{
  "id": "2205.15402",
  "version": "v1",
  "published": "2022-05-30T19:59:55.000Z",
  "updated": "2022-05-30T19:59:55.000Z",
  "title": "A generalization of cellular automata over groups",
  "authors": [
    "A. Castillo-Ramirez",
    "M. Sanchez-Alvarez",
    "A. Vazquez-Aceves",
    "A. Zaldivar-Corichi"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GR",
    "cs.FL"
  ],
  "abstract": "Let $G$ be a group and let $A$ be a finite set with at least two elements. A cellular automaton (CA) over $A^G$ is a function $\\tau : A^G \\to A^G$ defined via a finite memory set $S \\subseteq G$ and a local function $\\mu :A^S \\to A$. The goal of this paper is to introduce the definition of a generalized cellular automaton (GCA) $\\tau : A^G \\to A^H$, where $H$ is another arbitrary group, via a group homomorphism $\\phi : H \\to G$. Our definition preserves the essence of CA, as we prove analogous versions of three key results in the theory of CA: a generalized Curtis-Hedlund Theorem for GCA, a Theorem of Composition for GCA, and a Theorem of Invertibility for GCA. When $G=H$, we prove that the group of invertible GCA over $A^G$ is isomorphic to a semidirect product of $\\text{Aut}(G)^{op}$ and the group of invertible CA. Finally, we apply our results to study automorphisms of the monoid $\\text{CA}(G;A)$ consisting of all CA over $A^G$. In particular, we show that every $\\phi \\in \\text{Aut}(G)$ defines an automorphism of $\\text{CA}(G;A)$ via conjugation by the invertible GCA defined by $\\phi$, and that, when $G$ is abelian, $\\text{Aut}(G)$ is embedded in the outer automorphism group of $\\text{CA}(G;A)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-30T19:59:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B15",
      "68Q80"
    ],
    "keywords": [
      "generalization",
      "invertible gca",
      "outer automorphism group",
      "finite memory set",
      "local function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}