Skip to main content
SpringerLink
Log in

Modelling of Ring Geometry from von Neumann’s Point of View

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The idea to consider different unions of points, lines, planes, etc., is rather old. Many important configurations of such kinds are geometric (or matroidal) lattices. In this work, we study Desargues, Pappus, and Pasch configurations in D-semimodular lattices.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Gurtskaia and A. Lashkhi, “Classical configurations in D-semimodular lattices,” Bull. Georgian Acad. Sci., 170, No. 1, 33–36 (2004).

    MathSciNet  Google Scholar 

  2. S. Kemkhadze, “Geometries which are connected with general geometric lattices,” Bull. Georgian Acad. Sci., 158, No. 2, 183–184 (1998).

    MathSciNet  MATH  Google Scholar 

  3. S. Kemkhadze and Z. Rostomashvili, “General geometric lattices,” Bull. Georgian Acad. Sci., 158, No. 1, 14–17 (1998).

    MathSciNet  MATH  Google Scholar 

  4. T. Kvirikashvili and A. Lashkhi, “Affine geometry of modules over a ring with an invariant basis number,” J. Math. Sci., 137, No. 5, 5161–5173 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  5. A. A. Lashkhi, “Lattices with modular identity and Lie algebras,” J. Sov. Math., 38, No. 2, 1829–1853 (1987).

    Article  Google Scholar 

  6. A. Lashkhi, “Axiomatics of P1-projective geometry,” Soobshch. Akad. Nauk Gruz. SSR, 130, No. 1, 37–40 (1988).

    MathSciNet  MATH  Google Scholar 

  7. A. A. Lashkhi, “The fundamental theorem of projective geometry for modules and Lie algebras,” J. Sov. Math., 42, No. 5, 1991–2007 (1988).

    Article  MATH  Google Scholar 

  8. A. A. Lashkhi, “General geometric lattices and projective geometry of modules,” J. Math. Sci., 74, No. 3, 1044–1077 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Lashkhi, “Ring geometries and their related lattices,” Fundam. Prikl. Mat., 11, No. 3, 127–137 (2005).

    MathSciNet  MATH  Google Scholar 

  10. A. Lashkhi, “Coordinatization of PI-geometrical lattices,” Dokl. Ross. Akad. Nauk, 409, No. 2 (2006).

    Google Scholar 

  11. A. Lashkhi, P. Gurtskaia, and V. Sesadze, “Classical configurations in D-semimodular lattices,” Bull. Georgian Acad. Sci., 170, No. 1, 33–36 (2004).

    MathSciNet  Google Scholar 

  12. F. Maeda and S. Maeda, Theory of Symmetric Lattices, Grundlehren Math. Wiss., 173, Springer- Verlag, New York–Berlin (1970).

    Book  MATH  Google Scholar 

  13. J. von Neumann, Continuous Geometry, Princeton Univ. Press, Princeton, New Jersey (1960).

    MATH  Google Scholar 

  14. P. Pálfy and C. Szabo, “Congruence varieties of groups and Abelian groups,” in: Lattice Theory and Its Applications (Darmstadt, 1991), Res. Exp. Math., 23, Heldermann, Lemgo (1995), pp. 163–183.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Georgian Technical University, Tbilisi, Georgia

    P. Gurtskaia & A. Lashkhi

Authors
  1. P. Gurtskaia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Lashkhi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Gurtskaia.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 79, Proceedings of the International Conference “Modern Algebra and Its Applications” (Batumi, 2010), Part 2, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gurtskaia, P., Lashkhi, A. Modelling of Ring Geometry from von Neumann’s Point of View. J Math Sci 191, 757–763 (2013). https://doi.org/10.1007/s10958-013-1360-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1360-3

Keywords

Search

Navigation