Stable finiteness of monoid algebras and surjunctivity

TL;DR

This paper proves that monoid algebras of surjunctive monoids are stably finite, meaning invertibility properties of matrices over these algebras are well-behaved, using model theory techniques.

Contribution

It establishes the stable finiteness of monoid algebras for surjunctive monoids, connecting cellular automata properties with algebraic invertibility.

Findings

Monoid algebras of surjunctive monoids are stably finite.

One-sided invertible matrices over these algebras are two-sided invertible.

The proof employs first-order model theory.

Abstract

A monoid $M$ is said to be surjunctive if every injective cellular automaton with finite alphabet over $M$ is surjective. We show that monoid algebras of surjunctive monoids are stably finite. In other words, given any field $K$ and any surjunctive monoid $M$, every one-sided invertible square matrix with entries in the monoid algebra $K[M]$ is two-sided invertible. Our proof uses first-order model theory.