Algebra universalis

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

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



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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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