© 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.
Published inThe 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)
Pages112 - 125
CitationFLEISCHER, 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.
VersionAM (Accepted Manuscript)
Publisher statementThis 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/
NotesThis 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 .
Book seriesLecture Notes in Theoretical Computer Science and General Issues;10304