Skip to main content
SpringerLink
Log in

Parallel axiom in convexity lattices

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

The convexity lattices, introduced by Bennett and Birkhoff, generalize the lattices of convex sets. We present three forms of Parallel Axiom in such lattices and define Euclidean and two classes of non-Euclidean lattices via the number of parallel lines through a point. The paper deals with these three classes of 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. M. Aigner,Combinatorial Theory, Springer Verlag, New York, 1979.MR 80h:05002

    MATH  Google Scholar 

  2. M. K. Bennett, Lattices of convex sets,Trans. AMS,234 (1977), 279–288.MR 57:3014

    Article  MATH  Google Scholar 

  3. M. K. Bennett, Convexity closure operators,Algebra Universalis,10 (1980), 345–354.MR 81c:54002

    Article  MATH  Google Scholar 

  4. M. K. Bennett, Affine geometry: a lattice characterization,Proc. AMS,88 (1983), 21–26.MR 84G:51016

    Article  MATH  Google Scholar 

  5. M. K. Bennett, [1985], Separation condition on convexity lattices,Universal Algebra and Lattice Theory, ed. by S. Comer, Springer, New York.MR 87k:06017

    Google Scholar 

  6. M. K. Bennett, Biatomic lattices,Algebra Universalis,24 (1987), 60–73.MR 89a:06021

    Article  MATH  Google Scholar 

  7. M. K.Bennett, and G.Birkhoff, [1983], A Peano axiom for convexity lattices, Calcutta Math. Soc. Diamond Jubilee Commemorative Volume, 33–43.MR 87k:06014

  8. M. K. Bennett, andG. Birkhoff, Convexity lattices,Algebra Universalis,20 (1985), 1–26.MR 87k:06013

    Article  MATH  Google Scholar 

  9. G. Grätzer,General Lattice Theory, Academic Press, New York, 1978.MR 80c:06001

    Google Scholar 

  10. D. Hilbert,Grundlagen der Geometrie, Verlag und Druck von B. G. Teubner, Leipzig und Berlin, 1930.

    MATH  Google Scholar 

  11. L. O. Libkin, andI. B. Muchnik, Hyperplanes and halfspaces in convexity lattices, Preprint, Mathematical Institute, Budapest, 1989.

    Google Scholar 

  12. A. B.Romanowska, and J. D. H.Smith,Modal Theory: An Algebraic Approach to Order, Geometry and Convexity, Heldermann Verlag, 1985.MR 96k:08001

  13. U. Sasaki, Lattice theoretic characterization of affine geometry of arbitrary dimension,Hiroshima Math. J. (A),16 (1952), 223–238.MR 15:674

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer and Information Science, University of Pennsylvania, 19104, Philadelphia, PA, USA

    L. O. Libkin

Authors
  1. L. O. Libkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Libkin, L.O. Parallel axiom in convexity lattices. Period Math Hung 24, 1–12 (1992). https://doi.org/10.1007/BF02651082

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02651082

Mathematics subject classification numbers

Key words and phrases

Search

Navigation