Infinite finitely generated automata semigroups have infinite orbits

TL;DR

This paper establishes a fundamental equivalence between the infiniteness of automata-generated semigroups and the existence of infinite orbits of right-infinite words, deepening understanding of automata semigroup dynamics.

Contribution

It proves that a finitely generated automata semigroup is infinite if and only if some right-infinite word has an infinite orbit, a new characterization of automata semigroup infiniteness.

Findings

Automata semigroup is infinite iff some right-infinite word has infinite orbit

Provides a criterion for infiniteness based on infinite orbits

Deepens understanding of automata semigroup structure

Abstract

We prove that the semigroup generated by a finite state Mealy automaton $\mathcal{A}=(Q,A,\tau)$ is infinite if and only if there exists some right-infinite word in the alphabet $A$ with infinite orbit.