Advertisement

Algebraic Aspects of Orthomodular Lattices

  • Gunter Bruns
  • John Harding
Chapter
Part of the Fundamental Theories of Physics book series (FTPH, volume 111)

Abstract

In this survey article we try to give an up-to-date account of certain aspects of the theory of ortholattices (abbreviated OLs), orthomodular lattices (abbreviated OMLs) and modular ortholattices (abbreviated MOLs), not hiding our own research interests. Since most of the questions we deal with have their origin in Universal Algebra, we start with a section discussing the basic concepts and results of Universal Algebra without proofs. In the next three sections we discuss, mostly with proofs, the basic results and standard techniques of the theory of OMLs. In the remaining five sections we work our way to the border of present day research, with no or only sketchy proofs. Section 5 deals with products and subdirect products, section 6 with free structures and section 7 with classes of OLs defined by equations. In section 8 we discuss embeddings of OLs into complete ones. The last section deals with questions originating in Category Theory, mainly amalgamation, epimorphisms and monomorphisms. The later sections of this paper contain an abundance of open problems. We hope that this will initiate further research.

Keywords

Boolean Algebra Congruence Lattice Subdirect Product Orthomodular Lattice Algebraic Aspect 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Adams, D.H. (1969) The completion by cuts of an orthocomplemented mod-ular lattice, Bulletin of the Australian Mathematical Society 1, 279–280.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Amemiya, I. and Araki, H. (1966) A remark on Piron’s paper, Publications of the Research Institute of Mathematical Sciences Kyoto University A 2, 423–429.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Baer, R. (1946) Polarities in finite projective planes, Bulletin of the American Mathematical society 52, 77–93.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Balbes, D.R. and Dwinger, P. (1974) Distributive Lattices, University of Mis-souri Press, Missouri.zbMATHGoogle Scholar
  5. [5]
    Banaschewski, B. (1956) Hiillensysteme und erweiterungen von quasi-ordnungen, Zeitschrift fur Mathematische Logik und Grundlagen der Mathe-matik 2, 35–46.MathSciNetGoogle Scholar
  6. [6]
    Beran, L. (1984) Orthomodular Lattices, Algebraic Approach, Academia, Prague — D. Reidel, Dordrecht.Google Scholar
  7. [7]
    Birkhoff, G. (1967) Lattice Theory, Third edition, American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R. I.zbMATHGoogle Scholar
  8. [8]
    Bruns, G. (1976) Free ortholattices, Canadian Journal of Mathematics 5, 977–985.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Bruns, G. (1983) Varieties of modular ortholattices, Houston Journal of Mathematics 9, 1–7.MathSciNetADSzbMATHGoogle Scholar
  10. [10]
    Bruns, G. and Harding, J. (1998) Amalgamation of ortholattices, Order 14, 193–209.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Bruns, G. and Harding, J. (1998) Epimorphisms in certain varieties of algebras, Submitted to Order (October 1998 ).Google Scholar
  12. [12]
    Bruns, G. and Kalmbach, G. (1972) Varieties of orthomodular lattices II, Canadian Journal of Mathematics Vol. XXIV 2, 328–337.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Bruns, G. and Kalmbach, G. (1973) Some remarks on free orthomodular lattices, Proceedings of the Lattice Theory Conference, Houston 397–408.Google Scholar
  14. [14]
    Bruns G. and Roddy, M. (1992) A finitely generated modular ortholattice, Canadian Bulletin of Mathematics 35, 29–33.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Bruns G. and Roddy, M. (1994) Projective orthomodular lattices, Canadian Bulletin of Mathematics 37, 145–153.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Bruns G. and Roddy, M. (1997) Projective orthomodular lattices II, Algebra Universalis 37, 147–153.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Burris, S. and Sankappanavar, H.P. (1981) A Course in Universal Algebra,Springer-Verlag.Google Scholar
  18. [18]
    Crawley, P. and Dilworth, R.P. (1973) Algebraic Theory of Lattices,Prentice Hall.Google Scholar
  19. [19]
    Crown, G.D., Harding, J. and Janowitz, M.F. (1996) Boolean products of lattices, Order 13, 175–205.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Dichtl, M. (1984) Astroids and pastings, Algebra Universalis 18, 380–385.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Gehrke, M. and Harding, J. (1999) Bounded lattice expansions, Submitted to Transactions of the American Mathematical Society (August 1999 ).Google Scholar
  22. [22]
    Goldblatt, R.I. (1975) The Stone space of an ortholattice, Bulletin of the London Mathematical Society 7, 45–48.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Grätzer, G. (1978) General Lattice Theory,Academic Press.Google Scholar
  24. [24]
    Greechie, R.J. (1968) On the structure of orthomodular lattices satisfying the chain condition, Journal of Combinatorial Theory 4, 210–218.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Greechie, R.J. (1971) Orthomodular lattices admitting no states, Journal of Combinatorial Theory 10, 119–132.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Harding, J. (1991) Orthomodular lattices whose MacNeille completions are not orthomodular, Order 8, 93–103.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Harding, J. (1992) Irreducible orthomodular lattices which are simple, Algebra Universalis 29, 556–563.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Harding, J. (1993) Completions of orthomodular lattices II, Order 10, 283–294.MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. [29]
    Harding, J. (1993) Any lattice can be regularly embedded into the MacNeille completion of a distributive lattice, The Houston Journal of Mathematics 19, 39–44.MathSciNetzbMATHGoogle Scholar
  30. [30]
    Harding, J. (1998) Canonical completions of lattices and ortholattices, Tatra Mountains Mathematical Publications 15, 85–96.MathSciNetzbMATHGoogle Scholar
  31. [31]
    Kalmbach, G. (1977) Orthomodular lattices do not satisfy any special lattice equation, Archiv der Mathematik (Basel) 28, 7–8.MathSciNetCrossRefGoogle Scholar
  32. [32]
    Kalmbach, G. (1983) Orthomodular Lattices, Academic Press.Google Scholar
  33. [33]
    Kiss, E.W., Mirki, L., Pröhle, P. and Tholen, W. (1983) Categorical alge-braic properties, Compendium on amalgamation, congruence extension, epi-morphisms, residual smallness, and injectivity, Studia Scientiarum Mathe-maticarum Hungarica 18, 79–141.zbMATHGoogle Scholar
  34. [34]
    MacLaren, M.D. (1964) Atomic orthocomplemented lattices, Pacific Journal of Mathematics 14, 597–612.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    MacNeille, H.M. (1937) Partially ordered sets, Transactions of the American Mathematical Society 42, 416–460.MathSciNetCrossRefGoogle Scholar
  36. [36]
    Maeda, F. and Maeda, S. (1970) Theory of Symmetric Lattices, Springer.Google Scholar
  37. [37]
    Navara, M. (1997) On generating finite orthomodular sublattices, Tatra Mountains Mathematical Publications 10, 109–117.MathSciNetzbMATHGoogle Scholar
  38. [38]
    Ptak, P. and Pulmannova, S. (1991) Orthomodular Structures as Quantum Logics, Veda, Bratislava.zbMATHGoogle Scholar
  39. [39]
    Roddy, M. (1984) An orthomodular analogue of the Birkhoff-Menger theorem, Algebra Universalis 19, 55–60.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Roddy, M. (1987) Varieties of modular ortholattices, Order 3, 405–426.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    Roddy, M. (1989) On the word problem for orthocomplemented modular lat-tices, Canadian Journal of Mathematics Vol. XLI, 961–1004.MathSciNetCrossRefGoogle Scholar
  42. [42]
    Urquhart, A. (1978) A topological representation theory for lattices, Algebra Universalis 8, 45–58.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Gunter Bruns
    • 1
  • John Harding
    • 2
  1. 1.Department of MathematicsMcMaster UniversityHamiltonCanada
  2. 2.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

Personalised recommendations