A new hierarchy for automaton semigroups

TL;DR

This paper introduces a new hierarchical classification for automaton semigroups based on their asymptotic state-activity growth, extending existing group hierarchies and providing decidability results for the Order Problem.

Contribution

It defines a strict, computable hierarchy for automaton semigroups, linking growth behaviors to entropy and analyzing the decidability of the Order Problem within this framework.

Findings

Hierarchy extends Sidki's for automaton groups

Order Problem is decidable when state-activity is bounded

Order Problem remains open for linear state-activity

Abstract

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata. We prove that the Order Problem is decidable when the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family. The former results are implemented and will be available in the GAP package FR developed by the first author.