It is shown that any low linear order of the form \(\mathcal{L}\)+ω∗, where \(\mathcal{L}\) is some η-presentation, has a computable copy. This result contrasts with there being low η-presentations not having a computable copy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Feiner, “Hierarchies of Boolean algebras,” J. Symb. Log., 35, No. 3, 365-374 (1970).
R. G. Downey and C. G. Jockush, “Every low Boolean algebra is isomorphic to a recursive one,” Proc. Am. Math. Soc., 122, No. 3, 871-880 (1994).
L. Feiner, “Orderings and Boolean algebras not isomorphic to recursive ones,” Ph. D. Thesis, MIT, Cambridge, MA (1967); http://hdl.handle.net/1721.1/60738
C. G. Jockusch, Jun. and R. I. Soare, “Degrees of orderings not isomorphic to recursive linear orderings,” Ann. Pure Appl. Log., 52, Nos. 1/2, 39-64 (1991).
R. G. Downey and M. F. Moses, “On choice sets and strongly non-trivial self-embeddings of recursive linear orders,” Z. Math. Logik Grund. Math., 35, No. 3, 237-246 (1989).
R. G. Downey, “Computability theory and linear orderings,” in Handbook of Recursive Mathematics, Stud. Log. Found. Math., 139, Y. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel (eds.), Elsevier, Amsterdam (1998), pp. 823-976.
A. N. Frolov, “ \({\Delta }_{2}^{0}\)-copies of linear orderings,” Algebra and Logic, 45, No. 3, 201-209 (2006).
A. N. Frolov, “Linear orderings of low degree,” Sib. Mat. Zh., 51, No. 5, 913-925 (2010).
A. N. Frolov, “Low linear orderings,” J. Log. Comput., 22, No. 4, 745-754 (2012).
M. V. Zubkov, “Sufficient conditions for the existence of 0′’-limitwise monotonic functions for computable η-like linear orders,” Sib. Mat. Zh., 58, No. 1, 80-90 (2017).
A. N. Frolov, “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, Ser. Sovrem. Mat. Pril. Temat. Obz., 158, VINITI, Moscow (2018), pp. 81-115.
A. N. Frolov and M. V. Zubkov, “The simplest low linear order with no computable copies,” J. Symb. Log., DOI: https://doi.org/10.1017/jsl.2022.44
R. J. Coles, R. Downey, and B. Khoussainov, “On initial segments of computable linear orders,” Order, 14, No. 2, 107-124 (1998).
M. V. Zubkov, “Initial segments of computable linear orders with additional computable predicates,” Algebra and Logic, 48, No. 5, 321-329 (2009).
R. I. Soare, Recursively Enumerable Sets and Degrees, Perspect. Math. Log., Omega Ser., Springer-Verlag, Heidelberg (1987).
J. G. Rosenstein, Linear Orderings, Pure Appl. Math., 98, Academic Press, New York (1982).
M. V. Zubkov and A. N. Frolov, “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, Ser. Sovrem. Mat. Pril. Temat. Obz., 157, VINITI, Moscow (2018), pp. 70-105.
K. Harris, “ η-Representation of sets and degrees,” J. Symb. Log., 73, No. 4, 1097-1121 (2008).
M. V. Zubkov, “A theorem on strongly η-representable sets,” Izv. Vyssh. Uch. Zav., Mat., No. 7, 77-81 (2009).
M. V. Zubkov, “Strongly η-representable degrees and limitwise monotonic functions,” Algebra and Logic, 50, No. 4, 346-356 (2011).
A. N. Frolov and M. V. Zubkov, “Increasing η-representable degrees,” Math. Log. Q., 55, No. 6, 633-636 (2009).
A. M. Kach and D. Turetsky, “Limitwise monotonic functions, sets and degrees on computable domains,” J. Symb. Log., 75, No. 1, 131-154 (2010).
R. G. Downey, A. M. Kach, and D. Turetsky, “Limitwise monotonic functions and their applications,” in Proc. 11th Asian Logic Conf. in Honor of Prof. Chong Chitat on his 60th Birthday, Singapore, June 22-27, 2009, T. Arai et al. (eds.), National Univ. Singapore, World Scientific, Hackensack, NJ (2012), pp. 59-85.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 61, No. 5, pp. 552-570, September-October, 2022. Russian DOI: https://doi.org/10.33048/alglog.2022.61.503.
M. V. Zubkov Supported by Russian Scientific Foundation (project No. 20-31-70012) and carried out as part of realizing the Development Program for the Scientific and Educational Mathematical Center of the Volga Federal Region (Agreement No. 075-02-2022-882).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zubkov, M.V. A Class of Low Linear Orders Having Computable Presentations. Algebra Logic 61, 372–384 (2022). https://doi.org/10.1007/s10469-023-09706-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-023-09706-1