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THE SIMPLEST LOW LINEAR ORDER WITH NO COMPUTABLE COPIES

Part of: Computability and recursion theory Model theory

Published online by Cambridge University Press:  10 June 2022

ANDREY FROLOV Open the ORCID record for ANDREY FROLOV [Opens in a new window]  and
MAXIM ZUBKOV Open the ORCID record for MAXIM ZUBKOV [Opens in a new window]
ANDREY FROLOV
Affiliation:
INNOPOLIS UNIVERSITY UNIVERSITETSKAYA STREET 1, INNOPOLIS 420500, RUSSIA E-mail: a.frolov.kpfu@gmail.com
MAXIM ZUBKOV*
Affiliation:
N.I. LOBACHEVSKY INSTITUTE OF MATHEMATICS AND MECHANICS KAZAN FEDERAL UNIVERSITY KREMLEVSKAYA 18, KAZAN 420008, RUSSIA
*
E-mail: maxim.zubkov@kpfu.ru
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Abstract

A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to $2$. In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong $\eta $-representation with no computable copy that is the simplest possible example.

Keywords

linear orderlow Turing degreeη-representationcomputable presentation

MSC classification

Primary: 03C57: Effective and recursion-theoretic model theory 03D45: Theory of numerations, effectively presented structures

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 1 , March 2024 , pp. 97 - 111
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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References

Downey, R. G., Computability theory and linear orderings , Handbook of Recursive Mathematics, vol. 2 (Ershov, Yu. L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), Elsevier, Amsterdam, 1998, pp. 823–976.Google Scholar
Feiner, L. J., Orderings and Boolean algebras not isomorphic to recursive ones , Ph.D. thesis, Massachusetts Institute of Technology, 1967.Google Scholar
Feiner, L. J., Hierarchies of Boolean algebras, this Journal, vol. 35 (1970), pp. 365–374.Google Scholar
Fellner, S., Recursive and finite axiomatizability of linear orderings , Ph.D. thesis, Rutgers University, 1976.Google Scholar
Frolov, A. N., ${\Delta}_2^0$ -copies of linear orderings . Algebra and Logic , vol. 45 (2006), pp. 201–209.CrossRefGoogle Scholar
Frolov, A. N., Linear orders of low degree . Siberian Mathematical Journal , vol. 51 (2010), pp. 913–925.CrossRefGoogle Scholar
Frolov, A. N., Low linear orderings . Journal of Logic and Computation , vol. 44 (2012), pp. 745–754.CrossRefGoogle Scholar
Frolov, A. N., Scattered linear orderings with no computable presentation . Lobachevskii Journal of Mathematics, vol. 35 (2014), pp. 19–22.CrossRefGoogle Scholar
Frolov, A. N., Computable presentability of countable linear orders , Proceedings of the Seminar on Algebra and Mathematical Logic of the Kazan (Volga Region) Federal University , Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, vol. 158, VINITI, Moscow, 2018, pp. 81–115.Google Scholar
Frolov, A. N. and Zubkov, M. V., Computable linear orders and limitwise monotonic functions , Proceedings of the Seminar on Algebra and Mathematical Logic of the Kazan (Volga Region) Federal University , Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, vol. 157, VINITI, Moscow, 2018, pp. 70–105.Google Scholar
Jockusch, C. G. and Downey, R. G., Every low Boolean algebra is isomorphic to a recursive one . Proceedings of the American Mathematical Society , vol. 122 (1994), pp. 871–880.Google Scholar
Lerman, M., On recursive linear orderings , Logic Year 1979-80 (Lerman, M., Schmerl, J. H., and Soare, R. I., editors), Lecture Notes in Mathematics, vol. 859, Springer, Berlin, 1981, pp. 132–142.CrossRefGoogle Scholar
Montalban, A., Notes on the jump of a structure , Mathematical Theory and Computational Practice (Ambos-Spies, K., Löwe, B., and Merkle, W., editors), Lecture Notes in Computer Science, vol. 5635, Springer, Berlin, 2009, pp. 372–378.Google Scholar
Montalbán, A. and Kach, A., Cuts of linear orders . Order , vol. 28 (2011), pp. 593–600.Google Scholar
Moses, M. F. and Downey, R. G., On choice sets and strongly non-trivial self-embeddings of recursive linear orders . Mathematical Logic Quarterly , vol. 35 (1989), pp. 237–246.Google Scholar
Rosenstein, J. G., Linear Orderings , Academic Press, New York–London, 1982.Google Scholar
Soare, R. I. and Jockusch, C. G., Degrees of orderings not isomorphic to recursive linear orderings . Annals of Pure and Applied Logic , vol. 52 (1991), pp. 39–61.Google Scholar
Thurber, J., Alaev, P.E., and Frolov, A. N., Computability on linear orderings enriched with predicates . Algebra and Logic , vol. 48 (2009), pp. 313–320.Google Scholar
Turetsky, D., Downey, R. G., and Kach, A. M., Limitwise monotonic functions and their applications , Proceedings of the 11th Asian Logic Conference (Arai, T., Feng, Q., Kim, B., Wu, G., and Yang, Y., editors), World Scientific, Singapore, 2012, pp. 59–85.Google Scholar
Turetsky, D. and Kach, A. M., Limitwise monotonic functions, sets and degrees on computable domains, this Journal, vol. 75 (2010), pp. 131–154.Google Scholar
Zubkov, M. V., A theorem on strongly η-representable sets . Russian Mathematics (Iz. VUZ) , vol. 53 (2009), pp. 65–69.CrossRefGoogle Scholar
Zubkov, M. V., Strongly η-representable degrees and limitwise monotonic functions . Algebra and Logic , vol. 50 (2011), pp. 346–356.CrossRefGoogle Scholar
Zubkov, M. V., Sufficient conditions for the existence of 0 ′-limitwise monotonic functions for computable η-like linear orders . Siberian Mathematical Journal , vol. 58 (2017), pp. 80–90.CrossRefGoogle Scholar
Zubkov, M. V. and Frolov, A. N., Increasing η-representable degrees . Mathematical Logic Quarterly , vol. 55 (2009), pp. 633–636.Google Scholar