{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $L$ be a finite [[distributive lattice|0HM0HQCK6N]] with join $\\vee$ and meet $\\wedge$.\nLet $f_1, f_2, f_3, f_4 \\colon L \\to [0, \\infty)$ be functions such that\n$$f_1(x) \\, f_2(y) \\le f_3(x \\vee y) \\, f_4(x \\wedge y) \\quad \\text{for all } x, y \\in L.$$\nFor $A, B \\subseteq L$, let $A \\vee B = \\set{a \\vee b : a \\in A, b \\in B}$, let $A \\wedge B = \\set{a \\wedge b : a \\in A, b \\in B}$, and let $f(A) = \\sum_{a \\in A} f(a)$.\nThen\n$$f_1(A) \\, f_2(B) \\le f_3(A \\vee B) \\, f_4(A \\wedge B) \\quad \\text{for all } A, B \\subseteq L.$$\n\n## Remarks\n\nIt is also called the four functions theorem.\nEvery finite distributive lattice embeds in the lattice of subsets of a finite set, so it suffices to treat the power set of $\\set{1, \\ldots, n}$.\nThe proof is by induction on $n$; the case $n = 1$ is a direct check of an inequality between four products of nonnegative numbers.\n\nExample: with all four functions equal to $1$, the hypothesis holds and the conclusion is $|A| \\, |B| \\le |A \\vee B| \\, |A \\wedge B|$ for any subsets $A, B$ of $L$.\nTaking $f_1 = f_2 = f_3 = f_4 = \\mu$ for a function $\\mu$ with $\\mu(x)\\mu(y) \\le \\mu(x \\vee y)\\mu(x \\wedge y)$ gives the FKG inequality and, as a special case, the [[Harris inequality|W1YH3YD3S4]].\n\n## Explanations\n\n### L1 human\n\nThe theorem turns an inequality between single elements into the same inequality between whole sets of elements.\nSuppose the weights of any $x$ and $y$ are controlled by the weights of their join and meet.\nThen the total weights of two sets are controlled by the total weights of all their joins and meets.\n\n### L2 human\n\nThe four functions theorem is the common source of the correlation inequalities of combinatorics.\nIt contains the FKG inequality, the [[Harris inequality|W1YH3YD3S4]] and Kleitman's lemma on monotone families of sets.\nThese are used for random graphs, percolation and the statistical mechanics of spin systems, where increasing events tend to occur together.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "0HM0HQCK6N"
    ],
    "kind": "statement",
    "msc": [
      "05D05",
      "06D99",
      "60E15"
    ],
    "name": "Ahlswede–Daykin inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4695203",
      "wikipedia": "Ahlswede–Daykin inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ahlswede–Daykin_inequality"
      }
    ],
    "tag": "4085B8A528",
    "uses": [
      "0HM0HQCK6N"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/4085B8A528/",
  "links": [
    {
      "field": "about",
      "target": "0HM0HQCK6N"
    },
    {
      "field": "uses",
      "target": "0HM0HQCK6N"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/4085B8A528.md",
  "msc": [
    "05D05",
    "06D99",
    "60E15"
  ],
  "name": "Ahlswede–Daykin inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/408/4085B8A528.md",
  "status": null,
  "tag": "4085B8A528",
  "uri": "m2://4085B8A528",
  "url": "https://math2.tamnd.com/wiki/ahlswede-daykin-inequality/",
  "writing": {
    "tag": "4085B8A528",
    "kind": "statement",
    "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
      }
    ]
  }
}