On bi-embeddable categoricity of algebraic structures

Bazhenov, Nikolay; Rossegger, Dino; Zubkov, M. V.

doi:10.1016/j.apal.2021.103060

Cited by 3 publications

(2 citation statements)

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“…Informally speaking, the poset CR(L ) contains all computable copies of L , up to computable bi-embeddability. This observation provides a connection to the recent study of bi-embeddability spectra [14] and computable bi-embeddable categoricity [15,16].…”

Section: Pr(s ) = ({Deg Pr (A )supporting

confidence: 62%

Computable Reducibility for Computable Linear Orders of Type ω

2022

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We study computable reducibility for computable isomorphic copies of the standard ordering of natural numbers. Following Andrews and Sorbi, we isolate the class of self-full degrees inside the induced degree structure Ω. We show that, over an arbitrary degree from Ω, there exists an infinite antichain of self-full degrees. This fact implies that the poset Ω has continuum many automorphisms. We prove that any non-self-full degree from Ω has no minimal covers, which implies that, inside Ω, the self-full degrees are precisely those elements that have a minimal cover. Bibliography: 18 titles.The paper studies computable reducibility for computable linear orders. Let R and S be binary relations on the set of natural numbers ω. The relation R is computably reducible to S (denoted by R c S) if there is a total computable function f (x) such thatIn this case, one also says that the function f is a computable reduction from R to S.The systematic study of computable reducibility for positive (computably enumerable) equivalence relations was initiated by Ershov [1] who constructed one of the first examples of a universal positive equivalence. Further developments led to sophisticated classifications for various * To whom the correspondence should be addressed.

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Section: Pr(s ) = ({Deg Pr (A )supporting

confidence: 62%

Computable Reducibility for Computable Linear Orders of Type ω

2022

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“…In contrast to these results, there is a strong connection between the complexity of embeddings between bi-embeddable linear orders and Hausdorff rank of linear orders at least at finite levels. Namely, in [6] it was shown that if L is a computable linear order of Hausdorff rank n, then for every bi-embeddable copy of it there is an embedding computable in (2n − 1)-jumps from the atomic diagrams. Moreover, it has been shown that this is the best one can be done: let L be a computable linear order of Hausdorff rank n ≥ 1, then 0 (2n−2) does not compute embeddings between it and all its computable bi-embeddable copies.…”

Section: On Categoricity Of Scattered Linear Orders Of Constructive R...mentioning

confidence: 99%

On Categoricity of Scattered Linear Orders Of constructive Ranks

Frolov

Zubkov

2023

Preprint

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In this article we investigated the complexity of isomorphisms be-tween scattered linear orders of constructive ranks. We gave the general upperbound and proved that this bound is sharp. Also, we constructed examples show-ing that the categoricity level of a given scattered linear order can be an arbitraryordinal from 3 to the upper bound, except for the case when the ordinal is thesuccessor of a limit ordinal. The existence question of the scattered linear orderswhose categoricity level equals to the successor of a limit ordinal is still open.

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On categoricity of scattered linear orders of constructive ranks

Frolov,

Zubkov

2024

Arch. Math. Logic

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On bi-embeddable categoricity of algebraic structures

Cited by 3 publications

References 20 publications

Computable Reducibility for Computable Linear Orders of Type ω

Computable Reducibility for Computable Linear Orders of Type ω

On Categoricity of Scattered Linear Orders Of constructive Ranks

On categoricity of scattered linear orders of constructive ranks

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