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

Abstract 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 J -irreducible monoids according to the Dynkin diagrams. We obtain a similar recipe for the question to ( J , σ)-irreducible monoids (not J -irreducible) of type D 2 n . However, there is no similar answer for types A n ( n  ≥ 4) and E 2 6 . The fixed points of any ( J , σ)-irreducible monoid under σ is a finite reductive monoid. We obtain that any such finite reductive monoid is J -irreducible. Then we find the orbits of these monoids under the two sided action of their unit groups.