A theorem on strongly η-representable sets

Zubkov, M. V.

doi:10.3103/s1066369x09070081

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…sets, we derive (0 , 0 ). Below we need PROPOSITION 3.7 [14]. Let f : Q × ω −→ ω satisfy the following conditions: (1) for any q ∈ Q, there is a finite limit lim s→∞ f (q, s);…”

Section: A Turing Degree Is Called Amentioning

confidence: 99%

Strongly η-representable degrees and limitwise monotonic functions

Zubkov

2011

Algebra Logic

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It is proved that each strongly η-representable degree contains a set that is a range of values for some 0 -limitwise monotonic function pseudoincreasing on Q. Thus we obtain a description of strongly η-representable degrees in terms of 0 -limitwise monotonic functions. PRELIMINARIESWe deal with problems arising at the junction of computability theory and the theory of linear orders. The notation and definitions of computability theory are borrowed from [1]. The set of natural numbers {0, 1, 2, . . .} is denoted by ω. We write X − Y for the set-theoretic difference between sets X ⊆ ω and Y ⊆ ω and write X for the complement ω − X of a set X ⊆ ω. If f is some function, then rang(f ) = {y | (∃x)[f (x) = y]} is its range. Let f : A × ω −→ ω; then lim s→∞ f (x, s) is defined and is finite if there exists a number a x such that (∃s 0 )(∀s > s 0 )[f (x, s) = a x ].By writing x . y we mean a bounded difference; namely, x . y = x − y if x y and x . y = 0 otherwise. We fix an arbitrary computable bijective function N : ω −→ Q and define q i = N (i) for any i ∈ ω.A linear ordering L = L, < L is said to be computable if the universe L and the order relation < L are computable. A standard order relation on ω is denoted by <. For an order type of a dense linear ordering with no greatest and least elements, we write η. The set of rational numbers is denoted by Q.

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Section: A Turing Degree Is Called Amentioning

confidence: 99%

Strongly η-representable degrees and limitwise monotonic functions

Zubkov

2011

Algebra Logic

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“…Proof. The idea, just as in the proof that every ∆ 0 2 degree contains a limitwise monotonic set on ω (see [5] and [13]), is to show that S ⊕ ω is a support increasing limitwise monotonic set on Q for any S in a ∆ 0 2 degree. Towards this end, fix a ∆ 0 2 degree d, a set S ∈ d, and a ∆ 0 2 approximation {S s } s∈ω to S with S 0 = ∅ and |S s | < ∞ for all s.…”

Section: Verificationmentioning

confidence: 99%

Limitwise monotonic functions, sets, and degrees on computable domains

Kach¹,

Turetsky²

2010

J. symb. log.

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We extend the notion of limitwise monotonic functions to include arbitrary computable domains. We then study which sets and degrees aresupport increasing (support strictly increasing)limitwise monotonic on various computable domains. As applications, we provide a characterization of the setsSwith computableincreasing η-representationsusing support increasing limitwise monotonic sets on ℚ and note relationships between the class oforder-computablesets and the class of support increasing (support strictly increasing) limitwise monotonic sets on certain domains.

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“…On the other hand, Rosenstein [16] and Fellner [4] coded any -set and -set, correspondingly, into a computable strong -representation. This result was improved by Zubkov [21]: if A is and B is then is strongly -representable. Thus, there is a “gap” between the upper and lower bounds of the levels of the arithmetic hierarchy of sets that have computable -representations.…”

Section: Introductionmentioning

confidence: 99%

The Simplest Low Linear Order With No Computable Copies

Frolov¹,

Zubkov²

2022

J. symb. log.

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A low linear order with no computable copy constructed by C. Jockusch and R. Soare has Hausdorff rank equal to $2$ . In this regard, the question arises, how simple can be a low linear order with no computable copy from the point of view of the linear order type? The main result of this work is an example of a low strong $\eta $ -representation with no computable copy that is the simplest possible example.

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A theorem on strongly η-representable sets

Cited by 4 publications

References 3 publications

Strongly η-representable degrees and limitwise monotonic functions

Strongly η-representable degrees and limitwise monotonic functions

Limitwise monotonic functions, sets, and degrees on computable domains

The Simplest Low Linear Order With No Computable Copies

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