{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA *semifield* is a set $S$ with two binary operations $+$ and $\\cdot$ such that:\n\n1. $(S, +)$ is an [[abelian group|082GHNPJ53]], with identity element $0$.\n2. Both distributive laws hold: $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \\in S$.\n3. There is an element $1 \\ne 0$ with $1 \\cdot a = a \\cdot 1 = a$ for all $a \\in S$.\n4. For all $a, b \\in S$ with $a \\ne 0$, each of the equations $a x = b$ and $y a = b$ has exactly one solution $x \\in S$ and exactly one solution $y \\in S$.\n\nMultiplication is not required to be associative or commutative.\n\n## Remarks\n\nClauses 3 and 4 say that $S \\setminus \\set{0}$ is a loop under multiplication.\nA structure that satisfies clauses 1, 2 and 4 but not necessarily 3 is called a presemifield.\nIf $S$ is finite, clause 4 is equivalent to the absence of zero divisors: $ab = 0$ implies $a = 0$ or $b = 0$.\nAn associative semifield is a [[division ring|0EA5BTA0CX]], and a finite associative semifield is a finite [[field|04C5ZBG1WT]] by Wedderburn's little theorem.\n\nExample: every field is a semifield, since it satisfies all four clauses and is moreover associative.\nThe real octonions are a semifield in which multiplication is not associative.\n\nNon-example: the ring $\\Z$ satisfies clauses 1, 2 and 3 but fails clause 4, because $2x = 1$ has no solution $x \\in \\Z$.\n\nIn another usage, found for instance in tropical geometry, a semifield is a commutative semiring in which every nonzero element has a multiplicative inverse, such as $(\\R \\cup \\set{-\\infty}, \\max, +)$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "AHH97EJ6VH",
    "kind": "definition",
    "msc": [
      "12K10",
      "17A35"
    ],
    "name": "Semifield",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Semifield"
      }
    ],
    "tag": "1JCQTY728A",
    "uses": [
      "082GHNPJ53",
      "0EA5BTA0CX",
      "04C5ZBG1WT"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/1JCQTY728A/",
  "links": [
    {
      "field": "defines",
      "target": "AHH97EJ6VH"
    },
    {
      "field": "uses",
      "target": "082GHNPJ53"
    },
    {
      "field": "uses",
      "target": "0EA5BTA0CX"
    },
    {
      "field": "uses",
      "target": "04C5ZBG1WT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1JCQTY728A.md",
  "msc": [
    "12K10",
    "17A35"
  ],
  "name": "Semifield",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1JC/1JCQTY728A.md",
  "status": null,
  "tag": "1JCQTY728A",
  "uri": "m2://1JCQTY728A",
  "url": "https://math2.tamnd.com/wiki/semifield-definition/",
  "writing": {
    "tag": "1JCQTY728A",
    "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
      }
    ]
  }
}