Abstract
We find new sufficient conditions for the existence of a 0’-limitwise monotonic function defining the order for a computable η-like linear order L, i.e., of a function G such that L ∑ q∈ℚ G(q). Namely, we define the notions of left local maximal block and right local maximal block and prove that if the sizes of these blocks in a computable η-like linear order L are bounded then there is a 0’-limitwise monotonic function G with L = ∑ q∈ℚ G(q).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Frolov A. N. and Zubkov M. V., “Increasing ηrepresentable degrees,” Math. Log. Quart., vol. 55, 633–636 (2009).
Downey R. G., Kach A. M., and Turetsky D., “Limitwise monotonic functions and their applications,” in: Proc. 11th Asian Logic Conf., World Sci., Singapore, 2012, 59–85.
Downey R. G. and Jockusch C. G., “Every low Boolean algebra is isomorphic to a recursive one,” Proc. Amer. Math. Soc., vol. 122, no. 3, 871–880 (1994).
Jockusch C. G. and Soare R. I., “Degrees of orderings not isomorphic to recursive linear orderings,” Ann. Pure Appl. Logic., vol. 52, 39–61 (1991).
Downey R. G. and Moses M. F., “On choice sets and strongly non-trivial self-embeddings of recursive linear orders,” Math. Log. Quart., vol. 35, 237–246 (1989).
Frolov A. N., “?02 -copies of linear orderings,” Algebra and Logic, vol. 45, no. 3, 201–209 (2006).
Frolov A. N., “Low linear orderings,” J. Logic Comput., vol. 22, no. 4, 745–754 (2012).
Frolov A. N., “Linear orderings of low degree,” Sib. Math. J., vol. 51, no. 5, 913–925 (2010).
Soare R. I., Recursively Enumerable Sets and Degrees, Springer-Verlag, Heidelberg (1987).
Rosenstein J., Linear Orderings, Academic Press, New York (1982).
Downey R. G., “Computability theory and linear orderings,” in: Handbook of Computable Algebra, vol. 2, Elsevier, Amsterdam, 1998, 823–976.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by the Russian Foundation for Basic Research (Grants 15–41–02502, 15–01–08252, and 15–31–20607).
Kazan. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 1, pp. 107–121, January–February, 2017; DOI: 10.17377/smzh.2017.58.112.
Rights and permissions
About this article
Cite this article
Zubkov, M.V. Sufficient conditions for the existence of 0’-limitwise monotonic functions for computable η-like linear orders. Sib Math J 58, 80–90 (2017). https://doi.org/10.1134/S0037446617010128
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446617010128