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Green’s relations in finite transformation semigroups
conference contribution
posted on 2018-02-22, 13:48 authored by Lukas Fleischer, Manfred Kufleitner© Springer International Publishing AG 2017. 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 com-ponents. 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.
Funding
This work was supported by the DFG grants DI 435/5-2 and KU 2716/1-1.
History
School
- Science
Department
- Computer Science
Published in
The 12th International Computer Science Symposium in Russia (CSR) Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)Volume
10304 LNCSPages
112 - 125Citation
FLEISCHER, L. and KUFLEITNER, M., 2017. Green’s relations in finite transformation semigroups. IN: Weil, P. (ed.) Computer Science – Theory and Applications: The 12th International Computer Science Symposium in Russia (CSR 2017), Kazan, Russia, June 8-12th, 2017, Proceedings. Chaim: Springer, pp. 112-125.Publisher
© SpringerVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Acceptance date
2017-02-15Publication date
2017Notes
This is a pre-copyedited version of a contribution published in Computer Science – Theory and Applications: The 12th International Computer Science Symposium in Russia (CSR 2017) edited by Weil, P. published by Springer International. The definitive authenticated version is available online via https://doi.org/10.1007/978-3-319-58747-9_12 .ISBN
9783319587462ISSN
0302-9743eISSN
1611-3349Publisher version
Book series
Lecture Notes in Theoretical Computer Science and General Issues;10304Language
- en