{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "B85E71WZWA"
    }
  ],
  "body": "\n## Summary\n\nA fuzzy algebraic structure replaces a substructure of an algebra by a fuzzy subset.\nA fuzzy subset gives each element a degree of membership between $0$ and $1$.\nThe fuzzy subset must respect the operations: the result of an operation belongs at least as much as the least member among its arguments.\nThe first case studied was the fuzzy subgroup of a [[group|G5S1M9]], introduced by Rosenfeld in 1971.\nFuzzy subrings, fuzzy ideals and fuzzy subalgebras of general algebras followed.\nA fuzzy subalgebra is determined by its level sets, which form a nested family of ordinary subalgebras, one for each membership level.\n\n## Explanations\n\n### L1 human\n\nAn ordinary subgroup either contains an element or does not.\nA fuzzy subgroup lets an element belong only partly, with a degree such as $1/2$.\nThe rule is that combining elements never lowers membership below the weakest of them.\nFor example, in the integers let even numbers have degree $1$ and odd numbers degree $1/2$.\nAdding two numbers of degree at least $1/2$ gives a number of degree at least $1/2$, so this is a fuzzy subgroup.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08A72",
      "20N25",
      "03E72"
    ],
    "name": "Fuzzy algebraic structure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fuzzy_set"
      }
    ],
    "tag": "FSRST924GR"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/FSRST924GR/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/FSRST924GR.md",
  "msc": [
    "08A72",
    "20N25",
    "03E72"
  ],
  "name": "Fuzzy algebraic structure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FSR/FSRST924GR.md",
  "status": null,
  "tag": "FSRST924GR",
  "uri": "m2://FSRST924GR",
  "url": "https://tamnd.github.io/math2/wiki/fuzzy-algebraic-structure/",
  "writing": {
    "tag": "FSRST924GR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}