{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $T = \\R \\cup \\set{+\\infty}$.\nFor $a, b \\in T$, define\n$$a \\oplus b = \\min(a, b), \\qquad a \\otimes b = a + b,$$\nwith the conventions $\\min(a, +\\infty) = a$ and $a + (+\\infty) = +\\infty$.\nThen $(T, \\oplus, \\otimes)$ is a commutative [[semiring|D31S3GR51T]], the *tropical semiring* in its min-plus form.\n\nThe check of the semiring axioms:\n\n1. $\\oplus$ is associative and commutative, and $+\\infty$ is its neutral element, since $\\min(a, +\\infty) = a$.\n2. $\\otimes$ is associative and commutative, and $0$ is its neutral element, since $a + 0 = a$.\n3. $\\otimes$ distributes over $\\oplus$: $a + \\min(b, c) = \\min(a + b, a + c)$, because adding $a$ preserves the order.\n4. $+\\infty$ is absorbing for $\\otimes$: $a + (+\\infty) = +\\infty$.\n\nFor instance, $3 \\oplus 5 = 3$ and $3 \\otimes 5 = 8$, and $2 \\otimes (3 \\oplus 7) = 5 = (2 \\otimes 3) \\oplus (2 \\otimes 7)$.\n\n## Remarks\n\nThe semiring is [[idempotent|JCSAC9Y1CV]]: $a \\oplus a = \\min(a, a) = a$.\nIt is not a ring, since $3$ has no additive inverse: $\\min(3, b) = +\\infty$ has no solution $b$ in $T$.\nEvery element $a \\ne +\\infty$ has the inverse $-a$ for $\\otimes$, so the tropical semiring is a [[semifield|AHH97EJ6VH]].\nThe map $x \\mapsto -x$ is an isomorphism onto the max-plus semiring $(\\R \\cup \\set{-\\infty}, \\max, +)$, the base of [[max-plus algebra|64G0VSC5F4]].\nSome authors call that max-plus form the tropical semiring instead.\n\n## Explanations\n\n### L1 human\n\nIn tropical arithmetic, \"plus\" means \"take the smaller\" and \"times\" means \"add\".\nSo $3 \\oplus 5 = 3$ and $3 \\otimes 5 = 8$.\nThese rules still obey the usual laws of reordering, regrouping and distributing.\nThey describe shortest paths: along a route the lengths add, and between two routes one takes the shorter.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "D31S3GR51T"
    ],
    "kind": "example",
    "msc": [
      "16Y60",
      "15A80",
      "14T10"
    ],
    "name": "Tropical semiring",
    "same_as": {
      "wikidata": "Q60790098",
      "wikipedia": "Tropical semiring"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tropical_semiring"
      }
    ],
    "tag": "5PNBCRH2JG",
    "uses": [
      "JCSAC9Y1CV",
      "64G0VSC5F4",
      "AHH97EJ6VH"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/5PNBCRH2JG/",
  "links": [
    {
      "field": "instance_of",
      "target": "D31S3GR51T"
    },
    {
      "field": "uses",
      "target": "JCSAC9Y1CV"
    },
    {
      "field": "uses",
      "target": "64G0VSC5F4"
    },
    {
      "field": "uses",
      "target": "AHH97EJ6VH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5PNBCRH2JG.md",
  "msc": [
    "16Y60",
    "15A80",
    "14T10"
  ],
  "name": "Tropical semiring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5PN/5PNBCRH2JG.md",
  "status": null,
  "tag": "5PNBCRH2JG",
  "uri": "m2://5PNBCRH2JG",
  "url": "https://tamnd.github.io/math2/wiki/tropical-semiring/",
  "writing": {
    "tag": "5PNBCRH2JG",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}