## Defining totality in the enumeration degrees

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- by Mingzhong Cai, Hristo A. Ganchev, Steffen Lempp, Joseph S. Miller and Mariya I. Soskova
- J. Amer. Math. Soc.
**29**(2016), 1051-1067 - DOI: https://doi.org/10.1090/jams/848
- Published electronically: November 2, 2015
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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.## References

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## Bibliographic Information

**Mingzhong Cai**- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755-3551
- MR Author ID: 816369
- Email: Mingzhong.Cai@dartmouth.edu
**Hristo A. Ganchev**- Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
- MR Author ID: 873534
- Email: ganchev@fmi.uni-sofia.bg
**Steffen Lempp**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
- MR Author ID: 247988
- Email: lempp@math.wisc.edu
**Joseph S. Miller**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706-1388
- MR Author ID: 735102
- Email: jmiller@math.wisc.edu
**Mariya I. Soskova**- Affiliation: Faculty of Mathematics and Computer Science, Sofia University, 5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
- MR Author ID: 802392
- Email: msoskova@fmi.uni-sofia.bg
- 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. - © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 1051-1067 - MSC (2010): Primary 03D30
- DOI: https://doi.org/10.1090/jams/848
- MathSciNet review: 3522609