Semigroups Arising From Asynchronous Automata

TL;DR

This paper introduces expanding automaton semigroups derived from asynchronous automata, explores their properties, and demonstrates their complex dynamics and relationships with other automorphism classes.

Contribution

It defines expanding automaton semigroups, compares them with existing classes, and constructs free partially commutative monoids as synchronous automaton semigroups.

Findings

Expanding automaton semigroups are strictly larger than synchronous automaton semigroups.

Decidability issues arise in their actions on regular rooted trees.

The class is not closed under normal ideal extensions, unlike asynchronous automaton semigroups.

Abstract

We introduce a new class of semigroups arising from a restricted class of asynchronous automata. We call these semigroups "expanding automaton semigroups." We show that the class of synchronous automaton semigroups is strictly contained in the class of expanding automaton semigroups, and that the class of expanding automaton semigroups is strictly contained in the class of asynchronous automaton semigroups. We investigate the dynamics of expanding automaton semigroups acting on regular rooted trees, and show that undecidability arises in these actions. We show that this class is not closed under taking normal ideal extensions, but the class of asynchronous automaton semigroups is closed under taking these extensions. We construct every free partially commutative monoid as a synchronous automaton semigroup.