Rees matrix semigroup

Summary

A Rees matrix semigroup is a semigroup built from a group $G$ and two index sets $I$ and $\Lambda$. It also uses a matrix $P$ with entries in $G$. Its elements are triples $(i, g, \lambda)$, and the matrix $P$ is inserted in the middle of each product. Rees's theorem says that the completely simple semigroups are exactly the Rees matrix semigroups over groups. A variant with a zero element describes the completely $0$-simple semigroups, which are the building blocks of finite semigroups.