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.