{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $E$ be a finite set.\nA *signed subset* of $E$ is a pair $X = (X^+, X^-)$ of disjoint subsets of $E$; its support is $\\underline{X} = X^+ \\cup X^-$, and $-X = (X^-, X^+)$.\nAn *oriented matroid* on $E$, given by its signed circuits, is a set $\\mathcal{C}$ of signed subsets of $E$ such that:\n\n1. $(\\emptyset, \\emptyset) \\notin \\mathcal{C}$.\n2. If $X \\in \\mathcal{C}$, then $-X \\in \\mathcal{C}$.\n3. If $X, Y \\in \\mathcal{C}$ and $\\underline{X} \\subseteq \\underline{Y}$, then $X = Y$ or $X = -Y$.\n4. If $X, Y \\in \\mathcal{C}$, $X \\ne -Y$, and $e \\in X^+ \\cap Y^-$, then there is $Z \\in \\mathcal{C}$ with $Z^+ \\subseteq (X^+ \\cup Y^+) \\setminus \\set{e}$ and $Z^- \\subseteq (X^- \\cup Y^-) \\setminus \\set{e}$.\n\n## Remarks\n\nThe supports of the signed circuits are the circuits of a [[matroid|PX31ZNYS3Z]] on $E$, the underlying matroid.\nAxiom 4 is the signed form of circuit elimination.\nThere are equivalent axiom systems in terms of covectors, of vectors and of chirotopes, which are signed versions of the bases.\n\nExample: let $v_1, \\dots, v_n$ be vectors in $\\R^r$ and $E = \\set{1, \\dots, n}$.\nFor each linear dependence $\\sum_i \\lambda_i v_i = 0$ whose support $\\set{i : \\lambda_i \\ne 0}$ is minimal and nonempty, put $X^+ = \\set{i : \\lambda_i \u003e 0}$ and $X^- = \\set{i : \\lambda_i \u003c 0}$.\nThese signed sets satisfy the axioms; such oriented matroids are called realizable.\nFor $v_1 = e_1$, $v_2 = e_2$, $v_3 = e_1 + e_2$ in $\\R^2$, the only minimal dependence up to scaling is $v_1 + v_2 - v_3 = 0$, so $\\mathcal{C} = \\set{(\\set{1,2}, \\set{3}), (\\set{3}, \\set{1,2})}$.\n\nNon-example: on $E = \\set{1, 2}$, the set $\\mathcal{C} = \\set{(\\set{1}, \\set{2}), (\\set{1,2}, \\emptyset), (\\set{2}, \\set{1}), (\\emptyset, \\set{1,2})}$ fails axiom 3, since all four sets have the same support but $(\\set{1}, \\set{2}) \\ne \\pm (\\set{1,2}, \\emptyset)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "R5EB9NKQ5M",
    "kind": "definition",
    "msc": [
      "52C40",
      "05B35",
      "52B40"
    ],
    "name": "Oriented matroid",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Oriented_matroid"
      }
    ],
    "tag": "YWTT6FD50Y",
    "uses": [
      "PX31ZNYS3Z",
      "0CRMGPAH83"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YWTT6FD50Y/",
  "links": [
    {
      "field": "defines",
      "target": "R5EB9NKQ5M"
    },
    {
      "field": "uses",
      "target": "PX31ZNYS3Z"
    },
    {
      "field": "uses",
      "target": "0CRMGPAH83"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YWTT6FD50Y.md",
  "msc": [
    "52C40",
    "05B35",
    "52B40"
  ],
  "name": "Oriented matroid",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YWT/YWTT6FD50Y.md",
  "status": null,
  "tag": "YWTT6FD50Y",
  "uri": "m2://YWTT6FD50Y",
  "url": "https://tamnd.github.io/math2/wiki/oriented-matroid-definition/",
  "writing": {
    "tag": "YWTT6FD50Y",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}