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 $$\tau :{A^G} \to {A^G}$$ that preserve the operations AG of induced componentwise from the operations of A. We show τ that is an endomorphism of AG if and only if its local function is a homomorphism. When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting of all such endomorphic cellular automata, is isomorphic to the direct limit of Hom(AS, 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 $${\rm{End}}(A)[G]$$ . Moreover, when A is a finite Boolean algebra, we establish that the number of endomorphic cellular automata over AG admitting a memory set S is precisely $${(k|S|)^k}$$ , where k is the number of atoms of A.