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Mathematische Zeitschrift

, Volume 228, Issue 3, pp 435–450 | Cite as

Groups and actions in transformation semigroups

  • S.A. Linton
  • G. Pfeiffer
  • E.F. Robertson
  • N. Ruškuc

Abstract.

Let \(S\) be a transformation semigroup of degree \(n\). To each element \(s\in S\) we associate a permutation group \(G_R(s)\) acting on the image of \(s\), and we find a natural generating set for this group. It turns out that the \(\mathcal{R}\)-class of \(s\) is a disjoint union of certain sets, each having size equal to the size of \(G_R(s)\). As a consequence, we show that two \(\mathcal{R}\)-classes containing elements with equal images have the same size, even if they do not belong to the same \(\mathcal{D}\)-class. By a certain duality process we associate to \(s\) another permutation group \(G_L(s)\) on the image of \(s\), and prove analogous results for the \(\mathcal{L}\)-class of \(S\). Finally we prove that the Schützenberger group of the \(\mathcal{H}\)-class of \(s\) is isomorphic to the intersection of \(G_R(s)\) and \(G_L(s)\). The results of this paper can also be applied in new algorithms for investigating transformation semigroups, which will be described in a forthcoming paper.

Mathematics Subject Classification (1991):20M20, 20B40, 20M10 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • S.A. Linton
    • 1
  • G. Pfeiffer
    • 1
  • E.F. Robertson
    • 1
  • N. Ruškuc
    • 1
  1. 1. School of Mathematical and Computational Sciences, University of St Andrews, St Andrews KY16 9SS, Scotland GB

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