Monoids of O-type, subword reversing, and ordered groups

TL;DR

This paper introduces a simple scheme for constructing finitely generated monoids with linear left-divisibility orderings using subword reversing, connecting to Garside theory, and explores ordered groups with isolated points in their space of orderings.

Contribution

It presents a new approach to constructing and analyzing monoids with linear divisibility orderings via subword reversing, extending Garside theory to non-Noetherian contexts.

Findings

Constructed new families of ordered groups including torus knot groups.

Identified conditions for the existence of isolated points in the space of orderings.

Connected subword reversing with Garside theory in non-Noetherian settings.

Abstract

We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products.