Rees matrix semigroup

Let GG be a group and 00, a separate symbol. Then the set G0G\cup 0 is a semigroup with 00 with respect to the operation extending the multiplication in GG by the rule 0g=g0=00=00 g = g 0 = 0 0 = 0 for all gGg\in G. The same is actually true even if GG is a semigroup.

In his 1940 article, Rees introduced matrix semigroups starting with a semigroup with 00 two sets II and Λ\Lambda and a matrix pp with labels in Λ×I\Lambda\times I and values p λip_{\lambda i} in G0G \cup 0. A Rees matrix is a rectangular matrix with values in GG which has at most only one nonzero entry. The elements of the Rees matrix semigroup M(G,I,Λ,p)M(G, I,\Lambda,p) are all possible Rees I×ΛI\times \Lambda matrices with values in GG with multiplication ab=apba \circ b = a p b (with usual multiplication of matrices, where obvious auxiliary addition is taken, where never more than one element is nonzero).

In the same article he characterized a class of 0-simple regular semigroups and made steps useful for classification of some types of simple semigroups.

Important role in the theory of Rees matrix semigroups is played by primitive idempotents. A theorem of Rees in that context is an analogue of the second Wedderburn theorem in the theory of simple associative algebras where primitive idempotent?s play a major role.

Brandt groupoids (whose data are equiavlent to the connected groupoids in the modern sense) are very simple subcase of Rees matrix semigroups. In that case pp is the Kronecker matrix.

  • the current wikipedia article has errors (for example no condition on zeros in pp is made) Rees matrix semigroup
  • A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, vol. 1
  • David Rees, On semi-groups 3, Proc. Cambridge. Math. Soc. 387–400 (1940)
  • John M. Howie, Fundamentals of semigroup theory, Clarendon Press 1995
category: algebra

Last revised on February 19, 2016 at 14:30:37. See the history of this page for a list of all contributions to it.