Q 1-degrees of c.e. sets

Omanadze, R. Sh.; Chitaia, Irakli O.

doi:10.1007/s00153-012-0278-7

Cited by 14 publications

(13 citation statements)

References 14 publications

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“…It is shown in [4] that if C, D are hemimaximal sets then C ≡ Q 1 D if and only if C ≡ 1 D. For maximal sets we can prove the following theorem:…”

Section: Simplicity Maximality and Q 1 -Reducibilitymentioning

confidence: 96%

“…It is shown in that if

C, D

are hemimaximal sets then

C \equiv_{Q_{1}} D

if and only if

C \equiv_{1} D

. For maximal sets we can prove the following theorem: Theorem If

M_{1}, M_{2}

are maximal sets and

M_{1} \equiv_{Q_{1}} M_{2}

then

M_{1} \leq_{1} M_{2}

M_{2} \leq_{1} M_{1}

.…”

Section: Simplicity Maximality and Q1‐reducibilitymentioning

confidence: 97%

“…set such that B ≤ Q 1 A then B is hyperhypersimple (cf. [4]). Hence B is not of minimal Q 1 -degree as, by Lemma 3.1, B ∪ {m} < Q 1 B.…”

Section: Corollary 46mentioning

confidence: 99%

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Hyperhypersimple sets and Q₁ -reducibility

Chitaia

2016

Math. Log. Quart.

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“…It is shown in [4] that if C, D are hemimaximal sets then C ≡ Q 1 D if and only if C ≡ 1 D. For maximal sets we can prove the following theorem:…”

Section: Simplicity Maximality and Q 1 -Reducibilitymentioning

confidence: 96%

“…It is shown in that if

C, D

are hemimaximal sets then

C \equiv_{Q_{1}} D

if and only if

C \equiv_{1} D

. For maximal sets we can prove the following theorem: Theorem If

M_{1}, M_{2}

are maximal sets and

M_{1} \equiv_{Q_{1}} M_{2}

then

M_{1} \leq_{1} M_{2}

M_{2} \leq_{1} M_{1}

.…”

Section: Simplicity Maximality and Q1‐reducibilitymentioning

confidence: 97%

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Hyperhypersimple sets and Q₁ -reducibility

Chitaia

2016

Math. Log. Quart.

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“…There is a lot of work on understanding how s‐ and s 1 ‐reducibility behave in terms of immunity properties, particularily on

Π_{1}^{0}

sets (e.g., [5, 6, 22–25]). E.g., the following theorem implies that the s‐degree of any

Π_{1}^{0}

hh‐immune set contains infinitely many s 1 ‐degrees.…”

Section: Relationship Of ⩽Agree To Other Reducibilitiesmentioning

confidence: 99%

“…In § 3.2, we look at what s 1 ‐reducibility can tell us about agreement reducibility. Much of the work on s 1 ‐reducibility, e.g., that in [5, 22, 24], was done in terms of Q 1 ‐reducibility, where

A \leq_{Q_{1}} B

if there is a computable function

f : N \to N

such that

x \in A \Leftrightarrow {normalW}_{f (x)} \subseteq B

and if

x \neq y

, then

W_{f false(x false)}

and

W_{f false(y false)}

are disjoint. If

B \neq ⌀

, then

A \leq_{s_{1}} B \Leftrightarrow \bar{A} \leq_{Q_{1}} \bar{B}

, so results about one reducibility can be converted into results about the other.…”

Section: Introductionmentioning

confidence: 99%

Agreement reducibility

Epstein

Lange

2020

Mathematical Logic Qtrly

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We introduce agreement reducibility and highlight its major features. Given subsets A and B of N, we write A ≤ agree B if there is a total computable function f : N → N satisfying for all e, e , W e ∩ A = W e ∩ A if and only if W f (e) ∩ B = W f (e) ∩ B. We shall discuss the central role N plays in this reducibility and its connection to strong-hyper-hyper-immunity. We shall also compare agreement reducibility to other well-known reducibilities, in particular s 1-and s-reducibility. We came upon this reducibility while studying the computable reducibility of a class of equivalence relations on N based on set-agreement. We end by describing the origin of agreement reducibility and presenting some results in that context.

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