On Finite Monoids of Cellular Automata
Abstract
For any group G and set A, a cellular automaton over G and A is a transformation τ:AG→AG defined via a finite neighbourhood S⊆G (called a memory set of τ) and a local function μ:AS→A. In this paper, we assume that G and A are both finite and study various algebraic properties of the finite monoid CA(G,A) consisting of all cellular automata over G and A. Let ICA(G;A)G and A. In the first part, using information on the conjugacy classes of subgroups of G, we give a detailed description of the structure of ICA(G;A) in terms of direct and wreath products. In the second part, we study generating sets of CA(G;A). In particular, we prove that CA(G,A) cannot be generated by cellular automata with small memory set, and, when G is finite abelian, we determine the minimal size of a set V⊆CA(G;A) such that CA(G;A)=⟨ICA(G;A)∪V⟩.
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