Skip to main content
SpringerLink
Log in

Aspects of Finite Lattices

  • Conference paper
Higher Combinatorics

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 31))

Abstract

There has been increasing interest in finite lattices in the last few years. This fact is witnessed by the large number of papers in current journals and by the fact that various reports on specific topics in this area have already appeared, namely CRAPO & ROTA [8], RIVAL [35], JÓNSSON & NATION [23], and, of course, certain sections in BIRKHOFF [7] and CRAWLEY & DILWORTH [10]. The following report can be understood in this light; it describes topics in the theory of finite lattices from an algebraic point of view concerning research work done mainly by members of the Technische Hochschule Darmstadt.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Banaschewski, Hüllensysteme und Erweiterungen von Quasi-Ordnungen. Z.Math.Logik Grundlagen Math. 2 (1956), 117–130.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Birkhoff, On the combination of subalgebras. Proc. Cambridge Phil. Soc. 29 (1933), 441–464.

    Article  Google Scholar 

  3. G. Birkhoff, On the structure of abstract algebras. Proc. Cambridge Phil. Soc. 31 (1935), 433–454.

    Article  Google Scholar 

  4. G. Birkhoff, Rings of sets. Duke Math.J. 3. (1937), 442–454.

    Google Scholar 

  5. G. Birkhoff, Subdirect unions in universal algebras. Bull. Amer.Math.Soc. 50 (1944), 764–768.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Birkhoff, Lattice theory. 2nd ed. Providence: Amer.Math. Soc. 1948.

    MATH  Google Scholar 

  7. G. Birkhoff, Lattice theory. 3rd ed. Providence: Amer.Math. Soc. 1967.

    MATH  Google Scholar 

  8. H.H. Crapo, G.-C. Rota: On the foundations of combinatorial theory: Combinatorial geometries (preliminary edition). Cambridge-London: M.I.T. Press 1970.

    MATH  Google Scholar 

  9. P. Crawley, Decomposition theory of nonsemimodular lattices. Trans.Amer.Math.Soc. 99 (1961), 246–254.

    Article  MathSciNet  Google Scholar 

  10. P. Crawley, R.P. Dilworth, Algebraic theory of lattices. Englewood Cliffs, Prentice Hall 1973.

    MATH  Google Scholar 

  11. A. Day, C. Herrmann, R. Wille, On modular lattices with four generators. Alg. Universalis 2 (1972), 317–323.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Dedekind, Über die von drei Moduln erzeugte Dualgruppe. Math.Ann. 53 (1900), 371–403.

    Article  MathSciNet  MATH  Google Scholar 

  13. R.P. Dilworth, Lattices with unique irreducible decompositions. Ann.Math. 41 (1940), 771–777.

    Article  MathSciNet  Google Scholar 

  14. D. Dorninger, Modulare Polynomverbände über endlichen distributiven Verbänden. Monatsh.Math. 78 (1974), 305–310.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Dorninger, J. Wiesenbauer, Anzahlsätze für Polynomfunktionen auf Verbänden. Preprint.

    Google Scholar 

  16. B. Ganter, W. Poguntke, R. Wille: Finite sublattices of four-generated modular lattices. Preprint.

    Google Scholar 

  17. G. Gierz, K. Keimel, Topologische Darstellung von Verbänden. Math.Z. 150 (1976), 83–99.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. Grätzer, Lattice theory, First concepts and distributive lattices. San Francisco, Freeman 1971.

    Google Scholar 

  19. C. Herrmann, On the equational theory of submodule lattices. In: Proc. Lattice Theory Conf. Houston 1973, pp. 105–118.

    Google Scholar 

  20. C. Herrmann, M. Kindermann, R. Wille: On modular lattices generated by 1+2 + 2. Alg. Universalis 5 (1975), 243–251.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Herrmann, C.M. Ringel, R. Wille: On modular lattices with four generators. Not.Amer.Math.Soc. 20 (1973), A-418; 73T–A151.

    Google Scholar 

  22. B. Jónsson, Algebras whose congruence lattices are distributive. Math. Scand. 21 (1967), 110–121.

    MathSciNet  MATH  Google Scholar 

  23. B. Jónsson, J.B. Nation, A report on sublattices of a free lattice. In: Contributions to Universal Algebra. Szeged: J. Bolyai Mat. Soc. (to appear).

    Google Scholar 

  24. D. Kelly, The 3-irreducible partially ordered sets. Canad. J.Math, (submitted).

    Google Scholar 

  25. F. Klein, Über einen Zerlegungssatz in der Theorie der abstrakten Verknüpfungen. Math. Ann. 106 (1932), 114–130.

    Article  MathSciNet  Google Scholar 

  26. A. Kurosch, Durchschnittsdarstellungen mit irreduziblen Komponenten in Ringen u. sogen. Dualgruppen. Mat.Sb. 42 (1935), 613–616.

    MATH  Google Scholar 

  27. H. Lausch, W. Nöbauer, Algebra of polynomials. Amsterdam- London: North-Holland 1973.

    MATH  Google Scholar 

  28. G. Markowsky, Combinatorial aspects of lattice theory with applications to the enumeration of free distributive lattices. Ph.D.Thesis, Harvard University 1973.

    Google Scholar 

  29. G. Markowsky, Some combinatorial aspects of lattice theory. In: Proc. Lattice Theory Conf. Houston 1973, pp. 36–68.

    Google Scholar 

  30. R. McKenzie, Equational bases and non–modular lattice varieties. Trans. Amer. Math. Soc. 174 (1972), 1–43.

    Article  MathSciNet  Google Scholar 

  31. A. Mitschke, R. Wille, Freie modulare Verbände FM(DM3). In: Proc. Lattice Theory Conf. Houston 1973, pp. 383–396.

    Google Scholar 

  32. O. Ore, On the foundation of abstract algebra II. Ann.Math. 37 (1936), 265–292.

    Article  MathSciNet  Google Scholar 

  33. H.A. Priestley, Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2 (1970), 186–190.

    Article  MathSciNet  MATH  Google Scholar 

  34. H.A. Priestley, Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. III. Ser. 24 (1972), 507–530.

    Article  MathSciNet  Google Scholar 

  35. I. Rival, Contributions to combinatorial lattice theory. Ph.D. Thesis, Winnipeg 1973.

    Google Scholar 

  36. H.L. Rolf, The free lattice generated by a set of chains. Pacific J. Math. 8 (1958), 585–595.

    MathSciNet  MATH  Google Scholar 

  37. H.L. Rolf, The free modular lattice, FM (2 + 2 + 2), is infinite. Proc.Amer. Math.Soc. 17 (1966), 960–961.

    MathSciNet  MATH  Google Scholar 

  38. E.T. Schmidt, Kongruenzrelationen algebraischer Strukturen. Math.Forschungsber. 25, Berlin: VEB Dt. Verlag d. Wiss. 1969.

    Google Scholar 

  39. J. Schmidt, Zur Kennzeichnung der Dedekind-MacNeillschen Hülle einer geordneten Menge. Arch. Math. J 7 (1956), 241–249.

    Article  MATH  Google Scholar 

  40. J. Schmidt, Boolean duality extended. In: Theory of sets and topology. A collection of papers in honour of Felix Hausdorff. Berlin: VEB Dt. Verlag d. Wiss. 1972, pp. 435–453.

    Google Scholar 

  41. M.P. Schützenberger, Construction du treillis modulaire engendré par deux éléments et une chaîne finie discrète. C.R. Acad. Sci. Paris 235 (1952), 926–928.

    MathSciNet  MATH  Google Scholar 

  42. D. Schweigert, Über endliche ordnungspolynomvollständige Verbände. Monatsh. Math. 78 (1974), 68–76.

    Article  MathSciNet  MATH  Google Scholar 

  43. Yu.I. Sorkin, Free unions of lattices. Mat. Sb. 30 (72) (1952), 677–694 (russian).

    Google Scholar 

  44. M.H. Stone, The theory of representations of Boolean algebras. Trans.Amer.Math.Soc. 122 (1936), 379–398.

    Google Scholar 

  45. H. Strietz, Finite partition lattices are four-generated. In: Proc. Lattice Theory Conf. Ulm 1975, 257–259.

    Google Scholar 

  46. H. Strietz, Über Erzeugendensysteme endlicher Partitions-verbände. Preprint.

    Google Scholar 

  47. K. Takeuchi, On free modular lattices II. Tôhoku Math. J. 11 (1959), 1–12.

    Article  MATH  Google Scholar 

  48. R.M. Thrall, D.G. Duncan: Note on free modular lattices. Amer.J.Math. 75. (1953), 627–632.

    Article  MathSciNet  MATH  Google Scholar 

  49. A. Urquhart, A topological representation theory for lattices. Preprint.

    Google Scholar 

  50. A.G. Waterman: The free lattice with 3 generators over N5. Port. Math. 26 (1967), 285–288.

    MathSciNet  MATH  Google Scholar 

  51. R. Wille, Primitive Länge und primitive Weite Lei modularen Verbänden. Math.Z. 108 (1969), 129–136.

    Article  MathSciNet  Google Scholar 

  52. R. Wille, Subdirekte Produkte und konjunkte Summen. J. reine angew. Math. 239/240 (1970), 333–338.

    MathSciNet  Google Scholar 

  53. R. Wille, On free modular lattices generated by finite chains. Alg. Universalis 3 (1973), 131–138.

    Article  MathSciNet  MATH  Google Scholar 

  54. R. Wille, Über modulare Verbände, die von einer endlichen halbgeordneten Menge frei erzeugt werden. Math.Z. 131 (1973), 241–249.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Wille, Über freie Produkte endlicher modularer Verbände. Abh. Math. Sem. Hamburg 45 (1976), 218–224.

    Article  MathSciNet  MATH  Google Scholar 

  56. R. Wille, Subdirekte Produkte vollständiger Verbände. J. reine angew. Math. 283/284 (1976), 53–70.

    Article  MathSciNet  Google Scholar 

  57. R. Wille, Finite projective planes and equational classes of modular lattices. In: Atti. Conv. Geom. Comb. Roma 1973, (to appear).

    Google Scholar 

  58. R. Wille, A note on simple lattices. In: Contributions to Lattice Theory. Szeged: J. Bolyai Mat. Soc. (to appear).

    Google Scholar 

  59. R. Wille, On lattices freely generated by finite partially ordered sets. In: Contribution to Universal Algebra. Szeged: J. Bolyai Mat. Soc. (to appear).

    Google Scholar 

  60. R. Wille, Eine Charakterisierung endlicher, ordnungspolynomvollständiger Verbände. Arch. Math, (to appear).

    Google Scholar 

  61. R. Wille, A note on algebraic operations and algebraic functions of finite lattices. Preprint.

    Google Scholar 

  62. R. Wille, Jeder endlich erzeugte, modulare Verband endlicher Weite ist endlich. Math. čas. 24 (1974), 77–80.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Technische Hochschule, Darmstadt, Germany

    R. Wille

Authors
  1. R. Wille
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Freie Universität Berlin, Königin-Luise-Strasse 24/26, D-1000, Berlin 33, Germany

    Martin Aigner

Rights and permissions

Reprints and permissions

Copyright information

© 1977 D. Reidel Publishing Company, Dordrecht-Holland

About this paper

Cite this paper

Wille, R. (1977). Aspects of Finite Lattices. In: Aigner, M. (eds) Higher Combinatorics. NATO Advanced Study Institutes Series, vol 31. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1220-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-94-010-1220-1_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-1222-5

  • Online ISBN: 978-94-010-1220-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation