{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "JXBSN6E2X5"
    }
  ],
  "body": "\n## Summary\n\nThe opposite of a structure with a [[binary operation|0M418FE7BB]] $\\ast$ has the same elements and the reversed operation $a \\ast^{\\mathrm{op}} b = b \\ast a$.\nThe opposite of a [[group|G5S1M9]] is a group, and the opposite of a [[monoid|0N6R902FG1]] is a monoid.\nThe [[opposite ring|M8Q24XB0AF]] reverses the multiplication and keeps the addition.\nOpposite structures turn left notions into right notions, for example left modules over a ring into right modules over its opposite ring.\nA structure equals its opposite exactly when its operation is [[commutative|099E72KTF8]].\n\n## Explanations\n\n### L1 human\n\nThe opposite structure reads every product backwards.\nIf $a$ then $b$ gives $a \\ast b$, in the opposite structure $a$ then $b$ gives $b \\ast a$.\nFor the composition of moves of a puzzle, the opposite means doing the moves in the other order.\nFor addition of numbers, nothing changes, because $a + b = b + a$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "08A02",
      "20A05"
    ],
    "name": "Opposite structure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Opposite_group"
      }
    ],
    "tag": "D33V8HE0KY"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/D33V8HE0KY/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/D33V8HE0KY.md",
  "msc": [
    "08A02",
    "20A05"
  ],
  "name": "Opposite structure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D33/D33V8HE0KY.md",
  "status": null,
  "tag": "D33V8HE0KY",
  "uri": "m2://D33V8HE0KY",
  "url": "https://tamnd.github.io/math2/wiki/opposite-structure/",
  "writing": {
    "tag": "D33V8HE0KY",
    "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
      }
    ]
  }
}