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The Continuity of Cupping to 0
 Annals of Pure and Applied Logic
, 1993
"... It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the ..."
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It is shown that, if a, b are recursively enumerable degrees such that 0 ! a ! 0 0 and a [ b = 0 0 , then there exists a recursively enumerable degree c such that c ! a and c [ b = 0 0 . 1 Introduction By analogy with the notion of major subset in the context of the lattice of r.e. sets, the r.e. degree c is called a major subdegree of the r.e. degree a if c ! a and for every r.e. degree b a [ b = 0 0 ) c [ b = 0 0 : This paper represents modest progress towards answering the question: Does every r.e. degree which is neither 0 nor 0 0 have a major subdegree? This question was first posed by the second author in 1967, although it does not seem to have appeared in print. In the 70's and 80's efforts were made to answer the question but bore little fruit. In this paper we prove: The second author was supported by NSERC (Canada) Grant A3040, and the third author by National Science Foundation Grant DMS 8807389. The third author presented the results in this paper and tho...