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Ranks of finite semigroups of one-dimensional cellular automata

Castillo-Ramirez, Alonso; Gadouleau, Maximilien

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Alonso Castillo-Ramirez


Since first introduced by John von Neumann, the notion of cellular automaton has grown into a key concept in computer science, physics and theoretical biology. In its classical setting, a cellular automaton is a transformation of the set of all configurations of a regular grid such that the image of any particular cell of the grid is determined by a fixed local function that only depends on a fixed finite neighbourhood. In recent years, with the introduction of a generalised definition in terms of transformations of the form τ : AG → AG (where G is any group and A is any set), the theory of cellular automata has been greatly enriched by its connections with group theory and topology. In this paper, we begin the finite semigroup theoretic study of cellular automata by investigating the rank (i.e. the cardinality of a smallest generating set) of the semigroup CA(Zn; A) consisting of all cellular automata over the cyclic group Zn and a finite set A. In particular, we determine this rank when n is equal to p, 2k or 2kp, for any odd prime p and k ≥ 1, and we give upper and lower bounds for the general case.


Castillo-Ramirez, A., & Gadouleau, M. (2016). Ranks of finite semigroups of one-dimensional cellular automata. Semigroup Forum, 93(2), 347-362.

Journal Article Type Article
Acceptance Date Feb 22, 2016
Online Publication Date Mar 3, 2016
Publication Date Oct 1, 2016
Deposit Date Mar 17, 2016
Publicly Available Date Mar 18, 2016
Journal Semigroup Forum
Print ISSN 0037-1912
Electronic ISSN 1432-2137
Publisher Springer
Peer Reviewed Peer Reviewed
Volume 93
Issue 2
Pages 347-362


Published Journal Article (Advance online version) (476 Kb)

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Copyright Statement
Advance online version © The Author(s) 2016 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International
License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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