{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(L, \\vee, \\wedge)$ be a [[lattice|WH4SFAACAS]].\nA *prime ideal* of $L$ is an [[ideal|5GW2T2AM1A]] $I$ of $L$ such that:\n\n1. $I \\ne L$;\n2. for all $x, y \\in L$, if $x \\wedge y \\in I$, then $x \\in I$ or $y \\in I$.\n\n## Remarks\n\nA proper ideal $I$ is prime exactly when its complement $L \\setminus I$ is a [[filter|Z15FXM9WM1]]; such a filter is called a prime filter.\nExample: let $X$ be a set, $p \\in X$, and $I = \\set{A \\subseteq X : p \\notin A}$ in the power set of $X$.\n$I$ is an ideal, it is proper since $X \\notin I$, and if $p \\notin A \\cap B$ then $p \\notin A$ or $p \\notin B$.\nNon-example: in the power set of $\\set{1, 2, 3}$, the principal ideal $\\set{\\emptyset, \\set{1}}$ is proper but not prime.\nIndeed $\\set{2} \\cap \\set{3} = \\emptyset$ lies in it, while neither $\\set{2}$ nor $\\set{3}$ does.\nIn a [[distributive lattice|0HM0HQCK6N]], if an ideal $I$ and a filter $F$ are disjoint, then some prime ideal contains $I$ and is disjoint from $F$.\nThis is the prime ideal theorem for distributive lattices, and its proof uses a form of the axiom of choice.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QKWVXJWHP6",
    "kind": "definition",
    "msc": [
      "06B10",
      "06D05"
    ],
    "name": "Prime ideal of a lattice",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ideal_(order_theory)"
      }
    ],
    "tag": "984525NJS0",
    "uses": [
      "WH4SFAACAS",
      "5GW2T2AM1A",
      "J2T75ATRV5",
      "Z15FXM9WM1",
      "0HM0HQCK6N"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/984525NJS0/",
  "links": [
    {
      "field": "defines",
      "target": "QKWVXJWHP6"
    },
    {
      "field": "uses",
      "target": "WH4SFAACAS"
    },
    {
      "field": "uses",
      "target": "5GW2T2AM1A"
    },
    {
      "field": "uses",
      "target": "J2T75ATRV5"
    },
    {
      "field": "uses",
      "target": "Z15FXM9WM1"
    },
    {
      "field": "uses",
      "target": "0HM0HQCK6N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/984525NJS0.md",
  "msc": [
    "06B10",
    "06D05"
  ],
  "name": "Prime ideal of a lattice",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/984/984525NJS0.md",
  "status": null,
  "tag": "984525NJS0",
  "uri": "m2://984525NJS0",
  "url": "https://tamnd.github.io/math2/wiki/prime-ideal-of-a-lattice-definition/",
  "writing": {
    "tag": "984525NJS0",
    "kind": "definition",
    "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
      }
    ]
  }
}