Order

pp 1–39

# Orthogonal Countable Linear Orders

• Christian Delhommé
Article

## 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

## 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)
2. 2.
Albert, M., Atkinson, M., Klazar, M.: The enumeration of simple permutations. J. Integ. Seq. 6, Article 03.4.4 (2003)
3. 3.
Bezhanishvili, G., Morandi, P.J.: Order-compactifications of totally ordered spaces: Revisited. Order 28, 577–592 (2011)
4. 4.
Blatter, J.: Order compactifications of totally ordered topological spaces. J. Approxi. Theory 13, 56–65 (1975)
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)
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)
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)
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)
13. 13.
Kent, D.C., Richmond, T.A.: Ordered compactification of totally ordered spaces. Internat. J. Math. Math Sci. 11(4), 683–694 (1988)
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)
16. 16.
Länger, F., Pöschel, R.: Relational systems with trivial endomorphisms and polymorphisms. J. Pure Appl. Algebra 2, 129–142 (1984)
17. 17.
Laver, R.: On Fraïssé’s order type conjecture. Ann. Math. 93(1), 89–111 (1971)
18. 18.
Laver, R.: An order type decomposition theorem. Ann. of Math. 98, 96–119 (1973)
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)
21. 21.
Miyakawa, M., Nozaki, A., Pogosyan, G., Rosenberg, I.G.: The number of orthogonal permutations. Europ. J. Comb. 16, 71–85 (1995)
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)
24. 24.
Pouzet, M., Zaguia, N.: Ordered sets with no chains of ideals of a given type. Order 1, 159–172 (1984)
25. 25.
Rival, I., Zaguia, N.: Perpendicular orders. Discrete Math. 137(1–3), 303–313 (1995)
26. 26.
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)
28. 28.
Zaguia, I.: Prime two-dimensional orders and perpendicular total orders. Europ. J. Comb. 19, 639–649 (1998)