{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "GT687SNZMQ"
    }
  ],
  "body": "\n## Summary\n\nAn algebraic hyperstructure is a set with at least one hyperoperation, a binary operation whose value at two elements is a nonempty set of elements.\nThe main examples are hypergroups, introduced by Marty in 1934, which generalise [[groups|G5S1M9]] by replacing products with sets of products.\nHyperstructures are special [[multialgebras|3C930MVM0T]], and they arise naturally from quotients of groups by subgroups that are not normal.\n\n## Explanations\n\n### L1 human\n\nIn a group, two elements multiply to one element.\nIn a hypergroup, two elements multiply to a set of elements.\nOn the real line, let the product of $x$ and $y$ be every number between them; the product of $1$ and $3$ is then $[1, 3]$.\nThis product is associative in a natural sense.\nAlso every number $y$ lies in the product of $x$ and $y$, so the products of a fixed $x$ with all numbers cover the line.\nThese two facts make the line a hypergroup.\nHyperstructures study which facts about groups, rings and fields survive in this setting.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "20N20",
      "08A99"
    ],
    "name": "Hyperstructure",
    "same_as": {
      "wikidata": "Q5958653",
      "wikipedia": "Hyperstructure"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hyperstructure"
      }
    ],
    "specializes": [
      "3C930MVM0T"
    ],
    "tag": "1HJF40J5AK"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/1HJF40J5AK/",
  "links": [
    {
      "field": "specializes",
      "target": "3C930MVM0T"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1HJF40J5AK.md",
  "msc": [
    "20N20",
    "08A99"
  ],
  "name": "Hyperstructure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1HJ/1HJF40J5AK.md",
  "status": null,
  "tag": "1HJF40J5AK",
  "uri": "m2://1HJF40J5AK",
  "url": "https://tamnd.github.io/math2/wiki/hyperstructure/",
  "writing": {
    "tag": "1HJF40J5AK",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}