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Defining totality in the enumeration degrees


Authors: Mingzhong Cai, Hristo A. Ganchev, Steffen Lempp, Joseph S. Miller and Mariya I. Soskova
Journal: J. Amer. Math. Soc. 29 (2016), 1051-1067
MSC (2010): Primary 03D30
DOI: https://doi.org/10.1090/jams/848
Published electronically: November 2, 2015
MathSciNet review: 3522609
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Abstract: We show that if $ A$ and $ B$ form a nontrivial $ \mathcal {K}$-pair, then there is a semi-computable set $ C$ such that $ A\leq _e C$ and $ B\leq _e \overline {C}$. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal $ \mathcal {K}$-pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the ``c.e. in'' relation for total degrees in the enumeration degrees.


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Additional Information

Mingzhong Cai
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
Email: Mingzhong.Cai@dartmouth.edu

Hristo A. Ganchev
Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
Email: ganchev@fmi.uni-sofia.bg

Steffen Lempp
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: lempp@math.wisc.edu

Joseph S. Miller
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
Email: jmiller@math.wisc.edu

Mariya I. Soskova
Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
Email: msoskova@fmi.uni-sofia.bg

DOI: https://doi.org/10.1090/jams/848
Keywords: Enumeration degrees, total enumeration degrees, automorphisms of degree structures
Received by editor(s): January 27, 2014
Received by editor(s) in revised form: May 13, 2015, and August 26, 2015
Published electronically: November 2, 2015
Additional Notes: The first author was partially supported by NSF Grants DMS-1266214 and DMS-1458061.
The second and fifth authors were also partially supported by BNSF Grant No. DMU 03/07/12.12.2011 and by NSF Binational Grant DMS-1101123 from the United States, Russia, Kazakhstan, and Bulgaria.
The third author was partially supported by AMS-Simons Foundation Collaboration Grant 209087.
The fourth author was partially supported by NSF Grant DMS-1001847.
The fifth author was partially supported by a Marie Curie international outgoing fellowship STRIDE (298471) within the 7th European Community Framework Programme.
Article copyright: © Copyright 2015 American Mathematical Society