Advertisement

Order

pp 1–39 | Cite as

Orthogonal Countable Linear Orders

  • Christian Delhommé
Article
  • 11 Downloads

Abstract

Two linear orderings of a same set are perpendicular if every self-mapping of this set that preserves them both is constant or the identity. Two isomorphy types of linear orderings are orthogonal if there exist two perpendicular orderings of these types. Our main result is a characterisation of orthogonality to ω : a countably infinite type is orthogonal toω if and only if it is scattered and does not admit any embedding into the chain of infinite classes of its Hausdorff congruence. Besides we prove that a countable type is orthogonal toω + n (2 ≤ n < ω) if and only if it has infinitely many vertices that are isolated for the order topology. We also prove that a typeτ is orthogonal to ω + 1 if and only if it has a decomposition of the formτ = τ1 + 1 + τ2 withτ1 orτ2 orthogonal to ω, or one of them finite nonempty and the other one orthogonal toω + 2. Since it was previously known that two countable types are orthogonal whenever each one has two disjoint infinite intervals, this completes a characterisation of orthogonality of pairs of types of countable linear orderings. It follows that the equivalence relation of indistinguishability for the orthogonality relation on the class of countably infinite linear orders has exactly seven classes : the classes respectively of ω, ω + 1, ω + 2, ω + ω, ω ω , 3 ⋅ η and η, where η is the type of the ordering of rational numbers and 3 ⋅ η is the lexicographical sum along η of three element linear orders.

Keywords

Linearly ordered set Order preserving map Endomorphism Orthogonal orders Rigid relational structure Compactification of a linear order Indecomposable linear order Bichain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author thanks the referee, whose numerous remarks and suggestions greatly helped improve the manuscript.

References

  1. 1.
    Albert, M., Atkinson, M.: Simple permutations and pattern restricted permutations. Discret. Math. 300, 1–15 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Albert, M., Atkinson, M., Klazar, M.: The enumeration of simple permutations. J. Integ. Seq. 6, Article 03.4.4 (2003)MathSciNetMATHGoogle Scholar
  3. 3.
    Bezhanishvili, G., Morandi, P.J.: Order-compactifications of totally ordered spaces: Revisited. Order 28, 577–592 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Blatter, J.: Order compactifications of totally ordered topological spaces. J. Approxi. Theory 13, 56–65 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brignall, R.: A Survey of Simple Permutations. Permutation Patterns. London Math. Soc. Lecture Note, Ser., vol. 376, pp. 41–65. Cambridge Univ. Press, Cambridge (2010)Google Scholar
  6. 6.
    Delhommé, C., Zaguia, I.: Countable linear orders with disjoint infinite intervals are mutually orthogonal. Discret. Math. 341, 1885–1899 (2018)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Demetrovics, J., Miyakawa, M., Rosenberg, I., Simovici, D., Stojmenović, I.: Intersections of isotone clones on a finite set. In: Proc. 20th Internat. Symp. Multiple-valued Logic Charlotte, NC, pp. 248–253 (1990)Google Scholar
  8. 8.
    Demetrovics, J., Ronyai, L.: A note on intersections of isotone clones. Acta Cyberniteca 10(Szeged), 217–220 (1992)MathSciNetMATHGoogle Scholar
  9. 9.
    Ehrenfeucht, A., Harju, T., Rozenberg, G.: The Theory of 2-Structures: A Framework for Decomposition and Transformation. World Scientific (1999)Google Scholar
  10. 10.
    Fedorcuk, V.: Some questions on the theory of ordered spaces. Sib. Math. J. 10, 124–132 (1969)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fraïssé, R.: L’intervalle en théorie des relations ; ses généralisations ; filtre intervallaire et clôture d’une relation. (French) [The interval in relation theory; its generalizations; interval filter and closure of a relation]. Orders: Description and roles. (L’Arbresle, 1982), 313–341, North-Holland Math. Stud., 99. North-Holland, Amsterdam (1984)Google Scholar
  12. 12.
    Kaufman, R.: Ordered sets and compact spaces. Colloq. Math. 17, 35–39 (1967)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kent, D.C., Richmond, T.A.: Ordered compactification of totally ordered spaces. Internat. J. Math. Math Sci. 11(4), 683–694 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Klazar, M.: Some General Results in Combinatorial Enumeration. Permutation Patterns, 3–40, London Math. Soc Lecture Note Ser., p 376. Cambridge Univ. Press, Cambridge (2010)Google Scholar
  15. 15.
    Laflamme, C., Pouzet, M., Sauer, N., Zaguia, I.: Pairs of orthogonal countable ordinals. Discr. Math. 335, 35–44 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Länger, F., Pöschel, R.: Relational systems with trivial endomorphisms and polymorphisms. J. Pure Appl. Algebra 2, 129–142 (1984)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Laver, R.: On Fraïssé’s order type conjecture. Ann. Math. 93(1), 89–111 (1971)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Laver, R.: An order type decomposition theorem. Ann. of Math. 98, 96–119 (1973)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mac Lane, S.: Categories for the working mathematician. 2nd ed. Springer (1997)Google Scholar
  20. 20.
    Marcus, A., Tardös, G.: Excluded permutation matrices and the Stanley-Wilf conjecture. J. Combin. Theory Ser. Excluded Permut. A 107, 153–160 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Miyakawa, M., Nozaki, A., Pogosyan, G., Rosenberg, I.G.: The number of orthogonal permutations. Europ. J. Comb. 16, 71–85 (1995)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Nachbin, L.: Topology and Order. Van Nostrand Mathematical Studies, No. 4. Princeton (1965)Google Scholar
  23. 23.
    Pálfy, P.: Unary polynomial in algebra I. Algebra Universalis 18, 162–173 (1984)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pouzet, M., Zaguia, N.: Ordered sets with no chains of ideals of a given type. Order 1, 159–172 (1984)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rival, I., Zaguia, N.: Perpendicular orders. Discrete Math. 137(1–3), 303–313 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Rosenstein, J.G.: Linear Orderings. Academic Press (1982)Google Scholar
  27. 27.
    Sauer, N., Zaguia, I.: The order on the rationals has an orthogonal order with the same order type. Order 28, 377–385 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Zaguia, I.: Prime two-dimensional orders and perpendicular total orders. Europ. J. Comb. 19, 639–649 (1998)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Informatique et de Mathématiques (LIM-ERMIT), Faculté des Sciences et TechnologiesUniversité de la RéunionSainte-ClotildeFrance

Personalised recommendations