{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be an [[algebra|QJGPT2W0FT]].\nAn $n$-ary operation $f$ and an $m$-ary operation $g$ of $A$ *commute* if\n$$f\\bigl(g(a_{11}, \\dots, a_{1m}), \\dots, g(a_{n1}, \\dots, a_{nm})\\bigr) = g\\bigl(f(a_{11}, \\dots, a_{n1}), \\dots, f(a_{1m}, \\dots, a_{nm})\\bigr)$$\nfor all elements $a_{ij} \\in A$, $1 \\le i \\le n$, $1 \\le j \\le m$.\nThe algebra $A$ is *entropic* if every two of its operations commute, including each operation with itself.\n\n## Remarks\n\nEquivalently, $A$ is entropic if and only if each $n$-ary operation $f \\colon A^n \\to A$ is a [[homomorphism|KBXSZC6BV1]] from the [[direct power|WJ73XYBNE4]] $A^n$ to $A$.\nFor an algebra with a single binary operation, the condition for $f = g$ is exactly the [[medial law|VZHN6GA3TA]] $(xy)(uv) = (xu)(yv)$.\n\nExample: a module $M$ over a commutative ring $R$, with addition, the constant $0$, negation and one unary operation $x \\mapsto rx$ for each $r \\in R$.\nAddition commutes with itself because $(a + b) + (c + d) = (a + c) + (b + d)$.\nScalar multiplication commutes with addition because $r(a + b) = ra + rb$, and two scalars commute because $r(sa) = (rs)a = (sr)a = s(ra)$.\nThe checks for $0$ and negation, such as $r0 = 0$ and $-(a + b) = (-a) + (-b)$, are similar.\n\nNon-example: a nonabelian group $G$ under multiplication.\nMultiplication commutes with itself only if $(xy)(uv) = (xu)(yv)$ for all $x, y, u, v$; with $x = v = 1$ this gives $yu = uy$, which fails for some $y, u \\in G$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9T2XVJNBBY",
    "kind": "definition",
    "msc": [
      "08A62",
      "20N02"
    ],
    "name": "Entropic algebra",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Medial_magma"
      }
    ],
    "tag": "71346N77AX",
    "uses": [
      "QJGPT2W0FT",
      "KBXSZC6BV1",
      "WJ73XYBNE4",
      "VZHN6GA3TA"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/71346N77AX/",
  "links": [
    {
      "field": "defines",
      "target": "9T2XVJNBBY"
    },
    {
      "field": "uses",
      "target": "QJGPT2W0FT"
    },
    {
      "field": "uses",
      "target": "KBXSZC6BV1"
    },
    {
      "field": "uses",
      "target": "WJ73XYBNE4"
    },
    {
      "field": "uses",
      "target": "VZHN6GA3TA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/71346N77AX.md",
  "msc": [
    "08A62",
    "20N02"
  ],
  "name": "Entropic algebra",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/713/71346N77AX.md",
  "status": null,
  "tag": "71346N77AX",
  "uri": "m2://71346N77AX",
  "url": "https://tamnd.github.io/math2/wiki/entropic-algebra-definition/",
  "writing": {
    "tag": "71346N77AX",
    "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
      }
    ]
  }
}