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Green’s relations in finite transformation semigroups

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conference contribution
posted on 22.02.2018, 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 LNCS

Pages

112 - 125

Citation

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

© Springer

Version

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

15/02/2017

Publication date

2017

Notes

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

9783319587462

ISSN

0302-9743

eISSN

1611-3349

Book series

Lecture Notes in Theoretical Computer Science and General Issues;10304

Language

en

Location

Kazan, Russia