Abstract
We show: (1) for every noncomputable c.e. incomplete c-degree, there exists a nonspeedable c-degree incomparable with it; (2) The c-degree of a hypersimple set includes an infinite collection of \(c_1\)-degrees linearly ordered under \(\le _{c_1}\) with order type of the integers and consisting entirely of hypersimple sets; (3) there exist two c.e. sets having no c.e. least upper bound in the \(c_1\)-reducibility ordering; (4) the c.e. \(c_1\)-degrees are not dense.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belegradek, O.V.: On an algorithmically closed groups. Algeb. Logic 13(3), 135–143 (1974)
Blum, M.A.: A machine-independent theory of the complexity of recursive functions. J. Assoc. Comp. Mach. 14(2), 322–336 (1967)
Blum, M., Marques, I.: On complexity properties of recursively enumerable sets. J. Symbolic Logic 38(04), 579–593 (1973)
Chitaia, I.: Hyperhypersimple sets and \(Q_1\)-reducibility. Math. Logic Quart. 62(6), 590–595 (2016)
Degtev, A.N.: Several results on upper semilattices and \(m\)-degrees. Algeb. Logic 18(6), 420–430 (1979)
Degtev, A.N.: Peкypcивно пepeчислимыe множecтвa и cводимоcти тaблинчного типa. Hayкa, \(\Phi \)измaтлит (1998)
Denisov, S.D.: On \(m\)-degrees of recursively enumerable sets. Algeb. Logic 9(4), 254–256 (1970)
Dobritsa, V.N.: On word problem of recursively definable group. In: Proc. 3th. All-union Conf. Math. Logic, Novosibirsk, pp. 63–65, (Russian), (1974)
Jockusch, C.G., Jr.: Reducibilities in recursive function theory. PhD dissertation, Massachusetts Institute of Technology (1966)
Marchenkov, S.S.: Tabular powers of maximal sets. Math. Notes 20(3), 766–770 (1976)
Morozov, A.S.: A class of recursively enumberable sets. Siberian Math. J. 28(2), 278–282 (1987)
Omanadze, RSh.: On the completeness of recursively enumerable sets. Bull. Acad. Sci. Georg. SSR. 81(3), 529–532 (1976). ((in Russian))
Omanadze, RSh.: The upper semilattice of recursively enumerable \(Q\)-degrees. Algeb. Logic 23(2), 124–129 (1985)
Omanadze, RSh.: Quasi-degrees of recursively enumerable sets. Comput. Models Springer, US 51(5–6), 289–319 (2003)
Omanadze, RSh., Chitaia, I.O.: \(Q_1\)-degrees of c.e. sets. Arch. Math. logic. 51(5–6), 503–515 (2012)
Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc. 50, 284–316 (1944)
Rogers, H.: Theory of recursive functions and effective computability. McGraw-Hill, New York (1967)
Soare, R.I.: Computational complexity, speedable and levelable sets. J. Symbolic Logic 42(4), 545–563 (1977)
Soare, R.I.: Recursively enumerable sets and degrees. Springer, Verlag (1987)
Yates, C.M.: On the degrees of index sets. Trans. Amer. Math. Soc. 121(2), 309–328 (1966)
Acknowledgements
The authors would like to thank the anonymous referee for many suggestions and improvements throughout the paper. Chitaia’s work was supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) [Grant number: YS-18-168].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chitaia, I., Omanadze, R. Degree structures of conjunctive reducibility. Arch. Math. Logic 61, 19–31 (2022). https://doi.org/10.1007/s00153-021-00774-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-021-00774-7