The Identity Problem in ℤ ≀ ℤ Is Decidable

Author Ruiwen Dong

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Ruiwen Dong
  • Department of Computer Science, University of Oxford, UK


The author would like to thank Markus Schweighofer and David Sawall for useful discussions.

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Ruiwen Dong. The Identity Problem in ℤ ≀ ℤ Is Decidable. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 124:1-124:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic manipulation
  • wreath product
  • algorithmic group theory
  • identity problem
  • polynomial semiring
  • positive coefficients


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