The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity
<|reference_start|>The Finiteness Problem for Automaton Semigroups of Extended Bounded Activity: We extend the notion of activity for automaton semigroups and monoids introduced by Bartholdi, Godin, Klimann and Picantin to a more general setting. Their activity notion was already a generalization of Sidki's activity hierarchy for automaton groups. Using the concept of expandability introduced earlier by the current authors, we show that the language of $\omega$-words with infinite orbits is effectively a deterministic B\"uchi language for our extended activity. This generalizes a similar previous result on automaton groups by Bondarenko and the third author. By a result of Francoeur and the current authors, the description via a B\"uchi automaton immediately yields that the finiteness problem for complete automaton semigroups and monoids of bounded activity is decidable. In fact, we obtain a stronger result where we may consider sub-orbits under the action of a regular, suffix-closed language over the generators. This, in particular, also yields that it is decidable whether a finitely generated subsemigroup (or -monoid) of a bounded complete automaton semigroup is finite.<|reference_end|>
@article{d'angeli2024the,
    title={The Finiteness Problem for Automaton Semigroups of Extended Bounded
  Activity},
    author={Daniele D'Angeli and Emanuele Rodaro and Jan Philipp W"achter},
    journal={arXiv preprint arXiv:2406.20056},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.20056},
    primaryClass={cs.FL math.GR}
}