Ordered Sets pp 173-197

# Lexicographic Sums

• Bernd Schröder
Chapter

## 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  (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.
2. 34.
Brightwell, G. (1993). On the complexity of diagram testing. Order, 10, 297–303.
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.
6. 68.
Duffus, D., & Goddard, T. (1996). The complexity of the fixed point property. Order, 13, 209–218.
7. 83.
Ewacha, K., Li, W., & Rival, I. (1991). Order, genus and diagram invariance. Order, 8, 107–113.
8. 101.
Gallai, T. (1967). Transitiv orientierbare graphen. Acta Mathematica Hungarica, 18, 25–66.
9. 103.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: Freeman.
10. 139.
Höft, H., & Höft, M. (1976). Some fixed point theorems for partially ordered sets. Canadian Journal of Mathematics, 28, 992–997.
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.
14. 154.
Kelly, D. (1985). Comparability graphs. In I. Rival (Ed.), Graphs and order (pp. 3–40). Dordrecht: D. Reidel.
15. 169.
Köbler, J., Schöning, U., & Torán, J. (1993). The graph isomorphism problem: Its structural complexity. Progress in theoretical computer science. Boston: Birkhäuser.
16. 174.
Kratsch, D., & Rampon, J.-X. (1994). Towards the reconstruction of posets. Order, 11, 317–341.
17. 196.
Lubiw, A. (1981). Some NP-complete problems similar to graph isomorphisms. SIAM Journal of Computing, 10, 11–21.
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.
19. 215.
Nešetřil, J., & Rödl, V. (1987). Complexity of diagrams. Order, 3, 321–330.
20. 249.
Rival, I. (1985). Unsolved problems: The diagram. Order, 2, 101–104.
21. 252.
Rival, I. (Ed.). (1989). Algorithms and order. Dordrecht/Boston: Kluwer.
22. 259.
Rödl, V., & Thoma, L. (1995). The complexity of cover graph recognition for some varieties of finite lattices. Order, 12, 351–374.
23. 273.
Schröder, B. (1998). On cc-comparability invariance of the fixed point property. Discrete Mathematics, 179, 167–183.
24. 275.
Schröder, B. (2000). Reconstruction of the neighborhood deck of ordered sets. Order, 17, 255–269.
25. 276.
Schröder, B. (2001). Reconstruction of N-free ordered sets. Order, 18, 61–68.
26. 280.
Schröder, B. (2005). The automorphism conjecture for small sets and series parallel sets. Order, 22, 371–387.
27. 323.
Wild, M. (1992). Cover-preserving order embeddings into Boolean lattices. Order, 9, 209–232.