Semigroup Forum

, Volume 95, Issue 1, pp 192–221 | Cite as

The word problem for some classes of Adian inverse semigroups

  • Muhammad Inam
Research Article


We show that all of the Schützenberger complexes of an Adian inverse semigroup are finite if the Schützenberger complex of every positive word is finite. This enables us to solve the word problem for certain classes of Adian inverse semigroups (and hence for the corresponding Adian semigroups and Adian groups).


Inverse semigroups Positive presentation Cycle free presentation Baumslag-Solitar presentation 



The author of this paper is thankful to John Meakin and Robert Ruyle for their several useful suggestions.


  1. 1.
    Adian, S.I.: Defining relations and algorithmic problems for groups and semigroups. Proc. Steklov Inst. Math. 85, 1–152 (1966)Google Scholar
  2. 2.
    Inam, M., Meakin, J., Ruyle, R.: A structural property of Adian inverse semigroups. Semigroup Forum, 94(1), 93–103 (2017)Google Scholar
  3. 3.
    Lawson, M.V.: Inverse Semigroups. World Scientific Co. Pte. Ltd., Singapore (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Linblad, S.P.: Inverse monoids presented by a single relator. PhD thesis, Dept. of Math., University of Nebraska-Lincoln (2003)Google Scholar
  5. 5.
    Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106, 295–307 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Remmers, J.H.: On the geometry of semigroup presentations. Adv. Math. 36, 283–296 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Stephen, J.B.: Presentations of inverse monoids. J. Pure Appl. Algebra 63, 81–112 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Steinberg, B.: A topological approach to inverse and regular semigroups. Pac. J. Math. 208(2), 367–396 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Steinberg, B.: A Sampler of a Topological Approach to Inverse Semigroups, Algorithms, Automata and Languages. Word Scientific, Singapore (2002)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of West GeorgiaCarrolltonUSA

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