License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.124
URN: urn:nbn:de:0030-drops-181768
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18176/
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Dong, Ruiwen

The Identity Problem in ℤ ≀ ℤ Is Decidable

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LIPIcs-ICALP-2023-124.pdf (0.9 MB)

Abstract

We consider semigroup algorithmic problems in the wreath product ℤ ≀ ℤ. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain the neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of ℤ ≀ ℤ. We show that both problems are decidable. Our result complements the undecidability of the Semigroup Membership Problem (does a semigroup contain a given element?) in ℤ ≀ ℤ shown by Lohrey, Steinberg and Zetzsche (ICALP 2013), and contributes an important step towards solving semigroup algorithmic problems in general metabelian groups.

BibTeX - Entry

@InProceedings{dong:LIPIcs.ICALP.2023.124,
  author =	{Dong, Ruiwen},
  title =	{{The Identity Problem in \mathbb{Z} ≀ \mathbb{Z} Is Decidable}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{124:1--124:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18176},
  URN =		{urn:nbn:de:0030-drops-181768},
  doi =		{10.4230/LIPIcs.ICALP.2023.124},
  annote =	{Keywords: wreath product, algorithmic group theory, identity problem, polynomial semiring, positive coefficients}
}

Keywords: wreath product, algorithmic group theory, identity problem, polynomial semiring, positive coefficients
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023

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