Types of Reductive Monoids

Let M be a reductive monoid with a reductive unit group G. Clearly there is a natural G × G action on M. The orbits are the -classes (in the sense of semigroup theory) and form a finite lattice. The general problem of finding the lattice remains open. In this paper we study a new class of reductive monoids constructed by multilined closure. We obtain a general theorem to determine the lattices of these monoids. We find that the ( , σ)-irreducible monoids of Suzuki type and Ree type belong to this new class. Using the general theorem we then list all the lattices and type maps of the ( , σ)-irreducible monoids of Suzuki type and Ree type.