Advertisement

News about Semiantichains and Unichain Coverings

  • Bartłomiej Bosek
  • Stefan Felsner
  • Kolja Knauer
  • Grzegorz Matecki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×Q are equal. As a positive result we state conditions on P and Q that imply the min-max relation. However, we also have an example showing that in general the min-max relation is false. This disproves the Saks-West conjecture.

Keywords

Partial Order Weak Order Maximum Antichain Ferrers Diagram Chain Partition 
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.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. 51(2), 161–166 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Escamilla, E., Nicolae, A., Salerno, P., Shahriari, S., Tirrell, J.: On nested chain decompositions of normalized matching posets of rank 3. Order 28, 357–373 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Greene, C.: Some partitions associated with a partially ordered set. J. Combinatorial Theory Ser. A 20(1), 69–79 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Greene, C., Kleitman, D.J.: The structure of Sperner k-families. J. Combinatorial Theory Ser. A 20(1), 41–68 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Griggs, J.R.: Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math. 32(4), 807–809 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Griggs, J.R.: On chains and Sperner k-families in ranked posets. J. Combin. Theory Ser. A 28(2), 156–168 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Griggs, J.R.: Matchings, cutsets, and chain partitions in graded posets. Discrete Math. 144(1-3), 33–46 (1995); Combinatorics of ordered sets (Oberwolfach, 1991)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Liu, Q., West, D.B.: Duality for semiantichains and unichain coverings in products of special posets. Order 25(4), 359–367 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Saks, M.: A short proof of the existence of k-saturated partitions of partially ordered sets. Adv. in Math. 33(3), 207–211 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Tovey, C.A., West, D.B.: Networks and chain coverings in partial orders and their products. Order 2(1), 49–60 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Trotter Jr., L.E., West, D.B.: Two easy duality theorems for product partial orders. Discrete Appl. Math. 16(3), 283–286 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Trotter, W.T.: Partially ordered sets. In: Handbook of Combinatorics, vol. 1, pp. 433–480. Elsevier, Amsterdam (1995)Google Scholar
  13. 13.
    Viennot, X.G.: Chain and antichain families, grids and Young tableaux. In: Orders: Description and Roles (L’Arbresle 1982). North-Holland Math. Stud., vol. 99, pp. 409–463. North-Holland, Amsterdam (1984)Google Scholar
  14. 14.
    Wang, Y.: Nested chain partitions of LYM posets. Discrete Appl. Math. 145(3), 493–497 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    West, D.B.: Unichain coverings in partial orders with the nested saturation property. Discrete Math. 63(2-3), 297–303 (1987); Special issue: ordered sets (Oberwolfach, 1985)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    West, D.B., Tovey, C.A.: Semiantichains and unichain coverings in direct products of partial orders. SIAM J. Algebraic Discrete Methods 2(3), 295–305 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Wu, C.: On relationships between semi-antichains and unichain coverings in discrete mathematics. Chinese Quart. J. Math. 13(2), 44–48 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bartłomiej Bosek
    • 1
  • Stefan Felsner
    • 2
  • Kolja Knauer
    • 2
  • Grzegorz Matecki
    • 1
  1. 1.Theoretical Computer Science Department Faculty of Mathematics and Computer ScienceJagiellonian UniversityPoland
  2. 2.Diskrete Mathematik Institut für MathematikTechnische Universität BerlinGermany

Personalised recommendations