# Rees matrix semigroup

**Rees matrix semigroups** are a special class of semigroup 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

*(i,s,λ)(j,t,μ) = (i, sp*_{λ,j}t, μ).

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:

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)* where *G* is a group. 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*is regular then it is also completely 0-simple.

^{0};I,Λ;P)## References

- Rees, David (1940),
*On semi-groups*,**3**, Proc. Cambridge. Math. Soc., pp. 387–400. - Howie, John M. (1995),
*Fundamentals of Semigroup Theory*, Clarendon Press, ISBN 0-19-851194-9.