# Dense orderings in the space of left-orderings of a group

## TL;DR

This paper explores the structure of the space of left-orderings of a group, revealing that for bi-orderable groups, the closure of dense orderings can encompass the entire space, especially in free groups.

## Contribution

It demonstrates the relationship between dense and Conradian orderings and characterizes the closure of dense orderings in various groups.

## Key findings

- Closure of dense orderings contains all Conradian orderings
- In free groups, the closure of dense orderings equals the entire space of orderings
- Partition of ordering spaces into discrete and dense subsets

## Abstract

Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally partitioned into two subsets. This note investigates the structure of this partition, specifically the set of dense orderings of a group and its closure within the space of orderings. We show that for bi-orderable groups this closure will always contain the space of Conradian orderings---and often much more. In particular, the closure of the set of dense orderings of the free group is the entire space of left-orderings.