Date of Award
1997
Degree Type
Thesis
Degree Name
Doctor of Philosophy
Program
Mathematics
Supervisor
Lex E. Renner
Abstract
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.
Recommended Citation
Li, Zhuo, "Orbit Structure of Finite and Reductive Monoids" (1997). Digitized Theses. 3207.
https://ir.lib.uwo.ca/digitizedtheses/3207