Skip to main content
SpringerLink
Log in

Tight embedding of modular lattices into partition lattices: progress and program

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

A famous Theorem of Pudlak and Tůma states that each finite lattice L occurs as sublattice of a finite partition lattice. Here we derive, for modular lattices L, necessary and sufficient conditions for cover-preserving embeddability. Aspects of our work relate to Bjarni Jónsson.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Faigle, U., Herrmann, C.: Projective geometry on partially ordered sets. Trans Am. Math. Soc. 266, 319–332 (1981)

    Article  MathSciNet  Google Scholar 

  2. Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Boston (2011)

    Book  Google Scholar 

  3. Grätzer, G., Kiss, E.: A construction of semimodular lattices. Order 2, 351–365 (1986)

    Article  MathSciNet  Google Scholar 

  4. Herrmann, C., Pickering, D., Roddy, M.: A geometric description of modular lattices. Algebra Univ. 31, 365–396 (1994)

    Article  MathSciNet  Google Scholar 

  5. Herrmann, C., Wild, M.: Acyclic modular lattices and their representations. J. Algebra 136, 17–36 (1991)

    Article  MathSciNet  Google Scholar 

  6. Huhn, A.: Two notes on n-distributive lattices. Colloq. Math. Soc. János Bolyai 14, 137–147 (1977)

    MathSciNet  Google Scholar 

  7. Jónsson, B., Nation, J.B.: Representations of 2-distributive modular lattices of finite length. Acta Sci. Math 51, 123–128 (1987)

    MathSciNet  MATH  Google Scholar 

  8. Mighton, J.: A new characterization of graphic matroids. J. Comb. Theory B 98, 1253–1258 (2008)

    Article  MathSciNet  Google Scholar 

  9. Nation, J.B.: Notes on lattice theory (2018) (unpublished)

  10. Oxley, J.G.: Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 3 (1997)

  11. Pudlak, P., Tůma, J.: Every finite lattice can be embedded in a finite partition lattice. Algebra Univ. 10, 74–95 (1980)

    Article  MathSciNet  Google Scholar 

  12. Tesler, G.P.: Semi-primary lattices and Tableaux algorithms, Ph.D., MIT (1995)

  13. Tutte, W.T.: Matroids and graphs. Trans Am. Math. Soc. 90, 527–552 (1959)

    Article  MathSciNet  Google Scholar 

  14. Wild, M.: Modular lattices of finite length, p. 28 (1992) [unpublished but on ResearchGate. (See also Subsection 8.7.)]

  15. Wild, M.: Cover preserving embedding of modular lattices into partition lattices. Discrete Math. 112, 207–244 (1993)

    Article  MathSciNet  Google Scholar 

  16. Wild, M.: The minimal number of join irreducibles of a finite modular lattice. Algebra Univ. 35, 113–123 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Stellenbosch University, Stellenbosch, South Africa

    Marcel Wild

Authors
  1. Marcel Wild
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcel Wild.

Additional information

Dedicated to the memory of Bjarni Jónsson.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wild, M. Tight embedding of modular lattices into partition lattices: progress and program. Algebra Univers. 79, 79 (2018). https://doi.org/10.1007/s00012-018-0559-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-018-0559-z

Mathematics Subject Classification

Keywords

Search

Navigation