On surjunctive monoids

TL;DR

This paper investigates the class of surjunctive monoids, establishing broad conditions under which monoids are surjunctive and identifying specific monoids that are not, thereby advancing understanding of cellular automata behavior over monoids.

Contribution

The paper characterizes classes of monoids that are surjunctive and provides new examples and non-examples, expanding the theoretical framework of cellular automata on monoids.

Findings

All finite monoids are surjunctive.

Certain classes like finitely generated commutative and residually finite monoids are surjunctive.

The bicyclic monoid and its containing monoids are non-surjunctive.

Abstract

A monoid $M$ is called surjunctive if every injective cellular automata with finite alphabet over $M$ is surjective. We show that all finite monoids, all finitely generated commutative monoids, all cancellative commutative monoids, all residually finite monoids, all finitely generated linear monoids, and all cancellative one-sided amenable monoids are surjunctive. We also prove that every limit of marked surjunctive monoids is itself surjunctive. On the other hand, we show that the bicyclic monoid and, more generally, all monoids containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive.