Another note on the join property
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- by Mingzhong Cai PDF
- Proc. Amer. Math. Soc. 143 (2015), 4059-4072 Request permission
Abstract:
We first prove two theorems on the low$_2$ degrees and the join property in the local structure $\mathcal {D}(\leq \mathbf {0}’)$: An r.e. degree is low$_2$ if and only if it is bounded by an r.e. degree without the join property (in $\mathcal {D}(\leq \mathbf {0}’)$), and an FPF $\Delta ^0_2$ degree is low$_2$ if and only if it fails to have the join property. We also study the join property in the global structure and show that for every array recursive degree, there is a degree above it which fails to satisfy the join property.References
- M. M. Arslanov, Some generalizations of a fixed-point theorem, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1981), 9–16 (Russian). MR 630478
- Laurent Bienvenu, Noam Greenberg, Antonín Kučera, Joseph S. Miller, André Nies, and Dan Turetsky, Joining non-low c.e. sets with diagonally non-computable functions, J. Logic Comput. 23 (2013), no. 6, 1183–1194. MR 3144880, DOI 10.1093/logcom/ext039
- Rod Downey, Noam Greenberg, Andrew Lewis, and Antonio Montalbán, Extensions of embeddings below computably enumerable degrees, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2977–3018. MR 3034456, DOI 10.1090/S0002-9947-2012-05660-1
- Antonín Kučera, An alternative, priority-free, solution to Post’s problem, Mathematical foundations of computer science, 1986 (Bratislava, 1986) Lecture Notes in Comput. Sci., vol. 233, Springer, Berlin, 1986, pp. 493–500. MR 874627, DOI 10.1007/BFb0016275
- Andrew E. M. Lewis, A note on the join property, Proc. Amer. Math. Soc. 140 (2012), no. 2, 707–714. MR 2846340, DOI 10.1090/S0002-9939-2011-10908-0
- Richard A. Shore, Direct and local definitions of the Turing jump, J. Math. Log. 7 (2007), no. 2, 229–262. MR 2423951, DOI 10.1142/S0219061307000676
- Richard A. Shore, Biinterpretability up to double jump in the degrees below $0’$, Proc. Amer. Math. Soc. 142 (2014), no. 1, 351–360. MR 3119208, DOI 10.1090/S0002-9939-2013-11719-3
- Richard A. Shore and Theodore A. Slaman, Defining the Turing jump, Math. Res. Lett. 6 (1999), no. 5-6, 711–722. MR 1739227, DOI 10.4310/MRL.1999.v6.n6.a10
Additional Information
- Mingzhong Cai
- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics, Dartmouth College, Hanover, NH 03755
- MR Author ID: 816369
- Email: mingzhong.cai@dartmouth.edu
- Received by editor(s): November 11, 2013
- Received by editor(s) in revised form: April 19, 2014, and May 1, 2014
- Published electronically: March 18, 2015
- Additional Notes: Research partially supported by NSF Grant DMS-1266214.
- Communicated by: Mirna Dzamonja
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 4059-4072
- MSC (2010): Primary 03D28, 03D55
- DOI: https://doi.org/10.1090/S0002-9939-2015-12538-5
- MathSciNet review: 3359594