Computable Abelian Groups

Melnikov, Alexander G.

doi:10.1017/bsl.2014.32

Cited by 24 publications

(7 citation statements)

References 78 publications

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“…Во втором случае, когда цоколь конечен, имеем дело с прямой суммой конечного числа циклических и квазициклических групп. Каждая такая группа вычислимо категорична [24] и имеет пунктуальную копию. Таким образом, по крайней мере изоморфизм остается и в этом случае вычислимым.…”

Section: булевы алгебрыunclassified

Пунктуальные Копии Алгебраических Структур

Zubkov¹,

Kalimullin²,

Melnikov³

et al. 2019

Sib. Mat. Zh.

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Аннотация. Оценивается сложность построения пунктуальной онлайн копии алгебраической структуры. Установлены общая верхняя оценка, а также оптимальные оценки для классов булевых алгебр, абелевых p-групп и линейных порядков. Кроме того, развитые здесь методы применяются для решения открытой проблемы Монталбана (2013) о копируемых линейных порядках.

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Section: булевы алгебрыunclassified

Пунктуальные Копии Алгебраических Структур

Zubkov¹,

Kalimullin²,

Melnikov³

et al. 2019

Sib. Mat. Zh.

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“…In contrast to the number of computable copies and existence of a "good" copy, existence of a "bad" copy is a very recent fact that can be found in [Mel14]. We give a proof, using our metatheorem, in §4.4.…”

Section: Introductionmentioning

confidence: 97%

Independence in computable algebra

2015

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We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.1991 Mathematics Subject Classification. 03D45, 03C57, 12Y05.

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“…As we noted above, Mal cev initiated the study of computable abelian groups in the early 1960's, and indeed he was especially interested in the torsion-free ones (see his early paper [Mal38]). Over the past 50 years the topic has accumulated many results and techniques (see surveys [Khi98,Mel14]). In the case of computable torsion-free abelian groups (TFAGs) all results can be sub-divided into three categories:…”

Section: Introductionmentioning

confidence: 99%

“…As a corollary, a TFAG is (relatively) computable categorical iff its (Prüfer) rank is finite. See [Khi98,Mel14] for more results of this sort.…”

Section: Introductionmentioning

confidence: 99%

“…The proof exploits a certain vague analogy between worthy tree-groups and reduced abelian p-groups. A reader familiar with the technique of stripping [DMN14,Mel14] will perhaps see the (vague) analogy of this technique with the proof of Claim 9.6. We note that it is generally unusual to see even a slight technical connection between TFAGs and reduced abelian p-groups.…”

Section: Introductionmentioning

confidence: 99%

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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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Computable Abelian Groups

Abstract: We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.

Cited by 24 publications

References 78 publications

Пунктуальные Копии Алгебраических Структур

Пунктуальные Копии Алгебраических Структур

Independence in computable algebra

Torsion-free abelian groups with optimal Scott families

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