Highness and bounding minimal pairs

Downey, Rodney G.; Lempp, Steffen; Shore, Richard A.

doi:10.1002/malq.19930390151

Cited by 22 publications

(11 citation statements)

References 18 publications

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“…(1) t is odd and all the af (6 E d(P)), including a: , are become undefined (since rf (2) t is even and either Case 5.1 holds for some Q < L P or some node < L P receives Note that in either case, a; is destroyed and enumerated into B. By Lemma 2.4(v), a: = a: 5 a :…”

Section: ~( P a )mentioning

confidence: 99%

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Splittings of 0' into the Recursively Enumerable Degrees

1996

Mathematical Logic Qtrly

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Lachlan [9] proved that there exists a non-recursive recursively enumerable (I. e.) degree such that every non-recursive I. e. degree below it bounds a minimal pair. In this paper we first prove the dual of this fact. Second, we answer a question of Jockusch by showing that there exists a pair of incomplete r. e. degrees ao, a1 such that for every non-recursive r. e. degree w , there is a pair of incomparable r. e. degrees bo, bi such that w = bo V bl and bi 5 a; for : = 0 , l . Mathematics Subject Classiflcation: 03D25, 03D30.Keywords: Recursively enumerable degree, Relative splitting, Uniform splitting. a v b = 0' and for every x 5 a, either z 5 b or 0' = z V b.for every non-recursive r.e. degree c 5 a, there is a minimal pair of r.e. degrees ao, 01 below c. COOPER (see DOWNEY et al. [2]) showed that one can choose a high, and DOWNEY, LEMPP and SHORE [2] showed that one can choose a high and a o V a 1 = c. LACHLAN [9] also showed that there exists a non-recursive r. e. degree such that below it there is no minimal pair of r.e. degrees. For the dual case, ROBINSON [lo] showed that for every low degree there is a splitting of 0' above it. As an extension of LACHLAN'S Non-Splitting Theorem (see [a]), HARRINGTON [4] showed that there exists an incomplete r.e. degree such that above it there is no splitting (in the r.e. degrees) of 0'. Using ROBINSON'S trick, one can show that for each low r.e. degree c there exists an incomplete r.e. degree a 2 c such that there is no splitting of 0' above a. Both the results of non-splitting properties of 0' can be obtained from LACHLAN'S Non-Splitting Theorem by applying the Pseudo-jump operators (see JOCKUSCH and SHORE [S]).In the first part of the present paper we prove T h e o r e m 1.1. There ezists an incomplete r. e. degree b such that for every incomplete r. e. degree w 2 b, there ezist r. e. degrees 00, a1 such that w < 0 0 , a1 < 0' and 0' = a0 V a l .One should note that DOWNEY and SHORE [3] showed that for all incomparable r. e. degrees ao, 01, there exist incomparable r.e. degrees bo, bl such that aj bj for i = 0,1, a0 V 01 = bo V bl and 4 A bl exists. Therefore, we have C o r o l l a r y 1.2. There ezists an incomplete r. e. degree b such that for every incomplete r. e. degree w 2 b, there ezist r. e. degrees 0 0 , a1 such that w < ao, 01 < 0', 0' = a0 V 01 and a0 A 01 ezists.In the second part of this paper we consider the uniform splitting of r.e. sets.WELCH [17] showed that there exists a pair of incomplete r.e. degrees 0 0 , 01 such that for every r.e. degree w , there exists a pair of r.e. degrees bo, bl such that w = bo V bl and bj 5 ai for i = 0 , l . WELCH'S result is used to show that the class of r. e. degrees is definable with parameters in the Turing degrees structure, by applying a general result of coding which is developed by SLAMAN and WOODIN. The absolute definability of the r. e. degrees within the Turing degrees was proved by COOPER [l].JOCKUSCH [5] gave an easy argument for the WELCH'S result by applying the Sacks Splitting Theorem to KO, wher...

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Section: ~( P a )mentioning

confidence: 99%

“…degree c 5 a, there is a minimal pair of r.e. degrees ao, 01 below c. COOPER (see DOWNEY et al [2]) showed that one can choose a high, and DOWNEY, LEMPP and SHORE [2] showed that one can choose a high and a o V a 1 = c. LACHLAN [9] also showed that there exists a non-recursive r. e. degree such that below it there is no minimal pair of r.e. degrees.…”

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confidence: 99%

Splittings of 0' into the Recursively Enumerable Degrees

1996

Mathematical Logic Qtrly

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“…Cooper [2] showed that every high c. e. degree bounds a minimal pair. Downey, Lempp and Shore [3] showed that there exists a high 2 nonbounding degree. In relation to the noncupping property, Harrington [5] extended the Cooper-Yates Noncupping Theorem by showing that for each high c. e. degree h, there exists a high c. e. degree a < h such that for any c. e. degree x, if h ≤ a ∨ x, then h ≤ x; and Harrington [6] showed that there is a c. e. degree a = 0 such that for all c. e. degrees x, y, if 0 < x ≤ a ≤ y, then there is a c. e. degree z < y such that x ∨ z = y.…”

Section: Introductionmentioning

confidence: 99%

A minimal pair joining to a plus cupping Turing degree

Li¹,

2003

Mathematical Logic Qtrly

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A computably enumerable (c. e.) degree a is called nonbounding, if it bounds no minimal pair, and plus cupping, if every nonzero c. e. degree x below a is cuppable. Let NB and PC be the sets of all nonbounding and plus cupping c. e. degrees, respectively. Both NB and PC are well understood, but it has not been possible so far to distinguish between the two classes. In the present paper, we investigate the relationship between the classes NB and PC, and show that there exists a minimal pair which join to a plus cupping degree, so that PC ⊆ NB. This gives a first known difference between NB and PC.

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“…developed by Lerman [50], Downey, Lempp and Shore [22], with some technique to cope with the interactions between two kinds of strategies. This theorem implies several well-known results, such as Downey, Lempp and Shore's result [22] that there is a high 2 nonbounding c.e. degree.…”

Section: The Proof Of This Theorem Combines Seetapun's Construction Amentioning

confidence: 99%

“…degrees. Downey, Lempp and Shore [22] proved that nonhemimaximal sets can be high 2 . Furthermore, Downey and Stob [24] showed that jump inversion holds for hemimaximal sets.…”

Section: Downey and Stobmentioning

confidence: 99%

Local structure theory and the Ershov hierarchy

Fang¹

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(NUS). I sincerely appreciate their support of my attending the weekly logic seminars, annual workshops and summer schools at NUS and Chinese Academy of Sciences (CAS). I also thank Prof. Theodore A. Slaman from University of California at Berkeley for his invaluable advices and talks. My thanks also go to the Institute for Mathematical Sciences of NUS and Academy of Mathematics and Systems Science of CAS for their financial support of my attending the annual workshops and summer schools.

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Highness and bounding minimal pairs

Cited by 22 publications

References 18 publications

Splittings of 0' into the Recursively Enumerable Degrees

Splittings of 0' into the Recursively Enumerable Degrees

A minimal pair joining to a plus cupping Turing degree

Local structure theory and the Ershov hierarchy

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