Weak density and cupping in the d-r.e. degrees

Cooper, S. Barry; Lempp, Steffen; Watson, P.S.

doi:10.1007/bf02937291

Cited by 48 publications

(39 citation statements)

References 8 publications

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“…degree. The proof of this theorem uses an in nite injury argument and is essentially the same as in Cooper, Lempp, Watson 1989 where, given r.e. sets U T V , a d-r.e.…”

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

“…So x has to be permitted to enter A by U at step 4 and to leave A at step 6. The former permission is already given by the Uk-change, the latter we build into the strategy as in Cooper, Lempp, Watson 1989 8 Protect A u 0 from other strategies from now on. 9 Set V j = U s 2 j with use j = u 0 and start cycle j + 1 ; 0 simultaneously.…”

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

“…The veri cation now proceeds as in Cooper, Lempp, Watson 1989 , and we leave the details to the reader, except for the following item: When we remove x from A, w e also lose the U e -permission for x which m ust, of course, remain in U e . But note that the win on S e is global and so U e is no longer needed unless V u changes later.…”

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

“…Familiarity with the proof of the weak density result of Cooper, Lempp, Watson 1989 is frequently assumed throughout the paper.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On isolating r.e. and isolated d-r.e. degrees

Arslanov¹,

Lempp²,

Shore³

1996

Computability, Enumerability, Unsolvability

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“…degree. The proof of this theorem uses an in nite injury argument and is essentially the same as in Cooper, Lempp, Watson 1989 where, given r.e. sets U T V , a d-r.e.…”

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

Section: Non-isolated D-re Degreesmentioning

confidence: 99%

“…Familiarity with the proof of the weak density result of Cooper, Lempp, Watson 1989 is frequently assumed throughout the paper.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On isolating r.e. and isolated d-r.e. degrees

Arslanov¹,

Lempp²,

Shore³

1996

Computability, Enumerability, Unsolvability

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“…degrees stems from their affinity even for large n with the r.e. degrees, and a number of recent papers (see for example [1,2,4,6]) have sought to clarify to what extent there are similarities and differences at the individual levels of the n-r.e. degree hierarchy.…”

mentioning

confidence: 99%

A splitting theorem for the 𝑛-r.e. degrees

Cooper¹

1992

Proc. Amer. Math. Soc.

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Abstract.We prove a splitting theorem for the n-r.e. degrees, of which the Sacks Splitting Theorem [9] for the r.e. (= 1-r.e.) degrees is a special case. For background terminology and notation see [4] and [11].Interest in the n-r.e. (and more particularly, d-r.e. = 2-r.e.) degrees stems from their affinity even for large n with the r.e. degrees, and a number of recent papers (see for example [1,2,4,6]) have sought to clarify to what extent there are similarities and differences at the individual levels of the n-r.e. degree hierarchy. Expectations that the few global properties of the r.e. degrees (in particular density) might carry over to the n-r.e. degrees, n > 1, have been only partially fulfilled. Lachlan observed that n-r.e. degrees are not minimal in the n-r.e. degrees, essentially because they are n-REA (see [7]), but density fails, even under 0' (see [3]). On the other hand, some of the pathological properties of the r.e. degrees disappear in the wider context, such as in Downey's diamond theorem [5] for the d-r.e. degrees and Arslanov's use [1] (and more generally in [4]), of d-r.e. degrees for cupping. Therefore the theorem below is welcome in the sense that we now have a global splitting property at every level of the n-r.e. hierarchy of degrees. (We note that the splitting of the n-r.e. degrees in the A2 degrees is an immediate consequence of the n-REA property and the relativised Sacks Splitting Theorem.) Further information relating to the structure of the n-r.e. degrees may be found in [8].Theorem. Let d be n-r.e. Then there exist n-r.e. degrees a, b such that a|b and aub = d.In order to prove the theorem, we give a construction for the case d properly d-r.e. and then indicate how our construction can be adapted to give the inductive step in the proof of the theorem for all « > 1 . The case zz = 1 is just the Sacks Splitting Theorem, of course.Let D G d be d-r.e. We construct d-r.e. sets A, B, and partial recursive (p.r.) functionals T, A, and Q, satisfying the overall requirements A = YD, B = AD, D = QAB

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