Let be a group and , a separate symbol. Then the set is a semigroup with with respect to the operation extending the multiplication in by the rule for all . The same is actually true even if is a semigroup.
In his 1940 article, Rees introduced matrix semigroups starting with a semigroup with two sets and and a matrix with labels in and values in . A Rees matrix is a rectangular matrix with values in which has at most only one nonzero entry. The elements of the Rees matrix semigroup are all possible Rees matrices with values in with multiplication (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 is the Kronecker matrix.
Last revised on February 19, 2016 at 19:30:37. See the history of this page for a list of all contributions to it.