{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $G$ be a [[group|G5S1M9]], let $I$ and $\\Lambda$ be nonempty [[sets|0SJ3RX9TXA]], and let $P = (p_{\\lambda i})_{\\lambda \\in \\Lambda, i \\in I}$ be a $\\Lambda \\times I$ matrix with entries in $G$.\nThe *Rees matrix semigroup* $\\mathcal{M}(G; I, \\Lambda; P)$ is the set $I \\times G \\times \\Lambda$ with the product\n$$(i, g, \\lambda)(j, h, \\mu) = (i, \\, g \\, p_{\\lambda j} \\, h, \\, \\mu).$$\nThe matrix $P$ is called the sandwich matrix.\n\n## Remarks\n\nThe product is associative: both ways of bracketing $(i, g, \\lambda)(j, h, \\mu)(k, f, \\nu)$ give $(i, \\, g p_{\\lambda j} h p_{\\mu k} f, \\, \\nu)$.\nSo $\\mathcal{M}(G; I, \\Lambda; P)$ is a [[semigroup|0XMEM406JM]].\nIt is simple: given $x = (i, g, \\lambda)$ and $y = (j, h, \\mu)$, one can solve $(j, a, \\lambda) \\, x \\, (i, b, \\mu) = y$ for $a, b \\in G$.\nRees's theorem: a semigroup is completely simple if and only if it is isomorphic to some $\\mathcal{M}(G; I, \\Lambda; P)$.\n\nExample: if $I = \\Lambda = \\set{1}$ and $P = (e)$, then $(1, g, 1) \\mapsto g$ is an isomorphism onto $G$.\nExample: if $G$ is trivial, $I = \\set{1, 2}$ and $\\Lambda = \\set{1}$, the product is $(i, \\lambda)(j, \\mu) = (i, \\mu) = (i, 1)$, so $xy = x$ for all $x, y$: a left zero semigroup.\n\nNon-example: the monoid $(\\N, +)$ is not isomorphic to any Rees matrix semigroup over a group.\nIt has the proper ideal $\\set{1, 2, 3, \\dots}$, while a Rees matrix semigroup has no proper two-sided ideal.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "X54RV6SENQ",
    "kind": "definition",
    "msc": [
      "20M10",
      "20M17"
    ],
    "name": "Rees matrix semigroup",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Rees_matrix_semigroup"
      }
    ],
    "tag": "M4DP72HFE5",
    "uses": [
      "G5S1M9",
      "0XMEM406JM",
      "0SJ3RX9TXA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/M4DP72HFE5/",
  "links": [
    {
      "field": "defines",
      "target": "X54RV6SENQ"
    },
    {
      "field": "uses",
      "target": "G5S1M9"
    },
    {
      "field": "uses",
      "target": "0XMEM406JM"
    },
    {
      "field": "uses",
      "target": "0SJ3RX9TXA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/M4DP72HFE5.md",
  "msc": [
    "20M10",
    "20M17"
  ],
  "name": "Rees matrix semigroup",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/M4D/M4DP72HFE5.md",
  "status": null,
  "tag": "M4DP72HFE5",
  "uri": "m2://M4DP72HFE5",
  "url": "https://math2.tamnd.com/wiki/rees-matrix-semigroup-definition/",
  "writing": {
    "tag": "M4DP72HFE5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}