Limitwise monotonic functions, sets, and degrees on computable domains

Kach, Asher M.; Turetsky, Daniel

doi:10.2178/jsl/1264433912

Cited by 11 publications

(7 citation statements)

References 11 publications

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“…Khoussainov, Hirschfeldt and Semukhin in [14] used a variation of limitwise monotonicity to build an ℵ 1 -categorical but not ℵ 0 -categorical theory whose only computable model is saturated. Kach and Turetsky [17] introduced generalizations of limitwise monotonicity and studied them. Recently, Downey, Kach and Turetsky [6] found a nice connection between limitwise monotonicity and non-highness in c.e.…”

Section: Introductionmentioning

confidence: 99%

Limitwise monotonic sequences and degree spectra of structures

Kalimullin

Khoussainov

Melnikov

2013

Proc. Amer. Math. Soc.

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In this paper, we study effective monotonic approximations of sets and sequences of sets. We show that there is a sequence of sets which has no uniform computable monotonic approximation but has an x-computable monotonic approximation for every hyperimmune degree x. We also construct a Σ 0 2 set which is not limitwise monotonic but is x-limitwise monotonic relative to every non-zero Δ 0 2 degree x. We show that if a sequence of sets is uniformly limitwise monotonic in x for all except countably many degrees x, then it has to be uniformly limitwise monotonic. Finally, we apply these results to investigate degree spectra of abelian groups, equivalence relations, and ℵ 1categorical structures.

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Section: Introductionmentioning

confidence: 99%

Limitwise monotonic sequences and degree spectra of structures

Kalimullin

Khoussainov

Melnikov

2013

Proc. Amer. Math. Soc.

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“…Proof. For the left to right direction we observe that the usual construction (unrelativizing the one given in [8]) has computable block relation as the blocks that are created are never merged.…”

Section: Theorem 23 a Set A Is In Ssilm(q) If And Only If There Is A ...mentioning

confidence: 98%

“…(Harris [6]) There is a Δ 0 3 degree that does not contain a set with a computable strong -representation. [8] modified the notion of limitwise monotonic to give the following:…”

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confidence: 99%

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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“…In contrast to computable categoricity, already for α = 2 relative and plain ∆ 0 α -categoricity differ in many standard classes, e.g. quite surprisingly in the class of equivalence structures [KT10]. Thus, it is clearly more satisfying to have a relatively ∆ 0 α -categorical example that is not ∆ 0 βcategorical for β < α.…”

Section: Introductionmentioning

confidence: 94%

Torsion-free abelian groups with optimal Scott families

Melnikov¹

2018

J. Math. Log.

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We prove that for any computable successor ordinal of the form α = δ + 2k (δ limit and k ∈ ω) there exists computable torsion-free abelian group (TFAG) that is relatively ∆ 0 α -categorical and not ∆ 0 α−1 -categorical. Equivalently, for any such α there exists a computable TFAG whose initial segments are uniformly described by Σ c α infinitary computable formulae up to automorphism (i.e., has a c.e. uniformly Σ c α -Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits.As far as we are concerned, the problem of finding such optimal examples of (relatively) ∆ 0 α -categorical TFAGs for arbitrarily large α was first raised by Goncharov at least 10 years ago, but it has resisted solution (see, e.g., Problem 7.1 in survey [Mel14]). As a byproduct of the proof, we introduce an effective functor that transforms a 0 -computable worthy labeled tree (to be defined) into a computable TFAG. We expect this technical result will find further applications not necessarily related to categoricity questions.Proof of Lemma 5.2. First, we verify the seemingly obvious claim:

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Limitwise monotonic functions, sets, and degrees on computable domains

Cited by 11 publications

References 11 publications

Limitwise monotonic sequences and degree spectra of structures

Limitwise monotonic sequences and degree spectra of structures

A Characterization of the Strongly -Representable Many-One Degrees

Torsion-free abelian groups with optimal Scott families

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