Algebra universalis

, Volume 70, Issue 2, pp 163–174 | Cite as

A review of some of Bjarni Jónsson’s results on representation of arguesian lattices

  • Christian Herrmann


We review (and slightly extend) Bjarni Jónsson’s results on representations of arguesian lattices that are complemented, of low height, or of simple gluing structure.

2010 Mathematics Subject Classification

Primary: 06C05 Secondary: 06C20 

Key words and phrases

arguesian lattice complemented lattice commuting equivalences subspace lattice 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Birkhoff G.: Lattice Theory, 3rd edn. Amer. Math. Soc., Providence (1967)MATHGoogle Scholar
  2. 2.
    Călugăreanu G., Conţiu C.: Type I representable lattices of dimension at most 4. Algebra Universalis 67, 131–139 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Călugăreanu, G., Conţiu, C.: Corrigenda et Addenda: On type I representable lattices of dimension at most 4. Algebra Universalis, to appearGoogle Scholar
  4. 4.
    Cherlin G.: Model Theoretic Algebra. Selected Topics. Lecture Notes in Mathematics vol. 521. Springer, Berlin (1976)MATHGoogle Scholar
  5. 5.
    Cohn, P.M.: Skew fields. Theory of general division rings. Encyclopedia of Mathematics and its Applications vol. 57. Cambridge University Press, Cambridge (1995)Google Scholar
  6. 6.
    Crawley, P., Dilworth, R.P.: Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs (1973) C. Herrmann Algebra Univers.Google Scholar
  7. 7.
    Czédli G., Hutchinson G.: An irregular Horn sentence in submodule lattices. Acta Sci. Math. (Szeged) 51, 35–38 (1987)MathSciNetMATHGoogle Scholar
  8. 8.
    Day A., Herrmann C., Jónsson B., Nation J.B., Pickering D.: Small non-arguesian lattices. Algebra Universalis 31, 66–94 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Freese R.: Projective geometries as projective modular lattices. Trans. Amer. Math. Soc. 251, 329–342 (1979)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Freese R.: Finitely based modular congruence varieties are distributive. Algebra Universalis 32, 104–114 (1994)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Freese R., Herrmann C., Huhn A.: On some identities valid in modular congruence varieties. Algebra Universalis 12, 322–334 (1981)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Freese R., Jónsson B.: Congruence modularity implies the Arguesian identity. Algebra Universalis 6, 225–228 (1976)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Freese R., McKenzie R.: Commutator theory for congruence modular varieties. London Math. Soc. Lecture Note Series vol.125. Cambridge University Press, Cambridge (1987)MATHGoogle Scholar
  14. 14.
    Guidici, L.: Dintorni del teorema di coordinatizzatione di von Neumann. Tesi di Dottorato, Univ. degli studi di Milano (1995).
  15. 15.
    Hagemann J., Herrmann C.: A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity. Arch. Math. (Basel) 32, 234–245 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Haiman M.: Arguesian lattices which are not type-1. Algebra Universalis 28, 128–137 (1991)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Herrmann C., Huhn A.P.: Zum Begriff der Characteristik modularer Verbände. Math. Z. 44, 185–194 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Herrmann, C., Huhn, A.P.: Lattices of normal subgroups which are generated by frames. In: Lattice theory (Szeged 1974). Colloq. Math. Soc. Janos Bolyai, vol. 14, pp. 97–136. North-Holland, Amsterdam (1976)Google Scholar
  19. 19.
    Herrmann C., Takách G.: A characterization of subgroup lattices of finite abelian groups. Beitr. Algebra Geom. 46, 215–239 (2005)MATHGoogle Scholar
  20. 20.
    Hutchinson G., Czédli G.: A test for identities satisfied in lattices of submodules. Algebra Universalis 8, 269–309 (1978)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jónsson B.: On the representation of lattices. Math. Scand. 1, 193–206 (1953)MathSciNetMATHGoogle Scholar
  22. 22.
    Jónsson B.: Modular lattices and Desargues’ Theorem. Math. Scand. 2, 295–314 (1954)MathSciNetMATHGoogle Scholar
  23. 23.
    Jónsson B.: Arguesian lattices of dimension n ≤  4. Math. Scand. 7, 131–145 (1954)Google Scholar
  24. 24.
    Jónsson B.: Representations of complemented modular lattices. Trans. Amer. Math. Soc. 97, 64–94 (1960)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Jónsson B., Monk G.S.: Representations of primary Arguesian lattices. Pacific J. Math. 30, 95–139 (1969)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Monk G.S.: Desargues’ law and the representation of primary lattices. Pacific J. Math. 30, 175–186 (1969)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Nation J.B., Pickering D.: Arguesian lattices whose skeleton is a chain. Algebra Universalis 24, 91–199 (1987)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Pálfy P.P., Szabó Cs.: An identity for subgroup lattices of abelian groups. Algebra Universalis 33, 191–195 (1995)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tesler, G.P.: Semi-primary lattices and tableau algorithms. PhD thesis, M.I.T. (1995).

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.TUD FB4DarmstadtGermany

Personalised recommendations