Cellular automata over algebraic structures

Abstract

Let G be a group and A a set equipped with a collection of finitary operations. We study cellular automata τ : A G → A G that preserve the operations of A G induced componentwise from the operations of A. We establish that the set EndCA(G; A), consisting of all such cellular automata, is isomorphic to the direct limit of Hom(A S , A), where S runs among all finite subsets of G. In particular, when A is an R-module, we show that EndCA(G; A) is isomorphic to the group algebra End(A) [G]. Moreover, when A is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over A G admitting a memory set S is precisely (k|S|) k , where k is the number of atoms of A.