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Another note on the join property

Author: Mingzhong Cai
Journal: Proc. Amer. Math. Soc. 143 (2015), 4059-4072
MSC (2010): Primary 03D28, 03D55
Published electronically: March 18, 2015
MathSciNet review: 3359594
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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 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.

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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

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
Article copyright: © Copyright 2015 American Mathematical Society

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