# Orbits of Automaton Semigroups and Groups

@article{DAngeli2019OrbitsOA, title={Orbits of Automaton Semigroups and Groups}, author={D. D'Angeli and Dominik Francoeur and E. Rodaro and Jan Philipp W{\"a}chter}, journal={ArXiv}, year={2019}, volume={abs/2007.10273} }

We study the orbits of right infinite or $\omega$-words under the action of semigroups and groups generated by automata. We see that an automaton group or semigroup is infinite if and only if it admits an $\omega$-word with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton… Expand

#### 2 Citations

Orbit Expandability of Automaton Semigroups and Groups

- Computer Science, Mathematics
- Theor. Comput. Sci.
- 2020

This paper shows that, given a word u, an automaton T and a number k as input, it is decidable to check whether u is k-expandable with respect to the action of T , and improves the upper bound on the maximal orbit-increasing suffix length. Expand

Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness

- Computer Science, Mathematics
- ArXiv
- 2017

It is obtained that the finiteness problem for automaton subsemigroups of semig groups generated by invertible, yet partial automata, so called automaton-inverse semigroups, is also undecidable. Expand

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Orbit Expandability of Automaton Semigroups and Groups

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This paper shows that, given a word u, an automaton T and a number k as input, it is decidable to check whether u is k-expandable with respect to the action of T , and improves the upper bound on the maximal orbit-increasing suffix length. Expand

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