The computable dimension of ordered abelian groups

Goncharov, S. S.; Lempp, Steffen

doi:10.1016/s0001-8708(02)00042-7

Cited by 46 publications

(16 citation statements)

References 13 publications

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“…For over fifty years since that first result in [9], computable categoricity for fields has remained largely a mystery. For many other classes of structures, mathematicians have found structural definitions equivalent to computable categoricity: see for instance [13], [14], [20], [24], [26], [35], and [36]. As an example, Goncharov and Dzgoev, and independently Remmel, showed that a linear order is computably categorical if and only if it has only finitely many pairs of adjacent elements.…”

Section: Introductionmentioning

confidence: 99%

Categoricity properties for computable algebraic fields

Hirschfeldt

Kramer

Miller

et al. 2014

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Categoricity properties for computable algebraic fields

Hirschfeldt

Kramer

Miller

et al. 2014

Trans. Amer. Math. Soc.

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“…The following examples comes from Goncharov, Dzgoev ( [13]); Goncharov, Lempp, Solomon ( [14]); and Remmel ([29], [30]). …”

Section: Introductionmentioning

confidence: 99%

Computable categoricity of trees of finite height

2005

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We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ 0 3 -condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆ 0 n+1 -categorical but not ∆ 0 n -categorical.

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“…These related branches include effective field theory (see Frölich and Shepherdson [16], Rabin [39], Metakides and Nerode [37]), computable Boolean algebras (Goncharov [19], Remmel [40]) and computable linear orders (Downey [10]). Other closely related subjects are the study of effectively presented vector spaces [7,8,36] and the theory of computable ordered groups [12,20]. For early developments in the field of computable abelian groups, see Nurtazin [38], Smith [42], Lin [34] and Khisamiev [26].…”

Section: Introductionmentioning

confidence: 99%

Iterated effective embeddings of abelian p-groups

Downey

Melnikov

2014

Int. J. Algebra Comput.

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Communicated by O. Kharlampovich This paper contributes to the theory of recursively presented (see Higman [Subgroups of finitely presented groups, Proc. R. Soc. Ser. A 262 (1961) 455-475]) infinitely generated abelian groups with solvable word problem. Mal'cev [On recursive Abelian groups, Dokl. Akad. Nauk SSSR 146 (1962) 1009-1012] and independently Rabin [Computable algebra, general theory and theory of computable fields, Trans. Amer. Math. Soc. 95 (1960) 341-360] initiated the study of such groups in the early 1960's. In this paper, we develop a technique that we call iterated effective embeddings. The significance of our new technique is that it extends existing methods from the realm of iterated 0 arguments to iterated 0 ones. This is a new phenomenon in computable algebra. We use this technique to confirm a 30 year-old conjecture of Ash, Knight and

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The computable dimension of ordered abelian groups

Cited by 46 publications

References 13 publications

Categoricity properties for computable algebraic fields

Categoricity properties for computable algebraic fields

Computable categoricity of trees of finite height

Iterated effective embeddings of abelian p-groups

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