nLab Rees matrix semigroup

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

In his 1940 article, Rees introduced matrix semigroups starting with a semigroup with $0$ two sets $I$ and $\Lambda$ and a matrix $p$ with labels in $\Lambda\times I$ and values $p_{\lambda i}$ in $G \cup 0$. A Rees matrix is a rectangular matrix with values in $G$ which has at most only one nonzero entry. The elements of the Rees matrix semigroup $M(G, I,\Lambda,p)$ are all possible Rees $I\times \Lambda$ matrices with values in $G$ with multiplication $a \circ b = a p b$ (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 $p$ is the Kronecker matrix.

• the current wikipedia article has errors (for example no condition on zeros in $p$ 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