Cellular Automata on Racks

TL;DR

This paper introduces the concept of cellular automata on racks, exploring their properties, equivariance, and analogs of classical theorems, expanding cellular automata theory to algebraic structures called racks.

Contribution

It defines cellular automata on racks, studies their properties, and establishes an analog of Curtis-Hedlund's theorem for these automata, a novel extension of cellular automata theory.

Findings

Cellular automata on racks can be defined via local rules.

Equivariant cellular automata commute with rack actions under certain conditions.

An analog of Curtis-Hedlund's theorem is established for cellular automata on racks.

Abstract

In this paper we initiate the study of cellular automata on racks. A rack $R$ is a set with a self-distributive binary operation. The rack $R$ acts on the set $A^R$ of configurations from $R$ to a set $A$. We define the cellular automaton on a rack $R$ as a continuous self-mapping of $A^R$ defined from a system of local rules. The cellular automata on racks do not commute with the rack action. However, under certain conditions, the cellular automata on racks do commute with the rack action. We study the equivariant cellular automta (which commute with the rack action) on racks and prove several properties of these cellular automata including the analog of Curtis-Hedlund's theorem for cellular automata on groups.