Rees' theorem

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 G⁰ is the semigroup obtained from G by attaching a zero element, then M(G⁰; 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(G⁰; I, Λ; P) is regular, then it is also completely 0-simple.