Isomorphisms of Additive Cellular Automata on Finite Groups

TL;DR

This paper investigates the algebraic structure of isomorphisms in additive cellular automata on finite groups, revealing that many are reducible to underlying algebraic isomorphisms, with a complete characterization of linear automorphisms.

Contribution

It provides a comprehensive description of automorphisms of additive cellular automata on finite groups, including a classification of regular and non-regular isomorphisms and the characterization of linear automorphisms.

Findings

Most automata isomorphisms are covered by automorphisms induced by reversible matrices.

A complete description of linear automorphisms of the automata monoid is provided.

Regular automorphisms are reducible to isomorphisms of algebraic structures like the index-group.

Abstract

We study sources of isomorphisms of additive cellular automata on finite groups (called index-group). It is shown that many isomorphisms (called regular) of automata are reducible to the isomorphisms of underlying algebraic structures (such as the index-group, monoid of automata rules, and its subgroup of reversible elements). However for some groups there exist not regular automata isomorphisms. A complete description of linear automorphisms of the monoid is obtained. These automorphisms cover the most part of all automata isomorphisms for small groups and are represented by reversible matrices M such that for any index-group circulant C the matrix M^{-1}CM is an index-group circulant.