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Ordered Sets pp 173-197 | Cite as

Lexicographic Sums

  • Bernd Schröder
Chapter
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Abstract

In this chapter, we investigate a construction, lexicographic sums, that uses existing ordered sets to build new ordered sets. The pictorial idea is very simple: Take an ordered set T and replace each of its points t with an ordered set P t . The resulting structure will be a new, larger ordered set. It is then natural to ask how various order-theoretical properties and parameters behave under lexicographic constructions. We will revisit lexicographic sums in later chapters to specifically answer this question in a variety of contexts. At the end of this chapter, in Section 7.6, we will explain the definition of “hard” problems that was mentioned at the start of Section  5.4 (see Definition 7.30). We will then use a construction that is close to the lexicographic sum construction (see Section 7.5), to show that determining if an ordered set has the fixed point property is “hard.”

Keywords

Comparability Graph Point Property Canonical Decomposition Covering Graph Fixed Point Property 
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.

References

  1. 6.
    Alvarez, L. (1965). Undirected graphs realizable as graphs of modular lattices. Canadian Journal of Mathematics, 17, 923–932.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 34.
    Brightwell, G. (1993). On the complexity of diagram testing. Order, 10, 297–303.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 46.
    Clay Mathematics Institute web site. (2015). http://www.claymath.org/millennium-problems
  4. 49.
    Cook, S. A. (1971). The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing (pp. 151–158). New York: Association for Computing Machinery.Google Scholar
  5. 66.
    Dreesen, B., Poguntke, W., & Winkler, P. (1985). Comparability invariance of the fixed point property. Order, 2, 269–274.MathSciNetzbMATHGoogle Scholar
  6. 68.
    Duffus, D., & Goddard, T. (1996). The complexity of the fixed point property. Order, 13, 209–218.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 83.
    Ewacha, K., Li, W., & Rival, I. (1991). Order, genus and diagram invariance. Order, 8, 107–113.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 101.
    Gallai, T. (1967). Transitiv orientierbare graphen. Acta Mathematica Hungarica, 18, 25–66.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 103.
    Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: Freeman.zbMATHGoogle Scholar
  10. 139.
    Höft, H., & Höft, M. (1976). Some fixed point theorems for partially ordered sets. Canadian Journal of Mathematics, 28, 992–997.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 141.
    Höft, H., & Höft, M. (1991). Fixed point free components in lexicographic sums with the fixed point property. Demonstratio Mathematica, XXIV, 294–304.Google Scholar
  12. 144.
    Hughes, J. (2004). The Computation and Comparison of Decks of Small Ordered Sets. MS thesis, Louisiana Tech University.Google Scholar
  13. 145.
    Ille, P. (1993). Recognition problem in reconstruction for decomposable relations. In N. W. Sauer et al. (Eds.), Finite and infinite combinatorics in sets and logic (pp. 189–198). Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
  14. 154.
    Kelly, D. (1985). Comparability graphs. In I. Rival (Ed.), Graphs and order (pp. 3–40). Dordrecht: D. Reidel.CrossRefGoogle Scholar
  15. 169.
    Köbler, J., Schöning, U., & Torán, J. (1993). The graph isomorphism problem: Its structural complexity. Progress in theoretical computer science. Boston: Birkhäuser.CrossRefzbMATHGoogle Scholar
  16. 174.
    Kratsch, D., & Rampon, J.-X. (1994). Towards the reconstruction of posets. Order, 11, 317–341.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 196.
    Lubiw, A. (1981). Some NP-complete problems similar to graph isomorphisms. SIAM Journal of Computing, 10, 11–21.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 197.
    Luks, E. M. (1982). Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and System Science, 25, 42–65.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 215.
    Nešetřil, J., & Rödl, V. (1987). Complexity of diagrams. Order, 3, 321–330.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 249.
    Rival, I. (1985). Unsolved problems: The diagram. Order, 2, 101–104.MathSciNetCrossRefGoogle Scholar
  21. 252.
    Rival, I. (Ed.). (1989). Algorithms and order. Dordrecht/Boston: Kluwer.zbMATHGoogle Scholar
  22. 259.
    Rödl, V., & Thoma, L. (1995). The complexity of cover graph recognition for some varieties of finite lattices. Order, 12, 351–374.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 273.
    Schröder, B. (1998). On cc-comparability invariance of the fixed point property. Discrete Mathematics, 179, 167–183.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 275.
    Schröder, B. (2000). Reconstruction of the neighborhood deck of ordered sets. Order, 17, 255–269.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 276.
    Schröder, B. (2001). Reconstruction of N-free ordered sets. Order, 18, 61–68.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 280.
    Schröder, B. (2005). The automorphism conjecture for small sets and series parallel sets. Order, 22, 371–387.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 323.
    Wild, M. (1992). Cover-preserving order embeddings into Boolean lattices. Order, 9, 209–232.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Bernd Schröder
    • 1
  1. 1.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

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