{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $Q$ be a [[complete lattice|D7A0ZGZT5S]], with join $\\bigvee$ of arbitrary subsets.\nLet $\\cdot$ be a binary operation on $Q$ such that $(Q, \\cdot)$ is a [[semigroup|0V3CFG1KK9]].\nThe structure $(Q, \\cdot)$ is a *quantale* if for every $a \\in Q$ and every family $(b_i)_{i \\in I}$ of elements of $Q$,\n$$a \\cdot \\bigvee_{i \\in I} b_i = \\bigvee_{i \\in I} (a \\cdot b_i) \\quad \\text{and} \\quad \\Big(\\bigvee_{i \\in I} b_i\\Big) \\cdot a = \\bigvee_{i \\in I} (b_i \\cdot a).$$\nThe index set $I$ may be empty.\n\n## Remarks\n\nExample: the [[ideals|0MYSA6MYNA]] of a [[commutative ring|0GYGHKX834]] $R$, ordered by inclusion, with the product of ideals, form a quantale.\nThe join of a family of ideals is their sum, and $J \\sum_i K_i = \\sum_i J K_i$ because the product of ideals distributes over sums.\nExample: a complete lattice in which $\\wedge$ distributes over arbitrary joins is a quantale with $a \\cdot b = a \\wedge b$; these are the frames.\nNon-example: on the two-element lattice $\\set{0, 1}$, let $a \\cdot b = 1$ for all $a, b$.\nThis is associative, but for the empty family $a \\cdot \\bigvee \\emptyset = a \\cdot 0 = 1 \\ne 0 = \\bigvee \\emptyset$, so distributivity fails.\nThe case $I = \\emptyset$ says that $a \\cdot 0 = 0 \\cdot a = 0$, where $0$ is the least element.\nA quantale is *unital* if the semigroup has an identity element, and *commutative* if the multiplication is commutative.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DA619M38C3",
    "kind": "definition",
    "msc": [
      "06F07"
    ],
    "name": "Quantale",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quantale"
      }
    ],
    "tag": "ECB3W5EVK3",
    "uses": [
      "D7A0ZGZT5S",
      "0V3CFG1KK9",
      "0GYGHKX834",
      "0MYSA6MYNA"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ECB3W5EVK3/",
  "links": [
    {
      "field": "defines",
      "target": "DA619M38C3"
    },
    {
      "field": "uses",
      "target": "D7A0ZGZT5S"
    },
    {
      "field": "uses",
      "target": "0V3CFG1KK9"
    },
    {
      "field": "uses",
      "target": "0GYGHKX834"
    },
    {
      "field": "uses",
      "target": "0MYSA6MYNA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ECB3W5EVK3.md",
  "msc": [
    "06F07"
  ],
  "name": "Quantale",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ECB/ECB3W5EVK3.md",
  "status": null,
  "tag": "ECB3W5EVK3",
  "uri": "m2://ECB3W5EVK3",
  "url": "https://tamnd.github.io/math2/wiki/quantale-definition/",
  "writing": {
    "tag": "ECB3W5EVK3",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}