Effective categoricity of equivalence structures

Calvert, Wesley; Cenzer, Douglas; Harizanov, Valentina S.; Morozov, Andrey

doi:10.1016/j.apal.2005.10.002

Cited by 58 publications

(66 citation statements)

References 27 publications

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“…It is a very easy argument we leave to the reader to prove the following: Theorem 1.4 (Calvert, Cenzer, Harizanov, and Morozov [8]). An equivalence structure E with infinitely many classes is computable if and only if there is a limitwise monotonic function F (with range ω ∪ {∞}) for which there are exactly |{x : F (x) = κ}| many classes of size κ (for each κ ∈ ω ∪ {∞}) in E. Theorem 1.4 can be rephrased to say the computable isomorphism types of computable equivalence structures are specified by limitwise monotonic functions.…”

Section: Definition 13 a Function F Is Limitwise Monotonic If Therementioning

confidence: 99%

Limitwise Monotonic Functions and Their Applications

Downey¹,

Kach²,

Turetsky³

2011

Proceedings of the 11th Asian Logic Conference

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Abstract.We survey what is known about limitwise monotonic functions and sets and discuss their applications in effective algebra and computable model theory. Additionally, we characterize the computably enumerable degrees that are totally limitwise monotonic, show the support strictly increasing 0 -limitwise monotonic sets on Q do not capture the sets with computable strong η-representations, and study the limitwise monotonic spectra of a set.

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Section: Definition 13 a Function F Is Limitwise Monotonic If Therementioning

confidence: 99%

Limitwise Monotonic Functions and Their Applications

Downey¹,

Kach²,

Turetsky³

2011

Proceedings of the 11th Asian Logic Conference

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“…Limitwise monotonicity, and its relativised variants, have since had a number of other applications in effective algebra and computable model theory-see for example [KNS97,CCHM06,Hir01,HMP07] or [DKT] for a recent survey. The first apparent application to computable linear orderings was the result by Coles et al [CDK97] that there exists a computable linear ordering with a (η-like) Π 0 2 initial segment not isomorphic to any computable linear ordering.…”

Section: Introductionmentioning

confidence: 99%

On limitwise monotonicity and maximal block functions

Harris

2015

COM

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. We prove the existence of a limitwise monotonic function g : N → N \ {0} such that, for any Π 0 1 function f : N → N \ {0}, Ran f = Ran g. Relativising this result we deduce the existence of an η-like computable linear ordering A such that, for any Π 0 2 function F : Q → N \ {0}, and η-like B of order type { F (q) | q ∈ Q }, B A . We prove directly that, for any computable A which is either (i) strongly η-like or (ii) η-like with no strongly η-like interval, there exists 0 -limitwise monotonic G : Q → N \ {0} such that A has order type { G(q) | q ∈ Q }. In so doing we provide an alternative proof to the fact that, for every η-like computable linear ordering A with no strongly η-like interval, there exists computable B ∼ = A with Π 0 1 block relation. We also use our results to prove the existence of an η-like computable linear ordering which is ∆ 0 3 categorical but not ∆ 0 2 categorical.

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“…In computable model theory, equivalence relations have also been a subject of study, e.g., [3,7,23], etc. In these papers, equivalence relations of rather low complexity were studied (computable, in the Ershov hierarchy, Σ equivalence relations on hyperarithmetical subsets of ω as a whole.…”

Section: Introductionmentioning

confidence: 99%