The Turing degrees below generics and randoms

Journal of Symbolic Logic 79 (1):171-178 (2014)
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Abstract

If X0and X1are both generic, the theories of the degrees below X0and X1are the same. The same is true if both are random. We show that then-genericity orn-randomness of X do not suffice to guarantee that the degrees below X have these common theories. We also show that these two theories are different. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis.

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10.1017/jsl.2013.13

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Embedding and Coding below a 1-Generic Degree.Noam Greenberg & Antonio Montalbán - 2003 - Notre Dame Journal of Formal Logic 44 (4):200-216.

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