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Membership and Conjugacy in Inverse Semigroups

Authors Lukas Fleischer , Florian Stober , Alexander Thumm , Armin Weiß

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ACM Subject Classification
  • Theory of computation → Algebraic language theory
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Circuit complexity
Keywords
  • inverse semigroups
  • membership
  • conjugacy
  • finite automata

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Abstract

The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for finite inverse semigroups. The closely related membership problem for finite semigroups has been shown to be PSPACE-complete in the transformation model by Kozen (1977) and NL-complete in the Cayley table model by Jones, Lien, and Laaser (1976). More recently, both the membership and the conjugacy problem for finite inverse semigroups were shown to be PSPACE-complete in the partial bijection model by Jack (2023).
Here we present a more detailed analysis of the complexity of the membership and conjugacy problems parametrized by varieties of finite inverse semigroups. We establish dichotomy theorems for the partial bijection model and for the Cayley table model. In the partial bijection model these problems are in NC (resp. NP for conjugacy) for strict inverse semigroups and PSPACE-complete otherwise. In the Cayley table model we obtain general 𝖫-algorithms as well as NPOLYLOGTIME upper bounds for Clifford semigroups and 𝖫-completeness otherwise.
Furthermore, by applying our findings, we show the following: the intersection non-emptiness problem for inverse automata is PSPACE-complete even for automata with only two states; the subpower membership problem is in NC for every strict inverse semigroup and PSPACE-complete otherwise; the minimum generating set and the equation satisfiability problems are in NP for varieties of finite strict inverse semigroups and PSPACE-complete otherwise.

Cite As Get BibTex

Lukas Fleischer, Florian Stober, Alexander Thumm, and Armin Weiß. Membership and Conjugacy in Inverse Semigroups. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 156:1-156:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.156

Author Details

Lukas Fleischer
  • FMI, University of Stuttgart, Germany
Florian Stober
  • FMI, University of Stuttgart, Germany
Alexander Thumm
  • University of Siegen, Germany
Armin Weiß
  • FMI, University of Stuttgart, Germany

Funding

  • Weiß, Armin: Supported by the German Research Foundation (DFG) grant WE 6835/1-2.

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