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,jt, 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 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 which is not 0. If such an M(G⁰;I,Λ;P) is regular then it is also completely 0-simple.

See also

• Semigroup
• Completely simple semigroup
• David Rees (mathematician)

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.