Orbit Structure of Finite and Reductive Monoids

A reductive monoid is an algebraic monoid with a reductive unit group. We introduce a new class of reductive monoids, $({\cal T}, \sigma)$-irreducible monoids. 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 ${\cal T}$-irreducible monoids. Motivated by their construction of finite reductive monoids, the concept of (${\cal T}, \sigma)$-irreducible monoid arises naturally. We obtain that the ${\cal T}$-irreducible monoids turn out to be a special class of the $({\cal T}, \sigma)$-irreducible monoids. We obtain a similar recipe for the question to $({\cal T}, \sigma)$-irreducible monoids (not ${\cal T}$-irreducible) of type $D\sbsp{n}{2}.$ However, there is no similar answer for types $A\sb{n} (n \ge 4)$ and $E\sbsp{6}{2}.$ The fixed points of any $({\cal T}, \sigma)$-irreducible monoid under $\sigma$ is a finite reductive monoid. We obtain that any such finite reductive monoid is ${\cal T}$-irreducible. Then we find the orbits of these monoids under the two sided action of their unit groups