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.
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Definition
Let S be a semigroup, I and Λ non-empty sets and P a matrix indexed by I and Λ with entries pi,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, spλ,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.