Semigroup Forum

, Volume 96, Issue 3, pp 565–580 | Cite as

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

  • Bernd Billhardt
  • Yanisa Chaiya
  • Ekkachai Laysirikul
  • Nuttawoot Nupo
  • Jintana Sanwong
Research Article


Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(Xf) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(Xf) can be recaptured easily.


Birget–Rhodes expansion Margolis–Meakin expansion E-unitary inverse monoid Dual prehomomorphism 



The authors are grateful to the unknown referee for helpful comments and suggestions improving the presentation of the paper.


  1. 1.
    Auinger, K., Szendrei, M.B.: On \(F\)-inverse covers of inverse monoids. J. Pure Appl. Algebra 204, 493–506 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Billhardt, B.: Expansions of completely simple semigroups. Stud. Sci. Math. Hung. 41(1), 39–58 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Billhardt, B.: A factorization of dual prehomomorphisms and expansions of inverse semigroups. Stud. Sci. Math. Hung. 41(3), 295–308 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Birget, J.C., Rhodes, J.: Almost finite expansions of arbitrary semigroups. J. Pure Appl. Algebra 32, 239–287 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific, Singapore (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Margolis, S.W., Meakin, J.C.: \(E\)-unitary inverse monoids and the Cayley graph of a group presentation. J. Pure Appl. Algebra 58, 45–76 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Petrich, M.: Inverse Semigroups. Wiley, New York (1984)zbMATHGoogle Scholar
  8. 8.
    Szakács, N.: On the graph condition regarding the \(F\)-inverse cover problem. Semigroup Forum 92, 551–558 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Szendrei, M.B.: A note on Birget–Rhodes expansion of groups. J. Pure Appl. Algebra 58, 93–99 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Bernd Billhardt
    • 1
  • Yanisa Chaiya
    • 2
  • Ekkachai Laysirikul
    • 3
  • Nuttawoot Nupo
    • 2
  • Jintana Sanwong
    • 2
  1. 1.Fachbereich 10-Mathematik und Naturwissenschaften, Institut für MathematikUniversität KasselKasselGermany
  2. 2.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  3. 3.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand

Personalised recommendations