Doctor of Philosophy
The purpose of this work is to enhance the understanding regular algebraic semigroups by considering the structural influence of Green's relations. There will be three chief topics of discussion.
- Green's relations and the Adherence order on reductive monoids
- Renner’s conjecture on regular irreducible semigroups with zero
- a Green’s relation inspired construction of regular algebraic semigroups
Primarily, we will explore the combinatorial and geometric nature of reductive monoids with zero. Such monoids have a decomposition in terms of a Borel subgroup, called the Bruhat decomposition, which produces a finite monoid, R, the Renner monoid. We will explore the structure of R by way of Green's relations. In particular, we will be exploring the nature of the Adherence order poset combined with J-, R-, L-, and H-classes.
From reductive monoids we broaden the impact of Green's relations and explore regular algebraic semigroups. Specifically, we resolve Renner’s conjecture and show that the supports, Xl = J/R and Xr = J/L are projective varieties. Spurred on by the result, we use invariant theory to generalise the Rees matrix construction for algebraic semigroups to construct irreducible regular semigroups with 0. Our construction will start with specified maximal classes, Re, Le, and He and reconstruct an entire semigroup. In a lengthy example, we will use some of our previous combinatorial results to apply the construction to a natural generalisation of determinantal varieties.
Highlights include the unique "vanilla form" decomposition for elements of the Renner monoid (Definition 5.36), a proof of Renner's conjecture on the projectiveness of supports for irreducible regular semigroups with zero (Theorem 8.40), and the construction of irreducible regular semigroups from prespecified maximal R- and L-classes (Definition 9.6).
O'Hara, Allen, "A Study Of Green’s Relations On Algebraic Semigroups" (2015). Electronic Thesis and Dissertation Repository. 3047.