{
  "abstract": "Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \\mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \\neq \\mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \\times C_2$, then $M(G;A) \\neq \\mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \\mathrm{CA}(G;A)$ if and only if $3 \\leq |A| < \\infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \\times C_2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18480",
  "author_total": 2,
  "authors": [
    {
      "name": "Edgar Alcalá-Arroyo"
    },
    {
      "name": "Alonso Castillo-Ramirez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f27795xyj8pwd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18480",
  "fulltext_url": null,
  "id": "f27795xyj8pwd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18480",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "nlin.CG"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7221800702",
  "pdf_url": "https://arxiv.org/pdf/2610.18480",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f27795xyj8pwd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2610.18480",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18480"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CG",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "When do lazy and invertible cellular automata generate all cellular automata?",
  "title_text": "When do lazy and invertible cellular automata generate all cellular automata?",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/f27795xyj8pwd",
  "venue": "arXiv",
  "versions": []
}