Semigroup Forum

, Volume 95, Issue 1, pp 109–125 | Cite as

Enumerating transformation semigroups

  • James East
  • Attila Egri-Nagy
  • James D. Mitchell
Research Article


We describe general methods for enumerating subsemigroups of finite semigroups and techniques to improve the algorithmic efficiency of the calculations. As a particular application we use our algorithms to enumerate all transformation semigroups up to degree 4. Classification of these semigroups up to conjugacy, isomorphism and anti-isomorphism, by size and rank, provides a solid base for further investigations of transformation semigroups.


Transformation semigroup Computational enumeration Multiplication table Ideal structure 



This work was partially supported by the NeCTAR Research Cloud, an initiative of the Australian Government’s Super Science scheme and the Education Investment Fund; and by the EU Project BIOMICS (Contract Number CNECT-ICT-318202).


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • James East
    • 1
  • Attila Egri-Nagy
    • 2
  • James D. Mitchell
    • 3
  1. 1.School of Computing, Engineering and Mathematics, Centre for Research in MathematicsUniversity of Western Sydney (Parramatta Campus)PenrithAustralia
  2. 2.Akita International UniversityYuwa, Akita-CityJapan
  3. 3.Mathematical InstituteUniversity of St AndrewsFifeScotland, UK

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