Strongly sofic monoids, sofic topological entropy, and surjunctivity

TL;DR

This paper introduces strongly sofic monoids, extends properties of sofic groups to them, and proves that such monoids are surjunctive and have stably finite monoid algebras, with implications for cellular automata.

Contribution

It defines strongly sofic monoids, shows they are surjunctive, and proves their monoid algebras are stably finite, extending known results from sofic groups.

Findings

Strongly sofic monoids are surjunctive.

Sofic topological entropy is a conjugacy invariant for these monoids.

Monoid algebras of strongly sofic monoids are stably finite.

Abstract

We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of strongly sofic monoids on compact spaces. We show that sofic topological entropy is a topological conjugacy invariant for such actions and use this fact to prove that every strongly sofic monoid is surjunctive. This means that if $M$ is a strongly sofic monoid and $A$ is a finite alphabet set, then every injective cellular automaton $\tau \colon A^M \to A^M$ is surjective. As an application, we prove that the monoid algebra of a strongly sofic monoid with coefficients in an arbitrary field is always stably finite. Our results are extensions to strongly sofic monoids of two previously known properties of sofic groups. The first one is the celebrated Gromov-Weiss theorem asserting that every sofic group is surjunctive. The second is the Elek-Szab\'o theorem which says that group algebras of sofic groups satisfy Kaplansky's stable finiteness conjecture.