Skip to main content
Log in

Alan Day's work on modular and Arguesian lattices

  • Published:
algebra universalis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Bruns, G. andRoddy, M.,A finitely generated modular ortholattice, Canad. Math. Bull.35 (1992), 29–33.

    Google Scholar 

  2. Cylke, A. A.,Perfect and linearly equivalent elements in modular lattices, to appear in Proc. Int. Conf. Representations of Algebras VI, Ottawa, 1992. V. Dlab and H. Lenzing eds.

  3. Czédli, G. andHutchinson, G.,An irregular Horn sentence in submodule lattices, Acta Sci. Math.51 (1987), 35–38.

    Google Scholar 

  4. Day, A.,Splitting algebras and a weak notion of projectivity, Algebra Universalis5 (1975), 153–162.

    Google Scholar 

  5. Day, A.,Equational theories of projective geometries, Contributions to Lattice Theory, Coll. Math. Soc. J. Bolyai, Szeged, vol. 33, 1980, pp. 277–316.

    Google Scholar 

  6. Day, A.,In search of a Pappian lattice identity, Canadian Math. Bull.24 (1981), 187–198.

    Google Scholar 

  7. Day, A.,Geometrical applications in modular lattices, Universal Algebra and Lattice Theory (R. Freese and O. Garcia, eds.), Lecture Notes in Mathematics, vol. 1004, Springer-Verlag, New York, 1983, pp. 111–141.

    Google Scholar 

  8. Day, A.,A lemma on projective geometries as modular and/or Arguesian lattices, Canadian Math. Bull.26 (1983), 283–290.

    Google Scholar 

  9. Day, A.,A note on Arguesian lattices, Arch. Math. (Brno)19 (1983), 117–123.

    Google Scholar 

  10. Day, A.,On some geometric properties defining classes of rings and varieties of modular lattices. Algebra Universalis17 (1983), 21–33.

    Google Scholar 

  11. Day, A.,Applications of coordinatization in modular lattice theory: the legacy of J. von Neumann, Order1 (1985), 295–300.

    Google Scholar 

  12. Day, A.,Some independence results in the coordinutizution of Arguesian lattices, Universal Algebra and Lattice Theory, S. Comer, ed., Lecture Notes in Mathematics, vol. 1140, Springer-Verlag, New York, 1985, pp. 37–45.

    Google Scholar 

  13. Day, A.,Dimension equations in modular lattices, Algebra Universalis22 (1986), 14–26.

    Google Scholar 

  14. Day, A.,The lattice theory of functional dependencies and normal decompositions, Inter. J. Algebra and Computation2 (1992), 409–431.

    Google Scholar 

  15. Day, A.,A lattice interpretation of database dependencies, Semantics of Programming languages and model theory, M. Droste and Yu. Gurevich eds., Gordon and Breach, London.

  16. Day, A. andFreese, R.,The role of gluing in modular lattice theory. The Dilworth Theorems, Selected Papers of Robert P. Dilworth (K. Bogart, R. Freese and J. Kung, eds.), Birkhauser, Basel, 1990, pp. 251–260.

    Google Scholar 

  17. Day, A., Herrmann, C. andWille, R.,On modular lattices with four generators, Algebra Universalis2 (1972), 317–323.

    Google Scholar 

  18. Day, A. andHerrmann, C.,Gluings of modular lattices, Order5 (1988), 85–101.

    Google Scholar 

  19. Day, A., Herrmann, C., Jónsson. B., Nation, J. B. andPickering, D.,Small non-Arguesian lattices, Algebra Universalis31 (1994), 66–94.

    Google Scholar 

  20. Day, A. andJónsson, B.,The structure of non-Arguesian lattices, Bull. Amer. Math Soc.13 (1985), 157–159.

    Google Scholar 

  21. Day, A. andJónsson, B.,A structural characterization of non-Arguesian lattices, Order2 (986), 335–350.

  22. Day, A. andJónsson, B.,Non-Arguesian configurations in a modular lattice, Acta Sci. Math. (Szeged)51 (1987), 309–318.

    Google Scholar 

  23. Day, A. andJónsson, B.,Non-Arguesian configurations and gluings of modular lattices, Algebra Universalis26 (1989), 208–215.

    Google Scholar 

  24. Day, A. andKiss, E.,Frames and rings in congruence modular varieties. Journal of Algebra109 (1987), 479–507.

    Google Scholar 

  25. Day, A. andPickering, D.,A note on the Arguesian lattice identity, Proc. Visegrad Conference on Universal Algebra (1982).

  26. Day, A. andPickering, D.,The coordinatization of Arguesian lattices, Trans. Amer. Math. Soc.278 (1983), 507–522.

    Google Scholar 

  27. Day, A. andStephan, J.,Congruence relations on incidence structures, preprint 1990.

  28. Day, A. andTschantz, S.,Lattice identities between type I and the Arguesian law, preprint 1985.

  29. Dlab, V. andRingel, C. M.,Perfect elements in free modular lattices, Math. Ann.247, 95–100.

  30. Freese, R.,Breadth 2 modular lattices, Proc. Conf. Lattice Theory, Houston, 1973, 409–451.

  31. Freese, R.,Some varieties of modular lattices not generated by their finite dimensional members, in Coll. Math. Soc. J. Bolyai17, Contributions to Universal Algebra, B. Csákany and J. Schmidt ed., North-Holland Publ. Co., Amsterdam, 1977, pp. 133–134.

    Google Scholar 

  32. Freese, R.,Projective geometries as projective modular lattices, Trans. Amer. Math. Soc.251 (1979), 329–342.

    Google Scholar 

  33. Freese, R.,Finitely based modular congruence varieties are distributive, Algebra Universalis32 (1994), 104–114.

    Google Scholar 

  34. Freese, R. andJónsson, B.,Congruence modularity implies the Arguesian identity, Algebra Universalis6 (1976), 225–228.

    Google Scholar 

  35. Freese, R. andMcKenzie, R.,Commutator Theory for Congruence Modular Varieties, London Math. Soc. LN125, London, 1987.

  36. Gelfand, I. M. andPonomarev, V. A.,Problems of linear algebra and classification of quadruples of subspaces in a finite dimensional vector space, Coll. Math. Soc. J. Bolyai5, Hilbert Space Operators, Tihany 1970; B.Sz.-Nagy (ed.), North-Holland, Amsterdam, 1972, 163–237.

    Google Scholar 

  37. Gelfand, I. M. andPonomarev, V. A.,Free modular lattices and their representation, Russian Math. Surveys29 (1974), 1–56.

    Google Scholar 

  38. Greferath, M.,Zur Hyperebenenkoordinatisierung in Desarguesschen projektiven Verbands-geometrien, J. Geometry42 (1991), 100–108.

    Google Scholar 

  39. Gross, H., Herrmann, C. andMoresi, R.,The classification of subspaces in Hermitean vector spaces, J. Algebra105 (1987), 516–541.

    Google Scholar 

  40. Gumm, H. P. andHerrmann, C.,Algebras in modular varieties: Baer refinements, cancellation,and isotopy, Houston J. Math.5 (1979), 503–523.

    Google Scholar 

  41. Haiman, M.,Proof theory of linear lattices, Adv. in Math.58 (1985), 209–242.

    Google Scholar 

  42. Haiman, M.,Arguesian lattices which are not type-I, Algebra Universalis28 (1991), 128–137.

    Google Scholar 

  43. Herrmann, C.,On modular lattices generated by two complemented pairs, Houston J. Math.2 (1976), 513–523.

    Google Scholar 

  44. Herrmann, C.,A parameter for subdirectly irreducible modular lattices with four generators, Acta Sci. Math.43 (1981), 169–179.

    Google Scholar 

  45. Herrmann, C.,Rahmen und erzeugende Quadrupel in modularen Verbänden, Algebra Universalis4 (1982), 357–387.

    Google Scholar 

  46. Herrmann, C.,On the word problem for modular lattices with four free generators. Math. Ann.265 (1983), 513–527.

    Google Scholar 

  47. Herrmann, C.,On the arithmetic of projective coordinate systems, Trans. Amer. Math. Soc.284 (1984), 759–785.

    Google Scholar 

  48. Herrmann, C.,Frames of permuting equivalences, Acta Sci. Math.51 (1987), 93–101.

    Google Scholar 

  49. Herrmann, C.,On contractions of vectorial lattice representations, Order8 (1991), 275–281.

    Google Scholar 

  50. Herrmann, C.,On the undecidability of implications between embedded multivalued database dependencies, to appear in Information and Computation.

  51. Herrmann, C., Pickering, D. andRoddy, M.,Geometric description of modular lattices, Algebra Universalis31 (1994), 365–396.

    Google Scholar 

  52. Herrmann, C. andWild, M.,Acyclic modular lattices and their representations, J. Algebra136 (1991), 17–36.

    Google Scholar 

  53. Huhn, A. P.,Two notes on n-distributive lattices, Coll. Math. Soc. J. Bolyai14, Lattice Theory, A. P. Huhn and E. T. Schmidt eds., Budapest, 1976, pp. 137–147.

  54. Jónssón, B.,On the representation of lattices. Math. Scand.I (1953), 193–206.

    Google Scholar 

  55. Jónsson, B.,Arguesian lattices of dimension n ≤ 4, Math. Scand.7(1959), 133–145.

    Google Scholar 

  56. Jónsson, B. andMonk, G.,Representation of primary Arguesian lattices, Pacific J. Math.30 (1969), 95–130.

    Google Scholar 

  57. Jónsson, B. andNation, J. B.,A report on sublattices of a free lattice, Coll. Math. Soc. J. Bolyai17, Contributions to Universal Algebra, B. Csákany and J. Schmidt eds., North Holland, Amsterdam, 1977, pp. 223–257.

    Google Scholar 

  58. Kalmbach, G.,Orthomodular Lattices, -Academic Press, London, 1983.

    Google Scholar 

  59. Kanellakis, P. C.,Elements of relational database theory, inHandbook of Theoretical Computer Science, vol. B: Formal Methods and Semantics, J. van Leuwen ed., Eisevier, Amsterdam, 1990, pp. 1073–1156.

    Google Scholar 

  60. Kurinnoi, G. ch.,A necessary condition for the isometric embeddability of finite dimensional modular lattices into complemented modular lattices (Russian), Ordered sets and lattices (Saratov University)5(1978), 67–76.

    Google Scholar 

  61. McKenzie, R.,Equational bases and non-modular lattice varieties, Trans. Amer. Math. Soc.174 (1972), 1–43.

    Google Scholar 

  62. Mitschke, A. andWille, R.,Über freie modulare Verbände FM q D M, inContributions to General Algebra, H. Kautschitsch et al. eds., Verlag J. Heyn, Klagenfurt, 1979, pp. 219–233.

    Google Scholar 

  63. Nation, J. B. andPickering, D.,Arguesian lattices whose skeleton is a chain, Algebra Universalis24 (1987), 91–100.

    Google Scholar 

  64. V. Neumann, J.,Continuous Geometry, Princeton Univ. Press, Princeton, 1960.

    Google Scholar 

  65. Pickering, D.,Minimal non-Arguesian lattices, Ph.D. Thesis, University of Hawaii, 1984.

  66. Roddy, M.,Varieties of modular ortholattices, Order3 (1987), 405–426.

    Google Scholar 

  67. Schmidt, E. T.,On finitely generated simple modular lattices, Per. Math. Hung.6 (1975), 213–216.

    Google Scholar 

  68. Schmidt, E. T.,On splitting modular lattices, Coll. Math. Soc. J. Bolyai29, Universal Algebra, North Holland, Amsterdam, pp. 697–703.

  69. Schmidt, E. T.,Remarks on finitely projected modular lattices, Acta Sci. Math.41 (1979), 187–190.

    Google Scholar 

  70. Schmidt, E. T.,On finitely projected modular lattices, Acta Math. Acad. Sci. Hung.38 (1981), 45–51.

    Google Scholar 

  71. Schmidt, S. E. andGreferath, M.,A unified approach to projective lattice geometries, Geometriae Dedicata43 (1992), 243–264.

    Google Scholar 

  72. Wild, M.,On the congruence problem in infinite dimensional sesquilinear spaces, Order7 (1991), 387–400.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herrmann, C. Alan Day's work on modular and Arguesian lattices. Algebra Universalis 34, 35–60 (1995). https://doi.org/10.1007/BF01200489

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01200489

Keywords

Navigation