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  "body": "\n## Statement\n\nIn his 1940 paper Rees proved the following theorem characterising completely simple semigroups:\n\nA semigroup is completely simple if and only if it is isomorphic to a Rees matrix semigroup over a group.\n\nThat is, every completely simple semigroup is isomorphic to a semigroup of the form *M*(*G*; *I*, *Λ*; *P*) for some group *G*. Moreover, Rees proved that if *G* is a group and *G*\u003csup\u003e0\u003c/sup\u003e is the semigroup obtained from *G* by attaching a [[zero element|0XMEM406JM]], then *M*(*G*\u003csup\u003e0\u003c/sup\u003e; *I*, *Λ*; *P*) is a regular semigroup if and only if every row and column of the matrix *P* contains an element that is not 0. If such an *M*(*G*\u003csup\u003e0\u003c/sup\u003e; *I*, *Λ*; *P*) is regular, then it is also completely 0-simple.\n",
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