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Von Neumann Regular Cellular Automata

  • Alonso Castillo-RamirezEmail author
  • Maximilien Gadouleau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10248)

Abstract

For any group G and any set A, a cellular automaton (CA) is a transformation of the configuration space \(A^G\) defined via a finite memory set and a local function. Let \(\mathrm {CA}(G;A)\) be the monoid of all CA over \(A^G\). In this paper, we investigate a generalisation of the inverse of a CA from the semigroup-theoretic perspective. An element \(\tau \in \mathrm {CA}(G;A)\) is von Neumann regular (or simply regular) if there exists \(\sigma \in \mathrm {CA}(G;A)\) such that \(\tau \circ \sigma \circ \tau = \tau \) and \(\sigma \circ \tau \circ \sigma = \sigma \), where \(\circ \) is the composition of functions. Such an element \(\sigma \) is called a generalised inverse of \(\tau \). The monoid \(\mathrm {CA}(G;A)\) itself is regular if all its elements are regular. We establish that \(\mathrm {CA}(G;A)\) is regular if and only if \(\vert G \vert = 1\) or \(\vert A \vert = 1\), and we characterise all regular elements in \(\mathrm {CA}(G;A)\) when G and A are both finite. Furthermore, we study regular linear CA when \(A= V\) is a vector space over a field \(\mathbb {F}\); in particular, we show that every regular linear CA is invertible when G is torsion-free (e.g. when \(G=\mathbb {Z}^d, d \ge 1\)), and that every linear CA is regular when V is finite-dimensional and G is locally finite with \(\mathrm {char}(\mathbb {F}) \not \mid o(g)\) for all \(g \in G\).

Keywords

Cellular automata Linear cellular automata Monoids von Neumann regular elements Generalised inverses 

Notes

Acknowledgments

We thank the referees of this paper for their insightful suggestions and corrections. In particular, we thank the first referee for suggesting the references [1, 7, 12], which greatly improved the results of Sect. 4.

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Copyright information

© IFIP International Federation for Information Processing 2017

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Centro Universitario de Ciencias Exactas e IngenieríasUniversidad de GuadalajaraGuadalajaraMexico
  2. 2.School of Engineering and Computing SciencesDurham UniversityDurhamUK

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