Advertisement

Algebra universalis

, Volume 72, Issue 4, pp 349–357 | Cite as

On varieties of modular ortholattices that are generated by their finite-dimensional members

  • Christian Herrmann
  • Micheale S. Roddy
Article
  • 61 Downloads

Abstract

We prove that the following three conditions on a modular ortholattice L with respect to a given variety of modular ortholattices, \({\mathcal{V}}\), are equivalent: L is in the variety of modular ortholattices generated by the finite-dimensional members of \({\mathcal{V}}\); L can be embedded in an atomisticmember of \({\mathcal{V}}\); L has an orthogeometric representation in an anisotropic orthogeometry \({{(Q,\bot), \,\,{\rm where}\,\, [0, u] \in \mathcal{V} \,\,{\rm for\,\, all}\,\, u \in L_{\rm fin}(Q)}}\).

2010 Mathematics Subject Classification

Primary: 06C15 Secondary: 06C20 08B15 51A50 

Key words and phrases

modular ortholattice anisotropic orthogeometry variety 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baer R.: Polarities in finite projective planes, Bull. Amer. Math. Soc. 51, 77–93 (1946)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Birkhoff G, von Neumann J.: The logic of quantum mechanics, Ann. of Math. 37, 823–843 (1936)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bruns G.: Varieties of modular ortholattices, Houston J. Math. 9, 1–7 (1983)MathSciNetMATHGoogle Scholar
  4. 4.
    Faure C.A., Frölicher A.: Modern Projective Geometry. Kluwer, Dordrecht (2000)CrossRefMATHGoogle Scholar
  5. 5.
    Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)Google Scholar
  6. 6.
    Harding J.: Decidability of the equational theory of the continuous geometry CG(\({\mathbb{F}}\)), J. Philos. Logic 42, 461–465 (2013)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Herrmann C.: On representations of complemented modular lattices with involution, Algebra Universalis 61, 339–364 (2009)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Herrmann C.: On the equational theory of projection lattices of finite von Neumann factors, J. Symb. Logic 75, 1102–1110 (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Herrmann C., Roddy M.S.: Proatomic modular ortholattices: Representation and equational theory, Note di matematica e fisica 10, 55–88 (1999)Google Scholar
  10. 10.
    Herrmann C., Roddy M.S.: A note on the equational theory of modular ortholattices, Algebra Universalis 44, 165–168 (2000)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Herrmann C., Roddy M.S.: Three ultrafilters in a modular logic, Internat. J. Theoret. Phys. 50, 3821–3827 (2011)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Herrmann C., Roddy M.S.: On geometric representations of modular ortholattices, Algebra Universalis 71, 285–297 (2014)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Holland Jr., S.S.: The current interest in orthomodular lattices. In: Abbott, J.C. (ed.) Trends in Lattice Theory, pp. 41–126. van Norstrand, New York (1970)Google Scholar
  14. 14.
    Kalmbach, G: MR0699045 (84j:06010)Google Scholar
  15. 15.
    von Neumann J.: Continuous geometries and examples of continuous geometries. Proc. Nat. Acad. Acad. Sci. U.S.A 22, 707–713 (1936)CrossRefGoogle Scholar
  16. 16.
    Roddy M. S.: Varieties of modular ortholattices, Order 3, 405–426 (1987)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.TUD FB4DarmstadtGermany
  2. 2.Dept. of Mathematics and Computer ScienceBrandon UniversityBrandonCanada

Personalised recommendations