SgpDec: Cascade (De)Compositions of Finite Transformation Semigroups and Permutation Groups

  • Attila Egri-Nagy
  • James D. Mitchell
  • Chrystopher L. Nehaniv
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)


We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.


transformation semigroup permutation group wreath product Krohn-Rhodes Theory 


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  1. 1.
    Alperin, J.L., Bell, R.B.: Groups and Representations. Springer (1995)Google Scholar
  2. 2.
    Delgado, M., Egri-Nagy, A., Mitchell, J.D., Pfeiffer, M.: VIZ – GAP package for visualisation (2014),
  3. 3.
    Dini, P., Nehaniv, C.L., Egri-Nagy, A., Schilstra, M.J.: Exploring the concept of interaction computing through the discrete algebraic analysis of the belousov-zhabotinsky reaction. Biosystems 112(2), 145–162 (2013), Selected papers from the 9th International Conference on Information Processing in Cells and TissuesGoogle Scholar
  4. 4.
    Dömösi, P., Nehaniv, C.L.: Algebraic Theory of Finite Automata Networks: An Introduction. SIAM Series on Discrete Mathematics and Applications, vol. 11. Society for Industrial and Applied Mathematics (2005)Google Scholar
  5. 5.
    Egri-Nagy, A., Nehaniv, C.L.: Cascade Product of Permutation Groups. arXiv:1303.0091v3 [math.GR] (2013),
  6. 6.
    Egri-Nagy, A., Nehaniv, C.L., Mitchell, J.D.: SgpDec – software package for hierarchical decompositions and coordinate systems, Version 0.7+ (2013),
  7. 7.
    Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press (1976)Google Scholar
  8. 8.
    Ellson, J., Gansner, E.R., Koutsofios, E., North, S.C., Woodhull, G.: Graphviz and dynagraph static and dynamic graph drawing tools. In: Graph Drawing Software, pp. 127–148. Springer (2003)Google Scholar
  9. 9.
    The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.7.1 (2013),
  10. 10.
    Ginzburg, A.: Algebraic Theory of Automata. Academic Press (1968)Google Scholar
  11. 11.
    Holcombe, W.M.L.: Algebraic Automata Theory. Cambridge University Press (1982)Google Scholar
  12. 12.
    Holt, D., Eick, B., O’Brien, E.: Handbook of Computational Group Theory. CRC Press (2005)Google Scholar
  13. 13.
    Krasner, M., Kaloujnine, L.: Produit complet des groupes de permutations et problème d’ extension de groupes. Acta Scientiarium Mathematicarum (Szeged) 14, 39–66 (1951)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Krohn, K., Rhodes, J.: Algebraic Theory of Machines. I. Prime Decomposition Theorem for Finite Semigroups and Machines. Transactions of the American Mathematical Society 116, 450–464 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wells, C.: A Krohn-Rhodes theorem for categories. Journal of Algebra 64, 37–45 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Paul Zeiger, H.: Cascade synthesis of finite state machines. Information and Control 10, 419–433 (1967), plus erratumGoogle Scholar
  17. 17.
    Paul Zeiger, H.: Cascade Decomposition Using Covers. In: Arbib, M.A. (ed.) Algebraic Theory of Machines, Languages, and Semigroups, ch. 4, pp. 55–80. Academic Press (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Attila Egri-Nagy
    • 1
    • 3
  • James D. Mitchell
    • 2
  • Chrystopher L. Nehaniv
    • 3
  1. 1.Centre for Research in Mathematics, School of Computing, Engineering and MathematicsUniversity of Western SydneyAustralia
  2. 2.School of Mathematics and StatisticsUniversity of St AndrewsUnited Kingdom
  3. 3.Centre for Computer Science & Informatics ResearchUniversity of HertfordshireUnited Kingdom

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