On the orbits of automaton semigroups and groups

We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses.First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivale...

Full description

Saved in:
Bibliographic Details
Date:2022
Main Authors: D'Angeli, D., Francoeur, D., Rodaro, E., Wächter, J. Ph.
Format: Article
Language:English
Published: Lugansk National Taras Shevchenko University 2022
Subjects:
Online Access:https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1692
Tags: Add Tag
No Tags, Be the first to tag this record!
Journal Title:Algebra and Discrete Mathematics

Institution

Algebra and Discrete Mathematics
Description
Summary:We investigate the orbits of automaton semigroups and groups to obtain algorithmic and structural results, both for general automata but also for some special subclasses.First, we show that a more general version of the finiteness problem for automaton groups is undecidable. This problem is equivalent to the finiteness problem for left principal ideals in automaton semigroups generated by complete and reversible automata.Then, we look at \(\omega\)-word (i.\,e.\ right infinite words) with a finite orbit. We show that every automaton yielding an \(\omega\)-word with a finite orbit already yields an ultimately periodic one, which is not periodic in general, however. On the algorithmic side, we observe that it is not possible to decide whether a given periodic \(\omega\)-word has an infinite orbit and that we cannot check whether a given reversible and complete automaton admits an \(\omega\)-word with a finite orbit, a reciprocal problem to the finiteness problem for automaton semigroups in the reversible case.Finally, we look at automaton groups generated by reversible but not bi-reversible automata and show that many words have infinite orbits under the action of such automata.