Effectively categorical abelian groups

Downey, Rodney G.; Melnikov, Alexander G.

doi:10.1016/j.jalgebra.2012.09.020

Cited by 26 publications

(48 citation statements)

References 26 publications

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“…For (an incomplete) survey of results in the field see, e.g., Khisamiev [26] and Downey [12]. See also our paper [8] for a discussion of some recent results in the area.…”

Section: Computable Abelian Groupsmentioning

confidence: 98%

“…Recently Downey and Melnikov [8] studied computable homogeneous completely decomposable groups. These are completely decomposable groups in which all elementary summands are isomorphic.…”

Section: 5mentioning

confidence: 99%

“…It is rather unexpected that the specific recursion-theoretic notion of semi-low sets appears in a description of ∆ 0 2 -categorical groups. The purpose of this paper is to extend the results from [8] to the general case of arbitrary completely decomposable groups.…”

Section: 5mentioning

confidence: 99%

“…Every computable completely decomposable group is ∆ 0 5 -categorical. The proof of Theorem 3.1 exploits methods from [8] as well as a new algebraic notion of a labeled regular set which is central to the proof. A labeled regular set is a collection of elements which possess some nice properties (Definition 3.6) together with a function which indicates characteristics of elements of the set.…”

Section: 5mentioning

confidence: 99%

“…The whole enterprise of combinatorial group theory, looking at effective procedures in finitely presented groups, is another clear vindication of a program looking at effective aspects of algebraic structures. In this tradition, the results in this paper require significant new algebraic understanding which, we believe, is of an independent interest; as well as a further development of the notions (such as excellent S-bases) from our earlier paper [8].…”

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confidence: 92%

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Computable completely decomposable groups

Downey

Melnikov

2014

Trans. Amer. Math. Soc.

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Abstract. A completely decomposable group is an abelian group of the form i H i , where H i ≤ (Q, +). We show that every computable completely decomposable group is ∆ 0 5 -categorical. We construct a computable completely decomposable group which is not ∆ 0 4 -categorical, and give an example of a computable completely decomposable group G which is ∆ 0 4 -categorical but not ∆ 0 3 -categorical. We also prove that the index set of computable completely decomposable groups is arithmetical.

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“…For (an incomplete) survey of results in the field see, e.g., Khisamiev [26] and Downey [12]. See also our paper [8] for a discussion of some recent results in the area.…”

Section: Computable Abelian Groupsmentioning

confidence: 98%

“…Recently Downey and Melnikov [8] studied computable homogeneous completely decomposable groups. These are completely decomposable groups in which all elementary summands are isomorphic.…”

Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

mentioning

confidence: 92%

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Computable completely decomposable groups

Downey

Melnikov

2014

Trans. Amer. Math. Soc.

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The Classification Problem for Compact Computable Metric Spaces

Melnikov

Nies

2013

Lecture Notes in Computer Science

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We adjust methods of computable model theory to effective analysis. We use index sets and infinitary logic to obtain classificationtype results for compact computable metric spaces. We show that every compact computable metric space can be uniquely described, up to an isomorphism, by a computable Π3 formula, and that orbits of elements are uniformly given by computable Π2 formulas. We show that the index set for such spaces is Π 0 3 -complete, and the isomorphism problem is Π 0 2 -complete within its index set. We also give further classification results for special classes of compact spaces, and for other related classes of Polish spaces. Finally, as our main result we show that each compact computable metric space is ∆ 0 3 -categorical, and there exists a compact computable Polish space which is not ∆ 0 2 -categorical.

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Categoricity Spectra of Computable Structures

Bazhenov

2021

J Math Sci

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The categoricity spectrum of a computable structure S is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable presentations of S. The degree of categoricity of S is the least degree in the categoricity spectrum of S. This paper is a survey of results on categoricity spectra and degrees of categoricity for computable structures. We focus on the results about degrees of categoricity for linear orders and Boolean algebras. We construct a new series of examples of degrees of categoricity for linear orders.

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Effectively categorical abelian groups

Cited by 26 publications

References 26 publications

Computable completely decomposable groups

Computable completely decomposable groups

The Classification Problem for Compact Computable Metric Spaces

Categoricity Spectra of Computable Structures

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