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