View in EDS

Computably locally compact groups and their closed subgroups.

Saved in:
Bibliographic Details
Title: Computably locally compact groups and their closed subgroups.
Authors: Melnikov, Alexander G.1 (AUTHOR) alexander.g.melnikov@gmail.com, Nies, André2 (AUTHOR) andre@cs.auckland.ac.nz
Source: International Journal of Algebra & Computation. Feb2026, Vol. 36 Issue 1, p81-114. 34p.
Subjects: Compactification (Mathematics), Abelian groups, Compact groups, Computable functions, Topological spaces
Abstract: Given a computably locally compact Polish space M, we show that its 1-point compactification M ∗ is computably compact. Then, for a computably locally compact group G, we show that the Chabauty space (G) of closed subgroups of G has a canonical effectively-closed (i.e., Π 1 0 ) presentation as a subspace of the hyperspace (G ∗) of closed sets of G ∗ . We construct a computable discrete abelian group H such that (H) is not computably closed in (H ∗) ; in fact, the only computable points of (H) are the trivial group and H itself, while (H) is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of (G) in terms of the countable meet groupoid of G introduced recently by the authors (arXiv:2204.09878). We apply our results and techniques to show that the index set of the computable locally compact abelian groups that contain a closed subgroup isomorphic to (ℝ , +) is arithmetical. [ABSTRACT FROM AUTHOR]
Copyright of International Journal of Algebra & Computation is the property of World Scientific Publishing Company 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
Header DbId: egs
DbLabel: Engineering Source
An: 190554588
AccessLevel: 6
PubType: Academic Journal
PubTypeId: academicJournal
PreciseRelevancyScore: 0
IllustrationInfo
Items – Name: Title
  Label: Title
  Group: Ti
  Data: Computably locally compact groups and their closed subgroups.
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Melnikov%2C+Alexander+G%2E%22">Melnikov, Alexander G.</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> alexander.g.melnikov@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Nies%2C+André%22">Nies, André</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> andre@cs.auckland.ac.nz</i>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Algebra+%26+Computation%22">International Journal of Algebra & Computation</searchLink>. Feb2026, Vol. 36 Issue 1, p81-114. 34p.
– Name: Subject
  Label: Subjects
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Compactification+%28Mathematics%29%22">Compactification (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Abelian+groups%22">Abelian groups</searchLink><br /><searchLink fieldCode="DE" term="%22Compact+groups%22">Compact groups</searchLink><br /><searchLink fieldCode="DE" term="%22Computable+functions%22">Computable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Topological+spaces%22">Topological spaces</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Given a computably locally compact Polish space M, we show that its 1-point compactification M ∗ is computably compact. Then, for a computably locally compact group G, we show that the Chabauty space (G) of closed subgroups of G has a canonical effectively-closed (i.e., Π 1 0 ) presentation as a subspace of the hyperspace (G ∗) of closed sets of G ∗ . We construct a computable discrete abelian group H such that (H) is not computably closed in (H ∗) ; in fact, the only computable points of (H) are the trivial group and H itself, while (H) is uncountable. In the case that a computably locally compact group G is also totally disconnected, we provide a further algorithmic characterization of (G) in terms of the countable meet groupoid of G introduced recently by the authors (arXiv:2204.09878). We apply our results and techniques to show that the index set of the computable locally compact abelian groups that contain a closed subgroup isomorphic to (ℝ , +) is arithmetical. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of International Journal of Algebra & Computation is the property of World Scientific Publishing Company 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.)
PLink https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=190554588
RecordInfo BibRecord:
  BibEntity:
    Identifiers:
      – Type: doi
        Value: 10.1142/S0218196725500432
    Languages:
      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 34
        StartPage: 81
    Subjects:
      – SubjectFull: Compactification (Mathematics)
        Type: general
      – SubjectFull: Abelian groups
        Type: general
      – SubjectFull: Compact groups
        Type: general
      – SubjectFull: Computable functions
        Type: general
      – SubjectFull: Topological spaces
        Type: general
    Titles:
      – TitleFull: Computably locally compact groups and their closed subgroups.
        Type: main
  BibRelationships:
    HasContributorRelationships:
      – PersonEntity:
          Name:
            NameFull: Melnikov, Alexander G.
      – PersonEntity:
          Name:
            NameFull: Nies, André
    IsPartOfRelationships:
      – BibEntity:
          Dates:
            – D: 01
              M: 02
              Text: Feb2026
              Type: published
              Y: 2026
          Identifiers:
            – Type: issn-print
              Value: 02181967
          Numbering:
            – Type: volume
              Value: 36
            – Type: issue
              Value: 1
          Titles:
            – TitleFull: International Journal of Algebra & Computation
              Type: main
ResultId 1