{
  "abstract": "A lazy cellular automaton $τ:A^G\\to A^G$ is determined by a finite neighborhood $S\\subseteq G$ containing the group identity $e$, a pattern $p\\in A^S$, and a writing symbol $a\\in A\\setminus\\{p(e)\\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\\in T$. We show that, over every group, the existence of a compatible set is necessary for $τ$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \\in S_a$, $s \\in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11811",
  "author_total": 4,
  "authors": [
    {
      "name": "Edgar Alcalá-Arroyo"
    },
    {
      "name": "Alonso Castillo-Ramirez"
    },
    {
      "name": "Maria G. Magaña-Chávez"
    },
    {
      "name": "Luguis de los Santos Baños"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jg8y465wyq85.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11811",
  "fulltext_url": null,
  "id": "3jg8y465wyq85",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11811",
  "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.GR"
    ],
    [
      "arxiv",
      "nlin.CG"
    ],
    [
      "openalex",
      "1703|Cellular Automata and Applications"
    ]
  ],
  "openalex": "W7221080927",
  "pdf_url": "https://arxiv.org/pdf/2610.11811",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jg8y465wyq85.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2610.11811",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11811"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CG",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Idempotence criteria for lazy cellular automata",
  "title_text": "Idempotence criteria for lazy cellular automata",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/3jg8y465wyq85",
  "venue": "arXiv",
  "versions": []
}