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International Computer Science Symposium in Russia

CSR 2017: Computer Science – Theory and Applications pp 112–125Cite as

Green’s Relations in Finite Transformation Semigroups

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10304))

Abstract

We consider the complexity of Green’s relations when the semigroup is given by transformations on a finite set. Green’s relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.

This work was supported by the DFG grants DI 435/5-2 and KU 2716/1-1.

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Notes

  1. 1.

    When introducing transformation semigroups in terms of actions, then this is the framework of faithful actions.

References

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Acknowledgments

We thank the anonymous referees for several useful suggestions which helped to improve the presentation of this paper.

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Authors and Affiliations

  1. FMI, University of Stuttgart, Universitätsstraße 38, 70569, Stuttgart, Germany

    Lukas Fleischer & Manfred Kufleitner

Authors
  1. Lukas Fleischer
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  2. Manfred Kufleitner
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Corresponding author

Correspondence to Lukas Fleischer .

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Editors and Affiliations

  1. University of Bordeaux , Talence, France

    Pascal Weil

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Fleischer, L., Kufleitner, M. (2017). Green’s Relations in Finite Transformation Semigroups. In: Weil, P. (eds) Computer Science – Theory and Applications. CSR 2017. Lecture Notes in Computer Science(), vol 10304. Springer, Cham. https://doi.org/10.1007/978-3-319-58747-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-58747-9_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-58746-2

  • Online ISBN: 978-3-319-58747-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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