Advertisement

Order

pp 1–12 | Cite as

Boolean Dimension, Components and Blocks

  • Tamás MészárosEmail author
  • Piotr Micek
  • William T. Trotter
Article
  • 3 Downloads

Abstract

We investigate the behavior of Boolean dimension with respect to components and blocks. To put our results in context, we note that for Dushnik-Miller dimension, we have that if \(\dim (C)\le d\) for every component C of a poset P, then \(\dim (P)\le \max \limits \{2,d\}\); also if \(\dim (B)\le d\) for every block B of a poset P, then \(\dim (P)\le d+2\). By way of constrast, local dimension is well behaved with respect to components, but not for blocks: if ldim(C) = d for every component C of a poset P, then ldim(P) = d + 2; however, for every d = 4, there exists a poset P with ldim(P) = d and \(\dim (B)\le 3\) for every block B of P. In this paper we show that Boolean dimension behaves like Dushnik-Miller dimension with respect to both components and blocks: if bdim(C) = d for every component C of P, then bdim(P) = 2 + d + 4 · 2d; also if bdim(B) = d for every block of P, then bdim(P) = 19 + d + 18 · 2d.

Keywords

Posets Boolean dimension 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Bosek, B., Grytczuk, J., Trotter, W.T.: Local dimension is unbounded for planar posets, arXiv:1712.06099
  2. 2.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Am. J. Math. 63, 600–610 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gambosi, G., Nešetril, J., Talamo, M.: On locally presented posets. Theor. Comput. Sci. 70, 251–260 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Joret, G., Micek, P., Milans, K., Trotter, W.T., Walczak, B., Wang, R.: Tree-width and dimension. Combinatorica 36, 431–450 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Nešetril, J., Pudlák, P.: A note on Boolean dimension of posets. In: Halász, G., Sós, V. T. (eds.) Irregularities of Partitions, Algorithms and Combinatorics, vol. 8, pp 137–140. Springer, Berlin (1989)Google Scholar
  6. 6.
    Rényi, A.: On random generating elements of a finite Boolean algebra. Acta. Sci. Math. 22, 75–81 (1961)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Streib, N., Trotter, W.T.: Dimension and height for posets with planar cover graphs. Eur. J. Comb. 3, 474–489 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets:Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)Google Scholar
  9. 9.
    Trotter, W.T., Moore, J.I.: The dimension of planar posets. J. Combinatorial Theory Series B 21, 51–67 (1977)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Trotter, W.T., Walczak, B.: Boolean dimension and local dimension. Electron Notes Discrete Math. 61, 1047–1053 (2017)CrossRefzbMATHGoogle Scholar
  11. 11.
    Trotter, W.T., Walczak, B., Wang, R., et al.: Dimension and cut vertices: an application of Ramsey theory. In: Butler, S. (ed.) Connections in Discrete Mathematics, pp 187–199. Cambridge University Press (2018)Google Scholar
  12. 12.
    Ueckerdt, T. Personal communicationGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of Mathematics, Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  2. 2.Department of Theoretical Computer Science, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaGeorgia

Personalised recommendations