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General geometric lattices and projective geometry of modules

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Literature Cited

  1. E. Artin,Geometric algebra [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  2. K. I. Beydar, V. N. Latyushev, V. T. Markov, A. V. Mihalev, L. A. Skornyakov, and A. A. Tuganbayev, “Associative rings,” In:Algebra. Topologiya. Geometriya. Vol. 22, All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1984), pp. 3–115.

    Google Scholar 

  3. M. Berge,Geometry [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  4. G. Birkhoff,Structure Theory [Russian translation], IL, Moscow (1952).

    Google Scholar 

  5. G. Birkhoff,Lattice Theory [Russian translation], Nauka, Moscow (1984).

    Google Scholar 

  6. R. Baer,Linear Algebra and Projective Geometry [Russian translation], IL, Moscow (1955).

    Google Scholar 

  7. Gratzer,General Lattice Theory [Russian translation], Mir, Moscow (1982).

    Google Scholar 

  8. M. A. Javadov, “Projective and non-Eucliadean geometries over matrices,”Dokl. Akad. Nauk SSSR,97, 769–772 (1954).

    MathSciNet  Google Scholar 

  9. A. A. Lashkhi, “Fundamental theorem of affine geometry of Lie algebras and modules,” In:Problemy Geometrii, Vol. 18,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1986), pp. 165–187.

    Google Scholar 

  10. A. A. Lashkhi, “Projective geometry of Lie algebras and modules,” Doctoral thesis (1988).

  11. A. A. Lashkhi, “An axiomatization ofPI-projective geometry,”Dokl. Akad. Nauk GSSR,130, 37–40 (1988).

    MATH  MathSciNet  Google Scholar 

  12. A. A. Lashkhi, “Harmonic mappings of modules,”Mat. Zametki,47, 161–163 (1990).

    MATH  MathSciNet  Google Scholar 

  13. A. A. Lashkhi, “Projective geometry over the rings of principal ideals,” In:Mezhdunar. Algebraicheskaya Konf., Novosibirsk (1989), p. 77.

  14. A. A. Lashkhi, “Projective geometry of Lie algebras and modules,”Vestn. Mosk. Gos. Univ., Mat., Mekh., No. 6, 99 (1990).

    Google Scholar 

  15. A. A. Lashkhi, “General geometric lattices and their embeddings,” In:II Mezhdunar. Algebraicheskaya Konf., Barnaul (1991).

  16. V. T. Markov, A. V. Mihalev, L. A. Skornyakov, and A. A. Tuganbayev, “Module endomorphism rings and submodule lattices,” In:Algebra. Topologiya. Geometriya. Vol. 21,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1983), pp. 133–254.

    Google Scholar 

  17. A. V. Mihalev, “Module endomorphism rings and submodule structures,” In:Algebra. Topologiya. Geometriya. Vol. 12,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1975), pp. 51–71.

    Google Scholar 

  18. A. V. Mihalev and L. A. Skornyakov, “Modules,” In:Algebra. Topologiya. Geometriya, Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1968), pp. 58–100.

    Google Scholar 

  19. B. A. Rosenfeld, “Manifold geometry of the planes of projective space as pointwise geometry,” In:Tr. Seminara po Vectorn. i Tenzorn. Analizu,8, 219–222 (1950).

  20. B. A. Rosenfeld, “Projective geometries over quanternions and pseudoquanternions,”Dokl. Akad. Nauk SSSR,74, No. 3, 421–424 (1951).

    MathSciNet  Google Scholar 

  21. L. A. Skornyakov,Dedekind Structures with Complements and Regular Rings [in Russian], Nauka, Moscow (1961).

    Google Scholar 

  22. L. A. Skornyakov, “Rings,” In:Itogi Nauki i Tekhn. Geometriya. All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1965), pp. 59–79.

    Google Scholar 

  23. L. A. Skornyakov, “Modules,” In:Itogi Nauki i Tekhn. Algebra. Geometriya. All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1964), pp. 80–89.

    Google Scholar 

  24. L. A. Skornyakov and A. V. Mihalev, “Modules,” In:Algebra. Topologiya. Geometriya. Vol. 14,Itogi Nauki i Tekhn., All-Union Institute for Scientific and Technical Information (VINITI), Akad. Nauk SSSR, Moscow (1976), pp. 57–90.

    Google Scholar 

  25. T. S. Fofanova, “General lattice theory,” In:Uporyadochennye Mnozhestva i Reshetki, (Saratov), No.3, 22–40 (1975).

  26. M. Hall,Group Theory [Russian translation], IL, Moscow (1962).

    Google Scholar 

  27. A. I. Shishov and A. A. Nikitin,Algebraic Theory of Projective Planes [Russian translation], Novosibirsk (1987).

  28. Amemia Ichiro, “On the representation of complemented modular lattices,”J. Math. Soc. Jpn.,9, No. 2, 263–279 (1957).

    Google Scholar 

  29. Amemia Ichiro and I. Halperin, “Non-associative regular rings and von Neumann’s coordinatization theorem,”Abstr. Short. Communs. Intern. Math. Congr. Edinburg, (1958), pp. 11–12.

  30. E. Artin,Coordinates in Affine Geometry, Rep. Math. Colloq., Norte Dame, Indiana (1940).

  31. B. Artman, “Ein Structursatz für obene Verbande,”Arc. Math.,18, No. 1, 23–32 (1967).

    Google Scholar 

  32. B. Artman, “On coordinates in modular lattices,” III.J. Math.,12, No. 4, 626–648 (1968).

    Google Scholar 

  33. B. Artman, “Hjelmslev-eben und modulare Verbande,”Math. Z.,111, No. 1, 15–45 (1970).

    Google Scholar 

  34. B. Artman, “Geometric aspects of primary lattices,”Pacif. J. Math.,43, No. 1, 15–25 (1972).

    Google Scholar 

  35. R. Baer, “A unified theory of projective spaces and finite Abelian groups,”Trans. Amer. Math. Soc.,52, 283–343 (1942).

    MATH  MathSciNet  Google Scholar 

  36. R. Baer, “Null systems in projective space,”Bull. Amer. Math. Soc.,51, 903–906 (1945).

    MATH  MathSciNet  Google Scholar 

  37. D. Barbilian, “Zur Axiomatik der projectiven ebenen reing Geometrien, I, II,”Jahresbericht. DWV,50, 179–229 (1940);51, 34–76 (1941).

    MATH  MathSciNet  Google Scholar 

  38. G. Birkhoff, “Abstract linear dependence and lattices,”Amer. J. Math.,35, 800–804 (1935).

    MathSciNet  Google Scholar 

  39. G. Birkhoff, “Combinatorial restrictions in projective geometries,”Ann. Math.,36, 743–748 (1935).

    MATH  MathSciNet  Google Scholar 

  40. G. Birkhoff, “Von Neumann and lattice theory,”Bull. Amer. Math. Soc.,64, No. 3, 50–56 (1958).

    MATH  MathSciNet  Google Scholar 

  41. G. Birkhoff and O. Frink, “Representations of lattices by sets,”Trans. Amer. Math. Soc.,64, 299–316 (1948).

    MathSciNet  Google Scholar 

  42. L. Blumenthal and K. Menger,Studies in Geometry, Freeman, San Francisco (1970).

    Google Scholar 

  43. U. Brehm,Untermodulverbande Torsionfreier Modulen, Freiburg Br. (1983).

  44. U. Brehm, “coordinatization of lattices,” In:Rings and Geom. Reidel, (1985), pp. 511–550.

  45. H. H. Crapo and G. Rota, “Geometric lattices,” In:Trends in Lattice Theory, Van Nostrand-Reinhold, New York (1971), pp. 127–172.

    Google Scholar 

  46. P. Crawley and R. P. Dilworth,Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, New Jersey (1973).

    Google Scholar 

  47. A. Day, “Splitting algebras and a weak notion of projectivity,”Algebra Univ.,5, No. 2, 153–162 (1975).

    MATH  MathSciNet  Google Scholar 

  48. A. Day, “In search of a Pappian lattice identity,”Can. Math. Bull.,24, No. 2, 178–198 (1981).

    MathSciNet  Google Scholar 

  49. A. Day, “Geometrical applications in modular lattices,”Lect. Notes Math.,1004, 111–114 (1983).

    MATH  MathSciNet  Google Scholar 

  50. A. Day and D. Pickering, “The coordinatization of Arguesian lattices,”Trans. Amer. Math. Soc.,278, No. 2, 507–522 (1983).

    MathSciNet  Google Scholar 

  51. T. A. Dowlinc and R. M. Wilson, “The Slimmest geometric lattices,”Trans. Amer. Math. Soc.,196, 203–215 (1974).

    MathSciNet  Google Scholar 

  52. M.-L. Dubreil-Jacotin, L. Lesieur, and R. Croaset,Lecons Sur la Théorie des Treillis des Structures Algébriques Ordonnes et des Treilis Geometriques, Gauthiers-Villars, Paris (1953).

    Google Scholar 

  53. U. Faigle and C. Herrmann, “Projective geometry on partially ordered sets,”Trans. Amer. Math. Soc.,266, 319–322 (1981).

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  55. R. Freese, “The variety of modular lattices not generated by its finite members,”Trans. Amer. Math. Soc.,255, 277–300 (1980).

    MathSciNet  Google Scholar 

  56. E. Fried, G. Gratzer, and H. Lakser, “Amalgamation and weak injectives in the equational class of modular lattices,”Notices. Amer. Math. Soc.,18, 624 (1971).

    Google Scholar 

  57. O. Frink, “Complemented modular lattices and projective spaces of infinite dimension,”Trans. Amer. Math. Soc.,60, 452–467 (1964).

    MathSciNet  Google Scholar 

  58. K. Fryer and I. Halperin, “Coordinates in geometry,”Trans. Roy. Soc. Canada,48, 11–26 (1954).

    MathSciNet  Google Scholar 

  59. K. Fryer and I. Halperin, “On the coordinatization theorem of J. von Neumann,”Can. J. Math.,7, No. 4, 432–444 (1955).

    MathSciNet  Google Scholar 

  60. K. Fryer and I. Halperin, “The von Neumann coordinatization theorem for complemented modular lattices,”Acta Sci. Math.,17, No. 3–4, 203–249 (1958).

    MathSciNet  Google Scholar 

  61. S. Gorn, “On incidence geometry,”Bull. Amer. Math. Soc.,46, 158–167 (1940).

    MATH  MathSciNet  Google Scholar 

  62. G. Gratzer, H. Lasker, and B. Jonsson, “The amalgamation property in equational classes of modular lattices,”Pacif. J. Math.,45, 507–524 (1973).

    Google Scholar 

  63. I. Halperin, “On the coordinatization theory of J. von Neumann,”Can. J. Math.,7, 432–444 (1955).

    MATH  MathSciNet  Google Scholar 

  64. I. Halperin, “Kuntinuierliche Geometrien,”Bull. Amer. Math. Soc.,64, 386–390 (1958).

    Google Scholar 

  65. I. Halperin, “A simplified proof of von Neumann’s coordinatization theorem,”Proc. Natl. Acad. Sci. USA,47, No. 9, 1495–1498 (1961).

    MATH  MathSciNet  Google Scholar 

  66. I. Halperin, “Introduction to von Neumann algebras and continuous geometry,”Can. Math. Bull.,3, No. 3, 273–288 (1960).

    MATH  MathSciNet  Google Scholar 

  67. I. Halperin and I. Amemiya, “Coordinatization of complemented modular lattices,”Proc. Konikl. Nederl. Acad. Wet.,62, No. 2, 70–78 (1959).

    MathSciNet  Google Scholar 

  68. C. Herrmann, “On the equational theory of submodule lattices,” In:Proc. Houston Lattice Theory Conf. (1973), pp. 105–118.

  69. C. Herrmann, “On the arithmetic of projective coordinate systems,”Trans. Amer. Math. Soc.,284, No. 2, 759–785 (1973).

    MathSciNet  Google Scholar 

  70. A. Hoffman, “On the foundations of projective and affine geometry,”Trans. Amer. Math. Soc.,71, 218–242 (1951).

    MATH  MathSciNet  Google Scholar 

  71. Hsu Chen-Jung, “On lattice-theoretic characterization of the parallelism in affine geometry,”Ann. Math.,50, 1–7 (1949).

    Google Scholar 

  72. G. Hutchinson, “On the representation of lattices by modules,”Trans. Amer. Math. Soc.,209, No. 482, 311–351 (1975).

    MATH  MathSciNet  Google Scholar 

  73. G. Hutchinson, “On the class of lattices representable by modules,”Proc. Univ. Houston, Lattice Theory Conf., 69–94 (1977).

  74. E. Inaba, “On primary littices,”J. Fac. Sci. Hokkaido Univ.,11, 39–107 (1948).

    MathSciNet  Google Scholar 

  75. M. F. Janowitz,Projective Ideals and Congruence Relations, I, II, Techn. Reports Univ., New Mexico, No. 51, No. 63 (1964).

  76. M. F. Janowitz, “Section semicomplemented lattices,”Math. Z.,108, No. 1, 63–76 (1968).

    MATH  MathSciNet  Google Scholar 

  77. M. F. Janowitz, “Perspective properties of relatively complemented lattices,”Math. Z., 193–210 (1968).

  78. B. Jonsson, “On the representation of lattices,”Math. Scand.,1, No. 2, 193–206 (1953).

    MATH  MathSciNet  Google Scholar 

  79. B. Jonsson, “Modular lattices and the Desargues theorem,”Math. Scand.,2, No. 2, 295–314 (1954).

    MATH  MathSciNet  Google Scholar 

  80. B. Jonsson,Lattice-Theoretic Approach to Projective and Affine Geometry, Stud. Log. Amsterdam (1959).

  81. B. Jonsson, “Representations of modular lattices and relation algebra,”Trans. Amer. Math. Soc.,92, No. 3, 449–464 (1959).

    MATH  MathSciNet  Google Scholar 

  82. B. Jonsson, “Representations of complemented modules lattices,”Trans. Amer. Math. Soc.,97, No. 1, 64–94 (1960).

    MATH  MathSciNet  Google Scholar 

  83. B. Jonsson and G. Monk, “Representations of primary Arguesian lattices,”Pacif. J. Math.,30, 95–139 (1966).

    MathSciNet  Google Scholar 

  84. W. M. Kantor, “Characterization of finite projective and affine spaces,”Can. J. Math.,21, 64–75 (1969).

    MATH  MathSciNet  Google Scholar 

  85. I. Kaplansky, “Any orthocomplemented complete modular lattice in continuous geometry,”Ann. Mat.,61, No. 3, 524–541 (1955).

    MATH  MathSciNet  Google Scholar 

  86. A. N. Kolmogorov, “Zur Begründung der proektiven Geometrie,”Ann. Math.,33, 162–176 (1932).

    MathSciNet  Google Scholar 

  87. E. L. Lady, “On classifying torsion free modules over discrete valuation rings,”Lect. Notes Math.,616, 168–172 (1977).

    MATH  MathSciNet  Google Scholar 

  88. A. A. Lashkhi, “General geometric lattices and projective geometry of modules,”Bull. Acad. Sci. Georgian Rep. (1992) (in press).

  89. A. A. Lashkhi and T. M. Gelashvili, “Fundamental theorem of affine geometry for Lie algebras and modules,”Bull. Acad. Sci. Georgian Rep.,141, No. 2 (1991).

    Google Scholar 

  90. A. A. Lashkhi and T. M. Gelashvili, “Coset lattice and fundamental theorem of affine geometry,” In:Proc. Intern. Conf. Algebra, Barnaul (1991).

  91. H. H. Luck, “Projective Hjelmslevraume,”J. Reine Angew. Math.,243, 121–158 (1970).

    MathSciNet  Google Scholar 

  92. H. Lundenburg, “Über die Stractursätze der projektiven Geometrie,”Arch. Math.,17, 206–209 (1966).

    Google Scholar 

  93. R. C. Lyndon, “The representation of relational algebras. I, II,”Ann. Math.,51, 707–709 (1950).

    MATH  MathSciNet  Google Scholar 

  94. R. C. Lyndon, “Relational algebras and projective geometries,”Mich. Math. J.,8, No. 1, 21–28 (1961).

    MATH  MathSciNet  Google Scholar 

  95. S. MacLane, “Some interpretations of abstract linear dependence in terms of projective,”Amer. Math. J.,58, 236–240 (1936).

    MATH  MathSciNet  Google Scholar 

  96. S. MacLane, “A lattice formulation for transcendence degree andp-basis,”Duke Math. J.,4, 455–468 (1938).

    MATH  MathSciNet  Google Scholar 

  97. F. Maeda,Kontinuierliche Geometrien, Berlin-Gottingen-Heidelberg (1958).

  98. F. Maeda, “Lattice-theoretic characterization of abstract geometries,”J. Sci. Hiroshima Univ.,15, 87–98 (1956).

    Google Scholar 

  99. F. Maeda, “Matroid lattices of infinite length,”J. Sci. Hiroshima Univ.,15, 177–182 (1952).

    MATH  Google Scholar 

  100. F. Maeda, “Perspectivity of points in matroid lattices,”J. Sci. Hiroshima Univ.,28, 101–112 (1952).

    Google Scholar 

  101. F. Maeda,Theory of Symmetric Lattices, Springer-Verlag, New York (1970).

    Google Scholar 

  102. M. MakKay and G. McNulty, “Universal Horn axiom systems for lattices of submodules,”Algebra Univ.,7, No. 1, 25–31 (1977).

    Google Scholar 

  103. G. Markowsky and M. Petrich, “Subprojective lattices and projective geometry,”J. Algebra,48, No. 2, 305–320 (1977).

    MathSciNet  Google Scholar 

  104. G. Markowsky and M. Petrich, “Subprojective lattices projective geometry,”J. Algebra,48, No. 2, 305–320 (1977).

    MathSciNet  Google Scholar 

  105. K. Menger, “New foundations of projective and affine geometry,”Ann. Math.,37, 456–482 (1936).

    MATH  MathSciNet  Google Scholar 

  106. K. Menger, “The projective space,”Duke Math. J.,17, 1–14 (1950).

    Article  MATH  MathSciNet  Google Scholar 

  107. G. Monk, “Desargues law and the representation of primary lattices,”Pacif. J. Math.,30, No. 1, 175–186.

  108. J. Neumann, “Examples of continuous geometries,”Proc. Natl. Acad. Sci. USA,22, 101–108 (1936).

    MATH  Google Scholar 

  109. J. Neumann, “Algebraic theory of continuous geometries,”Proc. Natl. Acad. Sci. USA,23, 16–22 (1937).

    MATH  Google Scholar 

  110. J. Neumann,Continuous Geometry, Princeton Univ. Press, New York (1960).

    Google Scholar 

  111. J. Neumann and I. Halperin, “On the transitivity of perspective mappings,”Ann. Math.,41, 87–93 (1940).

    Google Scholar 

  112. U. Sasaki, “On an axiom of continuous geometry,”J. Sci. Hiroshima Univ.,14, 100–101 (1950).

    Google Scholar 

  113. U. Sasaki, “Lattice-theoretic characterization of geometries satisfying ‘Axiome der Verknüpfung’,”J. Sci. Hiroshima Univ.,16A, No. 3, 417–423 (1953).

    Google Scholar 

  114. U. Sasaki and S. Fujiwara, “The decomposition of a matroid lattice,”J. Sci. Hiroshima Univ.,15, 223–238 (1952).

    Google Scholar 

  115. C. M. Scoppola, “Sul reticolo di rando diverso da una caratterizzatione reticolare,”Rend. Semin. Math. Univ. Padova.,65, 205–211 (1981).

    MathSciNet  Google Scholar 

  116. B. Serge,Lectures on Modern Geometry, Roma-Gremonese (1961).

  117. W. Stephenson, “Lattice isomorphism between modules. I. Endomorphism rings,”J. London Math. Sci.,1, No. 1, 177–187 (1969).

    MATH  MathSciNet  Google Scholar 

  118. W. Stephenson,Characterization of Rings and Modules by Means of Lattices, Thesis, Univ. London (1969).

  119. F. Stevensen,Projective Planes, Freeman, San Francisco (1972).

    Google Scholar 

  120. W. T. Tutte, “Lectures on matroids,”J. Res. Nat. Sur. Standards Sect.,69, 1–47 (1965).

    MATH  MathSciNet  Google Scholar 

  121. O. Veblen and F. W. Young,Projective Geometry, Vols. 1–2, Boston (1918–1938).

  122. H. Whitney, “On the abstract properties of linear dependence,”Amer. J. Math.,57, 509–533 (1935).

    MATH  MathSciNet  Google Scholar 

  123. R. Wille, “Finite projective planes and equational classes of modular lattices,” In:Atti Coll. Intern. Teorie Combinatore, Roma (1973).

  124. R. Wille, “Verbandstheoretische Charakterisierungn-stufiger Geometrien,”Arch. Math.,18, 465–458 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  125. L. R. Wilox, “Modularity in Birkhoff lattices,”Bull. Amer. Math. Soc.,50, 135–138 (1944).

    MathSciNet  Google Scholar 

  126. R. J. Wilson, “An introduction to matroid theory,”Amer. Math. Month.,80, 500–525 (1973).

    MATH  Google Scholar 

  127. G. Pappus,Reticoli e Geometrie Finite, Libereria Editrice Liguori, Napoli (1952).

    Google Scholar 

  128. G. Pappus,Lezioni di Geometrie Moderna, Roma (1955).

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 8, Geometriya-1, 1993.

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Lashkhi, A.A. General geometric lattices and projective geometry of modules. J Math Sci 74, 1044–1077 (1995). https://doi.org/10.1007/BF02362832

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