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CELLULAR AUTOMATA OVER ALGEBRAIC STRUCTURES
Published online by Cambridge University Press: 26 April 2021
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.
MSC classification
- Type
- Research Article
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- Copyright
- © The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust
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