Corpus ID: 67855910

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
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References

SHOWING 1-10 OF 34 REFERENCES
Automaton semigroup constructions
TLDR
It is proved that the free product of two automaton semigroups that contain left identities is again an automaton Semigroup, and it is shown that the class of automatonSemigroups is closed under the combined operation of ‘free product followed by adjoining an identity’. Expand
Orbit Expandability of Automaton Semigroups and Groups
TLDR
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: New constructions results and examples of non-automaton semigroups
TLDR
It is proved that (semigroup) free products of finite semigroups always arise as automaton semig groups, and that the class of automaton monoids is closed under forming wreath products with finite monoids. Expand
On the structure theory of partial automaton semigroups
TLDR
It is shown that no semidirect product of an arbitrary semigroup with a (non-trivial) subsemigroup of the free monogenic semigroup is an automaton semigroup, and any inverse automatonSemigroups can be generated by such anAutomatonSemigroup, showing that automaton-inverse semigroups and inverse Automaton semig groups coincide. Expand
Automaton semigroups and groups: On the undecidability of problems related to freeness and finiteness
TLDR
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
On the Complexity of the Word Problem of Automaton Semigroups and Automaton Groups
TLDR
An intermediate concept between automaton semigroups and automaton groups is introduced, which are generated by partial, yet invertible automata, and there is an automaton-inverse semigroup with a PSpace -complete word problem. Expand
Freeness of automata groups vs boundary dynamics
TLDR
It is proved that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group, and it is shown that theproperty of being not free is equivalent to have a finite Schreier graph in the boundary of the enrichedDual pointed on some essentially non-trivial point. Expand
On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata
TLDR
This paper addresses the torsion problem for a class of automaton semigroups, defined asSemigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet and proves that for a wide subclass the generated semigroup is torsions-free. Expand
A geometric approach to (semi)-groups defined by automata via dual transducers
TLDR
A geometric approach to groups defined by automata via the notion of enriched dual of an inverse transducer is given, which provides some finiteness results and shows that examples of groups having essentially free actions without critical points lie in the class of groupsdefined by the transducers whose enriched duals generate torsion-free semigroup. Expand
Automaton semigroups
  • Alan J. Cain
  • Computer Science, Mathematics
  • Theor. Comput. Sci.
  • 2009
TLDR
The concept of an automaton group generalizes easily to semigroups, and the systematic study of this area is beginning, and various natural semig groups are shown to arise as automaton semiggroups. Expand
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