Rees matrix semigroup

In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940.^[1] 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_λ,i 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, s p_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) 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.

See also

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

Footnotes

1. Rees 1940.

References

• Rees, David (1940), "On semi-groups", Mathematical Proceedings of the Cambridge Philosophical Society, 36 (4): 387–400, doi:10.1017/S0305004100017436.
• Howie, John M. (1995), Fundamentals of Semigroup Theory, Clarendon Press, ISBN 0-19-851194-9.