
In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because they are used to classify certain classes of simple semigroups.
Definition.
Let "S" be a semigroup, "I" and "Λ" non-empty sets and "P" a matrix indexed by "I" and "Λ" with entries "p""i","λ" taken from "S".
Then the Rees matrix semigroup "M"("S"; "I", "Λ"; "P") is the set "I"×"S"×"Λ" together with the multiplication
Rees matrix semigroups are an important technique for building new semigroups out of old ones.
Rees' theorem.
In his 1940 paper Rees proved the following theorem characterising completely simple semigroups: 
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"0 is the semigroup obtained from "G" by attaching a zero element, then "M"("G"0; "I", "Λ"; "P") is a regular semigroup if and only if every row and column of the matrix "P" contains an element which is not 0. If such an "M"("G"0; "I", "Λ"; "P") is regular, then it is also completely 0-simple.