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The Imbedding Problem for Modular Lattices

  • M. Hall
  • R. P. Dilworth
Part of the Contemporary Mathematicians book series (CM)

Abstract

It is trivially true that an arbitrary lattice may be imbedded in a complemented lattice. We need only adjoin a unit and null elements if they do not already exist and a single element which is a complement of each of the elements not the unit of null element. For distributive lattices, the imbedding problem is not so trivial, but is contained in the representation theorem which asserts that any distributive lattice is isomorphic with a ring of sets (Birkhoff (1), Mac Neille (1)). The corresponding problem of imbedding a modular lattice in a complemented modular lattice is an outstanding problem in lattice theory. We exhibit here an example of a modular lattice which cannot be imbedded in any complemented modular lattice. However we will be concerned primarily with the isometric problem for finite dimensional modular lattices; that is, the problem of imbedding a finite dimensional modular lattice in a complemented modular lattice of the same dimensionality.

Keywords

Distributive Lattice Projective Geometry Projective Lattice Subdirect Product Modular Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    G. Birkhoff, Lattice Theory, Amer. Math. Soc. Coll. Pub., vol. 25.Google Scholar
  2. 2.
    R.P. Dilworth, The arithmetical theory of Birkhoff lattices, Duke Math. Journal vol. 8 (1941) pp. 286–299.CrossRefGoogle Scholar
  3. 3.
    On dependence relations in a semi-modular lattice, submitted to the Duke Math. Journal.Google Scholar
  4. 4.
    H.M. MacNeille, Partially ordered sets, Trans. Amer. Math. Soc. vol. 42 (1937) pp. 416–460.CrossRefGoogle Scholar
  5. 5.
    Veblen and Young, Projective Geometry, vol. I.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • M. Hall
    • 1
  • R. P. Dilworth
    • 1
  1. 1.Yale UniversityUSA

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