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Orthogonal Countable Linear Orders

  • Christian Delhommé


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.


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


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


  1. 1.
    Albert, M., Atkinson, M.: Simple permutations and pattern restricted permutations. Discret. Math. 300, 1–15 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albert, M., Atkinson, M., Klazar, M.: The enumeration of simple permutations. J. Integ. Seq. 6, Article 03.4.4 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bezhanishvili, G., Morandi, P.J.: Order-compactifications of totally ordered spaces: Revisited. Order 28, 577–592 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blatter, J.: Order compactifications of totally ordered topological spaces. J. Approxi. Theory 13, 56–65 (1975)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Länger, F., Pöschel, R.: Relational systems with trivial endomorphisms and polymorphisms. J. Pure Appl. Algebra 2, 129–142 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Laver, R.: On Fraïssé’s order type conjecture. Ann. Math. 93(1), 89–111 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Laver, R.: An order type decomposition theorem. Ann. of Math. 98, 96–119 (1973)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miyakawa, M., Nozaki, A., Pogosyan, G., Rosenberg, I.G.: The number of orthogonal permutations. Europ. J. Comb. 16, 71–85 (1995)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rival, I., Zaguia, N.: Perpendicular orders. Discrete Math. 137(1–3), 303–313 (1995)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zaguia, I.: Prime two-dimensional orders and perpendicular total orders. Europ. J. Comb. 19, 639–649 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

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

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