The density of the nonbranching degrees

Fejer, Peter A.

doi:10.1016/0168-0072(83)90028-3

Cited by 34 publications

(12 citation statements)

References 10 publications

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“…The construction of such a set B is based on ideas from Fejer's variant of the non-branching degree construction in [Fejer 1983]. [8-t-11; B(x)…”

Section: The Densities Of the P-generic Degrees And The Non-p-genericmentioning

confidence: 99%

The distribution of the generic recursively enumerable degrees

Ding

1992

Arch Math Logic

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Summary. In this paper we investigate problems about densities of e-generic, sgeneric and p-generic degrees. We, in particular, show that all p-generic degrees are non-branching, which answers an open question by Jockusch who asked: whether all s-generic degrees are non-branching and refutes a conjecture of Ingrassia; the set of degrees containing r.e. p-generic sets is the same as the set of r.e. degrees containing an r.e. non-antoreducible set.

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“…The construction of such a set B is based on ideas from Fejer's variant of the non-branching degree construction in [Fejer 1983]. [8-t-11; B(x)…”

Section: The Densities Of the P-generic Degrees And The Non-p-genericmentioning

confidence: 99%

The distribution of the generic recursively enumerable degrees

Ding

1992

Arch Math Logic

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“…As shown in [1], a result of Lachlan [2,Lemma 18] easily implies that in Definition 1.1 it does not matter if infima are taken with respect to all degrees or with respect to just the r.e. degrees.…”

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confidence: 98%

“…In the present paper we consider the branching degrees, the class complementary to the nonbranching degrees, and give certain uniformity results concerning this class of degrees. The techniques involved are more complicated than those required in [1]. Before describing our results, we give the definition and discuss previous results concerning branching degrees.…”

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confidence: 99%

Branching Degrees above low Degrees

Fejer¹

1982

Transactions of the American Mathematical Society

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Abstract. In this paper, we investigate the location of the branching degrees within the recursively enumerable (r.e.) degrees. We show that there is a branching degree below any given nonzero r.e. degree and, using a new branching degree construction and a technique of Robinson, that there is a branching degree above any given low r.e. degree. Our results extend work of Shoenfield and Soare and Lachlan on the generalized nondiamond question and show that the branching degrees form an automorphism base for the r.e. degrees.1. Introduction. In [1] we began a program of taking certain natural and important definable subclasses of the recursively enumerable (r.e.) degrees and studying their relation to the r.e. degrees as a whole. Our hope is that this program will tend to illuminate uniformities in the structure of the r.e. degrees, just as early work in the field (e.g. the Sacks' Splitting and Density Theorems) did, rather than demonstrate pathological aspects of the structure.In [1], the particular class of r.e. degrees considered was the nonbranching degrees. We were able to obtain a strong uniformity result there, namely, that the nonbranching degrees are dense in the r.e. degrees. In the present paper we consider the branching degrees, the class complementary to the nonbranching degrees, and give certain uniformity results concerning this class of degrees. The techniques involved are more complicated than those required in [1]. Before describing our results, we give the definition and discuss previous results concerning branching degrees. Definition 1.1. An r.e. degree is branching if it is the infimum of two incomparable r.e. degrees.

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“…Indeed, by the Sack Splitting theorem (Theorem 2.4) any dense set of degrees, such as the branching (i.e. the infima of pairs of incomparable degrees) (Slaman [1991a]) and nonbranching (Fejer [1983]) degrees, generates R and so is an automorphism base for it. More information on generating sets for R can be found in AmbosSpies [1985].…”

Section: Undecidability and Beyondmentioning

confidence: 99%

The Recursively Enumerable Degrees

Shore

1999

Handbook of Computability Theory

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The density of the nonbranching degrees

Cited by 34 publications

References 10 publications

The distribution of the generic recursively enumerable degrees

The distribution of the generic recursively enumerable degrees

Branching Degrees above low Degrees

The Recursively Enumerable Degrees

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