{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "T8MYNX54PS"
    }
  ],
  "body": "\n## Summary\n\nA residuated mapping is a monotone map $f \\colon P \\to Q$ between [[partially ordered sets|011C29Q1S1]] with a special property.\nFor each $y$, the inequality $f(x) \\le y$ has a largest solution $x$.\nEquivalently, $f$ is the lower adjoint of a [[Galois connection|MBVJ4JQXC5]]; its upper adjoint is called the residual of $f$.\nResiduated maps are the basis of residuated lattices, which model division in ordered algebra and implication in logic.\n\n## Explanations\n\n### L1 human\n\nFor a positive number $a$, the inequality $a x \\le y$ in the reals has the largest solution $x = y / a$.\nThe map $x \\mapsto a x$ is residuated, and its residual is division by $a$.\nIn general, a residuated map is one for which \"how large can $x$ be so that $f(x)$ stays below $y$?\" always has an answer.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "06A15"
    ],
    "name": "Residuated mapping",
    "same_as": {
      "wikidata": "Q7315523",
      "wikipedia": "Residuated mapping"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Residuated_mapping"
      }
    ],
    "tag": "V76Q9B6VS8"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/V76Q9B6VS8/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/V76Q9B6VS8.md",
  "msc": [
    "06A15"
  ],
  "name": "Residuated mapping",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/V76/V76Q9B6VS8.md",
  "status": null,
  "tag": "V76Q9B6VS8",
  "uri": "m2://V76Q9B6VS8",
  "url": "https://tamnd.github.io/math2/wiki/residuated-mapping/",
  "writing": {
    "tag": "V76Q9B6VS8",
    "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
      }
    ]
  }
}