Ranks of finite semigroups of one-dimensional cellular automata

Castillo-Ramirez, Alonso; Gadouleau, Maximilien

doi:10.1007/s00233-016-9783-z

Ranks of finite semigroups of one-dimensional cellular automata

Castillo-Ramirez, Alonso; Gadouleau, Maximilien

Authors

Alonso Castillo-Ramirez

Dr Maximilien Gadouleau m.r.gadouleau@durham.ac.uk
Associate Professor

Abstract

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.

Citation

Castillo-Ramirez, A., & Gadouleau, M. (2016). Ranks of finite semigroups of one-dimensional cellular automata. Semigroup Forum, 93(2), 347-362. https://doi.org/10.1007/s00233-016-9783-z

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
DOI	https://doi.org/10.1007/s00233-016-9783-z

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