On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata

TL;DR

This paper investigates the torsion problem in automaton semigroups generated by invertible reversible Mealy automata, establishing torsion-freeness for a broad subclass and contributing to understanding their algebraic structure.

Contribution

It proves that for a wide subclass of invertible reversible Mealy automata, the generated semigroup is torsion-free, advancing knowledge in automaton semigroup theory.

Findings

Torsion problem is undecidable in general automaton semigroups.

Torsion-freeness is established for a broad subclass of invertible reversible Mealy automata.

The results deepen understanding of algebraic properties of automaton semigroups.

Abstract

This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.