1972), 317-323. MR 0316331 (47:4879)
  • [7] R. Dedekind, Uber die drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900), 371-403. MR 1511094
  • [8] R. P. Dilworth, The arithmetical theory of Birkhoff lattices, Duke Math. J. 8 (1941), 286-299. MR 0005094 (3:100a)
  • [9] R. P. Dilworth and R. Freese, Generators for lattice varieties, Algebra Universalis 6 (1976), 263-267. MR 0427175 (55:210)
  • [10] T. Evans, Word problems, Bull. Amer. Math. Soc. 84 (1978), 789-802. MR 0498063 (58:16240)
  • [11] R. Freese, Breadth two modular lattices, Proc. Univ. Houston Lattice Theory Conf. (Houston, Texas, 1973), Dept. Math., Univ. Houston, Houston, Texas, 1973, pp. 409-451. MR 0398926 (53:2777)
  • [12] -, Planar sublattices of $ FM(4)$, Algebra Universalis 6 (1976), 69-72. MR 0398927 (53:2778)
  • [13] -, Some varieties of modular lattices not generated by their finite dimensional members, Contribution to Universal Algebra, Szeged, Hungary, Colloq. Math. Soc. Janos Bolyai 17 (1977), 133-144. MR 0485608 (58:5433)
  • [14] -, The variety of modular lattices is not generated by its finite members, Trans. Amer. Math. Soc. 255 (1979), 277-300. MR 542881 (81g:06003)
  • [15] -, Projective geometries as protective modular lattices, Trans. Amer. Math. Soc. 251 (1979), 329-342. MR 531987 (81j:06010)
  • [16] -, Free modular lattices, Notices Amer. Math. Soc. 26 (1979), #79T-A4.
  • [17] M. M. Gluhov, Algorithmic solvability of the word problem for completely free modular lattices, Sibirsk. Math. Ž. 5 (1964), 1027-1034. MR 0214517 (35:5367)
  • [18] M. Hall and R. P. Dilworth, The imbedding problem for modular lattices, Ann. of Math. (2) 45 (1944), 450-456. MR 0010541 (6:33c)
  • [19] C. Herrmann, On the equational theory of submodule lattices, Proc. Univ. Houston Lattice Theory Conf. (Houston, Texas, 1973), Dept. Math., Univ. Houston, Houston, Texas, 1973, pp. 105-118. MR 0406882 (53:10668)
  • [20] -, Concerning M. M. Gluhov's paper on the word problem for free modular lattices, Algebra Universalis 5 (1975), 445.
  • [21] -, On modular lattices generated by two complemented pairs, Houston J. Math. 2 (1976), 513-523. MR 0424632 (54:12591)
  • [22] C. Herrmann and A. Huhn, Zum Wortproblem fur freie Untermodulverbande, Arch. Math. 26 (1975), 449-453. MR 0401562 (53:5389)
  • [23] -, Lattices of normal subgroups which are generated by frames, Lattice Theory, Szeged, Hungary, Colloq. Math. Soc. Janos Bolyai 14 (1975), 97-136. MR 0447064 (56:5379)
  • [24] G. Hutchinson, Recursively unsolvable word problems for modular lattices and diagram-chasing, J. Algebra 26 (1973), 385-399. MR 0332599 (48:10925)
  • [25] -, Embedding and unsolvability theorems for modular lattices, Algebra Universalis 7 (1977), 47-84. MR 0441804 (56:198)
  • [26] G. Hutchinson and G. Czedli, A test for identities satisfied in lattices of submodules, Algebra Universalis 8 (1978), 269-309. MR 0469840 (57:9621)
  • [27] B. Jónsson, Arguesian lattices of dimension $ n\, \leqslant \,4$, Math. Scand. 7 (1959), 133-145. MR 0109800 (22:685)
  • [28] -, Varieties of lattices: some open problems, Colloq. Math. Soc. J. Bolyai, Universal Algebra Conference, Esztergom, Hungary, 1977.
  • [29] A. Macintyre, The word problem for division rings, J. Symbolic Logic 38 (1973), 428-436. MR 0337554 (49:2323)
  • [30] F. Maeda, Kontinuierliche Geometrien, Springer-Verlag, Berlin, 1958. MR 0090579 (19:833c)
  • [31] J. von Neumann, Continuous geometry, (I. Halperin, ed.), Princeton Univ. Press, Princeton, N. J., 1960. MR 0120174 (22:10931)
  • [32] G. Sauer, W. Seibert and R. Wille, On free modular lattices over partial lattices with four generators, Proc. Univ. Houston Lattice Theory Conf. (Houston, Texas, 1973), Dept. Math., Univ. Houston, Houston, Texas, 1973, pp. 332-382. MR 0406885 (53:10671)
  • [33] M. Schutzenberger, Le problème des mots dans les treillis modulaires libres, C. R. Acad. Sci. Paris Sér. A-B 237 (1953), 507-508. MR 0057222 (15:192e)
  • [34] K. Takeuchi, On free modular lattices, Japan J. Math. 21 (1951), 53-65. MR 0051814 (14:529c)
  • [35] -, On free modular lattices. II, Tôhoku Math. J. 11 (1959), 1-12. MR 0106859 (21:5589)
  • [36] P. Whitman, Free lattices, Ann. of Math. (2) 42 (1941), 325-330. MR 0003614 (2:244f)
  • [37] -, Free lattices. II, Ann. of Math. (2) 43 (1942), 104-115. MR 0006143 (3:261d)
  • [38] -, review of [33], Math. Rev. 15 (1954), 192.
  • [39] -, Status of word problems for lattices, Proc. Sympos. Pure Math., Vol. 2, Amer. Math. Soc., Providence, R. I., 1961, pp. 17-21. MR 0124246 (23:A1560)
  • [40] R. Wille, Über modulare Verbande die von einer endlichen halbgeordneten Menge frei erzeugt werden, Math. Z. 131 (1973), 241-249. MR 0360390 (50:12840)
  • [41] L. Lipshitz, The undecidability of the word problems for protective geometries and modular lattices, Trans. Amer. Math. Soc. 193 (1974), 171-180. MR 0364040 (51:295)

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DOI: https://doi.org/10.1090/S0002-9947-1980-0576864-X
Article copyright: © Copyright 1980 American Mathematical Society

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