Rees matrix semigroup
{{more footnotes|date=February 2014}}
In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940.{{sfn|Rees|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 pλ,i taken from S.
Then the Rees matrix semigroup M(S; I, Λ; P) is the set I×S×Λ together with the product formula
:(i, s, λ)(j, t, μ) = (i, s{{Hair space}}pλ,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:
{{quote|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
Footnotes
References
- {{Citation
| last1=Rees | first1=David | authorlink1=David Rees (mathematician)
| title=On semi-groups
| journal=Mathematical Proceedings of the Cambridge Philosophical Society
| volume=36
| issue=4
| year=1940
| pages=387–400
| doi=10.1017/S0305004100017436}}.
- {{citation|last= Howie|first= John M.|authorlink=John Mackintosh Howie|title=Fundamentals of Semigroup Theory|year=1995|publisher=Clarendon Press|isbn=0-19-851194-9}}.