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  "body": "\n## Summary\n\nA Rees matrix semigroup is a [[semigroup|0XMEM406JM]] built from a [[group|G5S1M9]] $G$ and two index sets $I$ and $\\Lambda$.\nIt also uses a matrix $P$ with entries in $G$.\nIts elements are triples $(i, g, \\lambda)$, and the matrix $P$ is inserted in the middle of each product.\nRees's theorem says that the completely simple semigroups are exactly the Rees matrix semigroups over groups.\nA variant with a zero element describes the completely $0$-simple semigroups, which are the building blocks of finite semigroups.\n\n## Explanations\n\n### L1 human\n\nA Rees matrix semigroup is like a grid of copies of a group.\nThe rows are labelled by $I$ and the columns by $\\Lambda$.\nWhen you multiply two elements, you keep the row of the first and the column of the second.\nYou multiply the group parts with a twist taken from the matrix $P$.\nIf the group is trivial, a product just keeps the row of the left factor and the column of the right factor.\n",
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