On (bi)reversible automata generating lamplighter groups

TL;DR

This paper constructs specific reversible automata that generate lamplighter groups, demonstrating how automata theory can model complex algebraic structures like wreath products.

Contribution

It introduces a method to build reversible automata with states and alphabet from abelian groups that generate lamplighter groups, expanding automata applications.

Findings

Automata generate lamplighter groups as wreath products

Reversible automata can model complex algebraic structures

Construction applies to any nontrivial abelian group

Abstract

For any nontrivial abelian group $\mathbb{X}$ we construct a reversible (bireversible in case the order of $\mathbb{X}$ is odd) automaton such that its set of states and alphabet are identified with $\mathbb{X}$, transition and output functions are defined via the left and the right regular actions correspondingly and its group splits into the restricted wreath product $\mathbb{X} \wr \mathbb{Z}$, i.e. is a lamplighter group.