On the arithmetic of projective coordinate systems

Herrmann, Christian

doi:10.1090/S0002-9947-1984-0743743-8

On the arithmetic of projective coordinate systems
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Trans. Amer. Math. Soc. 284 (1984), 759-785 Request permission

Abstract:

A complete list of subdirectly irreducible modular (Arguesian) lattices generated by a frame of order $n \geq 4\;(n \geq 3)$ is given. Also, it is shown that a modular lattice variety containing the rational projective geometries cannot be both finitely based and generated by its finite dimensional members.

References

Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
Benno Artmann, On coordinates in modular lattices with a homogeneous basis, Illinois J. Math. 12 (1968), 626–648. MR 236075
Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053

Algebraic theory of lattices

Modular lattices and projective geometry

A. Day, Equational theories of projective geometries, Contributions to lattice theory (Szeged, 1980) Colloq. Math. Soc. János Bolyai, vol. 33, North-Holland, Amsterdam-New York, 1983, pp. 277–316. MR 724266
Alan Day, A lemma on projective geometries as modular and/or Arguesian lattices, Canad. Math. Bull. 26 (1983), no. 3, 283–290. MR 703398, DOI 10.4153/CMB-1983-046-1
Alan Day and Douglas Pickering, The coordinatization of Arguesian lattices, Trans. Amer. Math. Soc. 278 (1983), no. 2, 507–522. MR 701508, DOI 10.1090/S0002-9947-1983-0701508-6
Ralph Freese, Breadth two modular lattices, Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973) Dept. Math., Univ. Houston, Houston, Tex., 1973, pp. 409–451. MR 0398926
Ralph Freese, Projective geometries as projective modular lattices, Trans. Amer. Math. Soc. 251 (1979), 329–342. MR 531987, DOI 10.1090/S0002-9947-1979-0531987-8
Ralph Freese, The variety of modular lattices is not generated by its finite members, Trans. Amer. Math. Soc. 255 (1979), 277–300. MR 542881, DOI 10.1090/S0002-9947-1979-0542881-0
Ralph Freese, Free modular lattices, Trans. Amer. Math. Soc. 261 (1980), no. 1, 81–91. MR 576864, DOI 10.1090/S0002-9947-1980-0576864-X
Ralph Freese and Bjarni Jónsson, Congruence modularity implies the Arguesian identity, Algebra Universalis 6 (1976), no. 2, 225–228. MR 472644, DOI 10.1007/BF02485830
K. D. Fryer and I. Halperin, The von Neumann coordinatization theorem for complemented modular lattices, Acta Sci. Math. (Szeged) 17 (1956), 203–249. MR 93488
László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
Christian Herrmann, On modular lattices generated by two complemented pairs, Houston J. Math. 2 (1976), no. 4, 513–523. MR 424632
Christian Herrmann, Rahmen und erzeugende Quadrupel in modularen Verbänden, Algebra Universalis 14 (1982), no. 3, 357–387 (German). MR 654403, DOI 10.1007/BF02483938
Christian Herrmann, On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983), no. 4, 513–527. MR 721885, DOI 10.1007/BF01455951
Christian Herrmann and András Huhn, Zum Wortproblem für freie Untermodulverbände, Arch. Math. (Basel) 26 (1975), no. 5, 449–453. MR 401562, DOI 10.1007/BF01229766

Lattices of normal subgroups generated by frames

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Christian Herrmann, Margarete Kindermann, and Rudolf Wille, On modular lattices generated by $1+2+2$, Algebra Universalis 5 (1975), no. 2, 243–251. MR 389695, DOI 10.1007/BF02485258
A. P. Huhn, Schwach distributive Verbände. I, Acta Sci. Math. (Szeged) 33 (1972), 297–305 (German). MR 337710
A. P. Huhn, On G. Grätzer’s problem concerning automorphisms of a finitely presented lattice, Algebra Universalis 5 (1975), 65–71. MR 392713, DOI 10.1007/BF02485232
George Hutchinson, Embedding and unsolvability theorems for modular lattices, Algebra Universalis 7 (1977), no. 1, 47–84. MR 441804, DOI 10.1007/BF02485417
Bjarni Jónsson, Distributive sublattices of a modular lattice, Proc. Amer. Math. Soc. 6 (1955), 682–688. MR 72851, DOI 10.1090/S0002-9939-1955-0072851-X
Bjarni Jónsson and George S. Monk, Representations of primary Arguesian lattices, Pacific J. Math. 30 (1969), 95–139. MR 258685
B. Jónsson, The class of Arguesian lattices is self-dual, Algebra Universalis 2 (1972), 396. MR 316325, DOI 10.1007/BF02945054
Fumitomo Maeda, Kontinuierliche Geometrien, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 95, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1958 (German). MR 0090579
Bernard R. McDonald, Geometric algebra over local rings, Pure and Applied Mathematics, No. 36, Marcel Dekker, Inc., New York-Basel, 1976. MR 0476639
Ralph McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1–43. MR 313141, DOI 10.1090/S0002-9947-1972-0313141-1
John von Neumann, Continuous geometry, Princeton Mathematical Series, No. 25, Princeton University Press, Princeton, N.J., 1960. Foreword by Israel Halperin. MR 0120174
R. Wille, On free modular lattices generated by finite chains, Algebra Universalis 3 (1973), 131–138. MR 329987, DOI 10.1007/BF02945112
R. Wille, Primitive subsets of lattices, Algebra Universalis 2 (1972), 95–98. MR 311524, DOI 10.1007/BF02945015