The Lattice of -Classes of ( , σ)-Irreducible Monoids, II

A reductive monoid is an algebraic monoid with a reductive unit group. Generally, we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid. Putcha and Renner give a recipe to determine the orbits for -irreducible monoids according to the Dynkin diagrams. We obtain a similar recipe for the question to ( , σ)-irreducible monoids (not -irreducible) of type D2n. However, there is no similar answer for types An (n ≥ 4) and E26. The fixed points of any ( , σ)-irreducible monoid under σ is a finite reductive monoid. We obtain that any such finite reductive monoid is -irreducible. Then we find the orbits of these monoids under the two sided action of their unit groups.