{
  "id": "2305.07905",
  "version": "v1",
  "published": "2023-05-13T12:06:50.000Z",
  "updated": "2023-05-13T12:06:50.000Z",
  "title": "Semiaffine sets in Abelian groups",
  "authors": [
    "Iryna Banakh",
    "Taras Banakh",
    "Maria Kolinko",
    "Alex Ravsky"
  ],
  "comment": "5 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "A subset $X$ of an Abelian group $G$ is called $semiaf\\!fine$ if for every $x,y,z\\in X$ the set $\\{x+y-z,x-y+z\\}$ intersects $X$. We prove that a subset $X$ of an Abelian group $G$ is semiaffine if and only if one of the following conditions holds: (1) $X=(H+a)\\cup (H+b)$ for some subgroup $H$ of $G$ and some elements $a,b\\in X$; (2) $X=(H\\setminus C)+g$ for some $g\\in G$, some subgroup $H$ of $G$ and some midconvex subset $C$ of the group $H$. A subset $C$ of a group $H$ is $midconvex$ if for every $x,y\\in C$, the set $\\frac{x+y}2:=\\{z\\in H:2z=x+y\\}$ is a subset of $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-13T12:06:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E16",
      "20K99",
      "52A01"
    ],
    "keywords": [
      "abelian group",
      "semiaffine sets",
      "midconvex subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}