On algebraic cellular automata

TL;DR

This paper studies algebraic cellular automata over groups with affine algebraic set alphabets, proving their images are closed and they are reversible if bijective, under certain field conditions.

Contribution

It establishes fundamental properties of algebraic cellular automata, including image closure and reversibility, in the context of uncountable algebraically closed fields.

Findings

Cellular automata have closed images in the prodiscrete topology.

Bijective algebraic cellular automata are reversible.

Results depend on the ground field being uncountable and algebraically closed.

Abstract

We investigate some general properties of algebraic cellular automata, i.e., cellular automata over groups whose alphabets are affine algebraic sets and which are locally defined by regular maps. When the ground field is assumed to be uncountable and algebraically closed, we prove that such cellular automata always have a closed image with respect to the prodiscrete topology on the space of configurations and that they are reversible as soon as they are bijective.