Generically computable Abelian groups and isomorphisms.
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| Title: | Generically computable Abelian groups and isomorphisms. |
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| Authors: | Calvert, Wesley1 (AUTHOR), Cenzer, Douglas2 (AUTHOR), Harizanov, Valentina3 (AUTHOR) |
| Source: | Journal of Logic & Computation. Sep2025, Vol. 35 Issue 6, p1-21. 21p. |
| Subjects: | Abelian groups, Isomorphism (Mathematics), Recursion theory, Computable analysis, Cyclic groups, Equivalence classes (Set theory) |
| Abstract: | Approximate computability, in the form of generically computable sets introduced by Jockusch and Schupp, was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions of generically computable structures, and studied in particular equivalence structures and injection structures. We also introduced a graded family of elementarity conditions for substructures, in which the dense substructures more strongly resemble the original structure by being |$\varSigma _{n}$| elementary substructures for a given |$n$|. We now return to group theory, as we explore the generic computability of torsion Abelian groups. We show that any Abelian |$p$| -group has a generically computable copy, and that, for an important family of Abelian |$p$| -groups, a group |$\mathcal{G}$| has a |$\varSigma _{2}$| -generically c.e. copy if and only if it has a computable copy. We also give a partial characterization of the |$\varSigma _{1}$| -generically c.e. Abelian |$p$| -groups, and give a non-trivial characterization of the generically computable torsion Abelian groups. Coarsely computable and |$\varSigma _{n}$| -coarsely c.e. groups are also studied. It is well known that there are computable Abelian groups that are countable sums of cyclic groups of order |$p$| and |$p^{2}$| that are not computably isomorphic. We present a notion of generically computable isomorphism and give conditions under which two such groups will be generically computably isomorphic. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 187976839 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Generically computable Abelian groups and isomorphisms. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Calvert%2C+Wesley%22">Calvert, Wesley</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Cenzer%2C+Douglas%22">Cenzer, Douglas</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Harizanov%2C+Valentina%22">Harizanov, Valentina</searchLink><relatesTo>3</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Logic+%26+Computation%22">Journal of Logic & Computation</searchLink>. Sep2025, Vol. 35 Issue 6, p1-21. 21p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Abelian+groups%22">Abelian groups</searchLink><br /><searchLink fieldCode="DE" term="%22Isomorphism+%28Mathematics%29%22">Isomorphism (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Recursion+theory%22">Recursion theory</searchLink><br /><searchLink fieldCode="DE" term="%22Computable+analysis%22">Computable analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Cyclic+groups%22">Cyclic groups</searchLink><br /><searchLink fieldCode="DE" term="%22Equivalence+classes+%28Set+theory%29%22">Equivalence classes (Set theory)</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Approximate computability, in the form of generically computable sets introduced by Jockusch and Schupp, was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions of generically computable structures, and studied in particular equivalence structures and injection structures. We also introduced a graded family of elementarity conditions for substructures, in which the dense substructures more strongly resemble the original structure by being |$\varSigma _{n}$| elementary substructures for a given |$n$|. We now return to group theory, as we explore the generic computability of torsion Abelian groups. We show that any Abelian |$p$| -group has a generically computable copy, and that, for an important family of Abelian |$p$| -groups, a group |$\mathcal{G}$| has a |$\varSigma _{2}$| -generically c.e. copy if and only if it has a computable copy. We also give a partial characterization of the |$\varSigma _{1}$| -generically c.e. Abelian |$p$| -groups, and give a non-trivial characterization of the generically computable torsion Abelian groups. Coarsely computable and |$\varSigma _{n}$| -coarsely c.e. groups are also studied. It is well known that there are computable Abelian groups that are countable sums of cyclic groups of order |$p$| and |$p^{2}$| that are not computably isomorphic. We present a notion of generically computable isomorphism and give conditions under which two such groups will be generically computably isomorphic. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Logic & Computation is the property of Oxford University Press / USA and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1093/logcom/exaf044 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 21 StartPage: 1 Subjects: – SubjectFull: Abelian groups Type: general – SubjectFull: Isomorphism (Mathematics) Type: general – SubjectFull: Recursion theory Type: general – SubjectFull: Computable analysis Type: general – SubjectFull: Cyclic groups Type: general – SubjectFull: Equivalence classes (Set theory) Type: general Titles: – TitleFull: Generically computable Abelian groups and isomorphisms. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Calvert, Wesley – PersonEntity: Name: NameFull: Cenzer, Douglas – PersonEntity: Name: NameFull: Harizanov, Valentina IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Text: Sep2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 0955792X Numbering: – Type: volume Value: 35 – Type: issue Value: 6 Titles: – TitleFull: Journal of Logic & Computation Type: main |
| ResultId | 1 |
