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CELLULAR AUTOMATA OVER ALGEBRAIC STRUCTURES

Part of: Algebraic structures Topological dynamics

Published online by Cambridge University Press:  26 April 2021

ALONSO CASTILLO-RAMIREZ ,
O. MATA-GUTIÉRREZ  and
ANGEL ZALDIVAR-CORICHI
ALONSO CASTILLO-RAMIREZ
Affiliation:
Department of Mathematics, University Centre of Exact Sciences and Engineering, University of Guadalajara, Guadalajara, México e-mails: [email protected], [email protected], [email protected]
O. MATA-GUTIÉRREZ
Affiliation:
Department of Mathematics, University Centre of Exact Sciences and Engineering, University of Guadalajara, Guadalajara, México e-mails: [email protected], [email protected], [email protected]
ANGEL ZALDIVAR-CORICHI
Affiliation:
Department of Mathematics, University Centre of Exact Sciences and Engineering, University of Guadalajara, Guadalajara, México e-mails: [email protected], [email protected], [email protected]
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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

Primary: 08A62: Finitary algebras 37B15: Cellular automata
Secondary: 06E99: None of the above, but in this section 08A35: Automorphisms, endomorphisms
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 306 - 319
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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