{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "Q07RA7AMDE"
    }
  ],
  "body": "\n## Summary\n\nA semiring is a set with an addition and a multiplication that obey the axioms of a [[ring|0B1261B5ED]], except that negatives need not exist.\nThe natural numbers with the usual operations are the first example.\nEvery ring is a semiring, but semirings also include the Boolean semiring and the tropical semirings.\nSemirings appear in combinatorics, in tropical geometry and in computer science, for example in shortest path problems and formal languages.\n\n## Explanations\n\n### L1 human\n\nA semiring is a place where you can add and multiply as usual, but you may not be able to subtract.\nThe counting numbers $0, 1, 2, \\dots$ are the model: $3 - 5$ is not a counting number.\nAnother example uses \"minimum\" as addition and ordinary $+$ as multiplication on the real numbers together with $\\infty$.\nThere, the \"sum\" of $3$ and $5$ is $3$, and their \"product\" is $8$.\nThis strange arithmetic is exactly what you need to compute shortest paths in a road network.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "16Y60"
    ],
    "name": "Semiring",
    "same_as": {
      "wikidata": "Q1333055",
      "wikipedia": "Semiring"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Semiring"
      }
    ],
    "tag": "D31S3GR51T"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/D31S3GR51T/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/D31S3GR51T.md",
  "msc": [
    "16Y60"
  ],
  "name": "Semiring",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/D31/D31S3GR51T.md",
  "status": null,
  "tag": "D31S3GR51T",
  "uri": "m2://D31S3GR51T",
  "url": "https://math2.tamnd.com/wiki/semiring/",
  "writing": {
    "tag": "D31S3GR51T",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}