Geodesic
CELLULAR AUTOMATA OVER ALGEBRAIC STRUCTURES
Glasgow mathematical journal, Tome 64 (2022) no. 2, pp. 306-319

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DOI

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. 
CASTILLO-RAMIREZ, ALONSO; MATA-GUTIÉRREZ, O.; ZALDIVAR-CORICHI, ANGEL. CELLULAR AUTOMATA OVER ALGEBRAIC STRUCTURES. Glasgow mathematical journal, Tome 64 (2022) no. 2, pp. 306-319. doi: 10.1017/S0017089521000112
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     author = {CASTILLO-RAMIREZ, ALONSO and MATA-GUTI\'ERREZ, O. and ZALDIVAR-CORICHI, ANGEL},
     title = {CELLULAR {AUTOMATA} {OVER} {ALGEBRAIC} {STRUCTURES}},
     journal = {Glasgow mathematical journal},
     pages = {306--319},
     year = {2022},
     volume = {64},
     number = {2},
     doi = {10.1017/S0017089521000112},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089521000112/}
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