---
tag: CRGPCYJVE7
kind: statement
role: theorem
name: Rees' theorem
sources:
  - wikipedia: en:41983973@r1336933452
    locator: Rees matrix semigroup, section Rees' theorem
    note: adopted
    mention: wikipedia://en/41983973#C1M4
    sections: [Statement]
inventory: [F6V5G6PCHX]
---

## Statement

In his 1940 paper Rees proved the following theorem characterising completely simple semigroups:

A semigroup is completely simple if and only if it is isomorphic to a Rees matrix semigroup over a group.

That 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*<sup>0</sup> is the semigroup obtained from *G* by attaching a [[zero element|0XMEM406JM]], then *M*(*G*<sup>0</sup>; *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*<sup>0</sup>; *I*, *Λ*; *P*) is regular, then it is also completely 0-simple.