Increasing <i>η</i> ‐representable degrees

Frolov, A. N.; Zubkov, M. V.

doi:10.1002/malq.200810031

Cited by 17 publications

(10 citation statements)

References 5 publications

(5 reference statements)

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“…Similarly to the case of -representable degrees, Kach and Turetsky proved that every Δ 0 3 degree has a computable increasing -representation (proved earlier in [5]). Like in the case of Theorem 1.9, the proof of Theorem 1.12 gives us that if A ∈ SSILM 0 (Q) then A is strongly -representable.…”

Section: Theorem 112 a Set A Has A Computable Increasing -Representat...mentioning

confidence: 90%

A Characterization of the Strongly -Representable Many-One Degrees

Jacobsen-Grocott¹

2021

J. symb. log.

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$\eta $ -representations are a way of coding sets in computable linear orders that were first introduced by Fellner in his thesis. Limitwise monotonic functions have been used to characterize the sets with $\eta $ -representations, and give characterizations for several variations of $\eta $ -representations. The one exception is the class of sets with strong $\eta $ -representations, the only class where the order type of the representation is unique. We introduce the notion of a connected approximation of a set, a variation on $\Sigma ^0_2$ approximations. We use connected approximations to give a characterization of the many-one degrees of sets with strong $\eta $ -representations as well new characterizations of the variations of $\eta $ -representations with known characterizations.

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Section: Theorem 112 a Set A Has A Computable Increasing -Representat...mentioning

confidence: 90%

A Characterization of the Strongly -Representable Many-One Degrees

Jacobsen-Grocott¹

2021

J. symb. log.

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“…This implies that rang(H) ≤ T rang(F ). P Proofs for the next theorem and for a more general result can be found in [8] and [9], respectively. THEOREM 2.9.…”

Section: Strongly η-Representable Degrees and Limitwise Monotonic Funmentioning

confidence: 97%

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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“…Turning our attention to the class of such functions, we firstly note that, if an order type τ is determined by 0 -limitwise monotonic G : Q → N\{0} in the above sense, then τ has a computable presentation. Proposition 3.7 (Frolov,Zubkov;[9]) For any 0 -limitwise monotonic F : Q → N\{0} there exists a computable linear ordering A with order type τ = {F(q) | q ∈ Q}.…”

Section: Rigidity and η-Like Computable Linear Orderingsmentioning

confidence: 99%

Automorphisms ofη-like computable linear orderings and Kierstead's conjecture

2016

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Abstract. We develop an approach to the longstanding conjecture of H.A. Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering B, such that B has no interval of order type η, and such that the order type of B is determined by a 0 ′ -limitwise monotonic maximal block function, there exists computable L ∼ = B such that L has no nontrivial Π 0 1 automorphism.

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Increasing η ‐representable degrees

Abstract: In this paper we prove that any Δ 0 3 degree has an increasing η-representation. Therefore, there is an increasing η-representable set without a strong η-representation.

Cited by 17 publications

References 5 publications

A Characterization of the Strongly -Representable Many-One Degrees

A Characterization of the Strongly -Representable Many-One Degrees

Strongly η-representable degrees and limitwise monotonic functions

Automorphisms ofη-like computable linear orderings and Kierstead's conjecture

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