Advertisement

Lobachevskii Journal of Mathematics

, Volume 39, Issue 9, pp 1453–1459 | Cite as

On a Computable Presentation of Low Linear Orderings

  • A. N. FrolovEmail author
Selected Articles from the Journal Uchenye Zapiski Kazanskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki
  • 3 Downloads

Abstract

In 1998, R. Downey in his review paper outlined a research agenda for the study and description of sufficient conditions for computable representability of linear orderings, namely, the problem of describing order properties P such that, for any low linear ordering Limplies that, the feasibility of the condition P(L) implies that L has a computable presentation. This paper is a part of the research program initiated by R. Downey. It is shown in the paper that any low linear ordering whose factor order (in other words, condensation) is η (the order type of naturally ordered natural numbers) has a computable presentation via a 0—computable isomorphism if this ordering does not contain a strongly η-like infinite interval. A countable linear ordering is said to be strongly η-like if there exists some natural number k such that each maximal block of the ordering is of cardinality no more than k. It is also proved that the above result does not hold for a 0—computable isomorphism instead of the 0—computable one. Namely, we construct a low linear ordering L with condensation η and without strongly η-like infinite interval so that L has no computable presentation via a 0— computable isomorphism.

Keywords and phrases

linear ordering computable presentation low degree strongly η-like linear odering 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. I. Soare, Recursively Enumerable Sets and Degrees (Springer, Heidelberg, 1987), Vol. 18.CrossRefzbMATHGoogle Scholar
  2. 2.
    R. G. Downey and C. G. Jockusch, Jr., “Every low Boolean algebra is isomorphic to a recursive one,” Proc. Am. Math. Soc. 122, 871–880 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. Thurber, “Every low2 Boolean algebra has a recursive copy,” Proc. Am. Math. Soc. 123, 3859–3866 (1995).zbMATHGoogle Scholar
  4. 4.
    J. F. Knight and M. Stob, “Computable Boolean algebras,” J. Symb. Logic. 65, 1605–1623 (2000). doi 10.2307/2695066MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    K. Harris and A. Montalban, “Boolean algebra approximations,” Trans. Am. Math. Soc. 366, 5223–5256 (2014). doi 10.1090/S0002-9947-2014-05950-3MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    C. G. Jockusch and R. I. Soare, “Degrees of orderings not isomorphic to recursive linear orderings,” Ann. Pure Appl. Logic. 52, 39–64 (1991). doi 10.1016/0168-0072(91)90038-NMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    R. G. Downey and M. F. Moses, “On choice sets and strongly nontrivial self-embeddings of recursive linear orderings,” Z. Math. Logik Grund. Math. 35, 237–246 (1989).CrossRefzbMATHGoogle Scholar
  8. 8.
    R. G. Downey, “Computability theory and linear orderings,” in Handbook of Recursive Mathematics, Ed. by Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (Elsevier, Amsterdam, 1998), pp. 823–976.Google Scholar
  9. 9.
    A. N. Frolov, “Δ02 -copies of linear orderings,” Algebra Logic. 45, 201–209 (2006). doi 10.1007/s10469-006-0017-4MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. N. Frolov, “Linear orderings of low degree,” Sib. Math. J. 51, 913–925 (2010). doi 10.1007/s11202-010-0091-7MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    A. N. Frolov, “Low linear orderings,” J. Logic Comput. 22, 745–754 (2012). doi 10.1093/logcom/exq040MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Frolov, “Scattered linear orderings with no computable presentation,” Lobachevskii J. Math. 35, 19–22 (2014). doi 10.1134/S199508021401003XMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Kach and A. Montalbén, “Cuts of linear orders,” Order. 28, 593–600 (2011). doi 10.1007/s11083-010-9194-9MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    P. E. Alaev, A. N. Frolov, and J. Thurber, “Computability on linear orderings enriched with predicates,” Algebra Logic. 48, 313–320 (2009). doi 10.1007/s10469-009-9067-8MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Frolov and M. Zubkov, “Increasing η-representable degrees,” Math. Logic Q. 55, 633–636 (2009). doi 10.1002/malq. 200810031MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Kach, “Computable shuffle sums of ordinals,” Archive Math. Logic. 47, 211–219 (2008). doi 10.1007/s00153-008-0077-3MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Montalban, “Notes on the jump of a structure,” Lect. Notes Comput. Sci. 5635, 372–378 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Kazan (Volga Region) Federal UniversityKazanRussia

Personalised recommendations