The Myhill property for cellular automata on amenable semigroups

TL;DR

This paper proves that for cellular automata on cancellative left-amenable semigroups, pre-injectivity implies surjectivity, extending classical results to a broader algebraic setting.

Contribution

It establishes the Myhill property for cellular automata on cancellative left-amenable semigroups, generalizing known results from groups to semigroups.

Findings

Pre-injective cellular automata are surjective on these semigroups.

Extends the Myhill property beyond groups to semigroups.

Provides a new algebraic framework for cellular automata analysis.

Abstract

Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $\tau \colon A^S \to A^S$ is surjective.