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Algebra universalis

, 79:79 | Cite as

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

  • Marcel Wild
Article
  • 16 Downloads
Part of the following topical collections:
  1. In memory of Bjarni Jónsson

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.

Keywords

Modularity 2-distributivity Partition lattice Cover-preserving lattice embedding Kinds of projective spaces and their cycles Graphs and their (chordless) circuits Binary matroids 

Mathematics Subject Classification

06C05 05A18 05B35 51A05 51D25 

References

  1. 1.
    Faigle, U., Herrmann, C.: Projective geometry on partially ordered sets. Trans Am. Math. Soc. 266, 319–332 (1981)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  3. 3.
    Grätzer, G., Kiss, E.: A construction of semimodular lattices. Order 2, 351–365 (1986)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Herrmann, C., Pickering, D., Roddy, M.: A geometric description of modular lattices. Algebra Univ. 31, 365–396 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Herrmann, C., Wild, M.: Acyclic modular lattices and their representations. J. Algebra 136, 17–36 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Huhn, A.: Two notes on n-distributive lattices. Colloq. Math. Soc. János Bolyai 14, 137–147 (1977)MathSciNetGoogle Scholar
  7. 7.
    Jónsson, B., Nation, J.B.: Representations of 2-distributive modular lattices of finite length. Acta Sci. Math 51, 123–128 (1987)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mighton, J.: A new characterization of graphic matroids. J. Comb. Theory B 98, 1253–1258 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Nation, J.B.: Notes on lattice theory (2018) (unpublished)Google Scholar
  10. 10.
    Oxley, J.G.: Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 3 (1997)Google Scholar
  11. 11.
    Pudlak, P., Tůma, J.: Every finite lattice can be embedded in a finite partition lattice. Algebra Univ. 10, 74–95 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Tesler, G.P.: Semi-primary lattices and Tableaux algorithms, Ph.D., MIT (1995)Google Scholar
  13. 13.
    Tutte, W.T.: Matroids and graphs. Trans Am. Math. Soc. 90, 527–552 (1959)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wild, M.: Modular lattices of finite length, p. 28 (1992) [unpublished but on ResearchGate. (See also Subsection 8.7.)]Google Scholar
  15. 15.
    Wild, M.: Cover preserving embedding of modular lattices into partition lattices. Discrete Math. 112, 207–244 (1993)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wild, M.: The minimal number of join irreducibles of a finite modular lattice. Algebra Univ. 35, 113–123 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wille, R.: Über modulare Verbände, die von einer endlichen halbgeordneten Menge frei erzeugt werden. Math. Z. 131, 241–249 (1973)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsStellenbosch UniversityStellenboschSouth Africa

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