Classifying the closure of standard orderings on $\mathrm{Homeo}_{+}{(\mathbb{R})}$

TL;DR

This paper characterizes the closure of certain orderings on the group of orientation-preserving homeomorphisms of the real line and demonstrates that there are orderings outside this closure.

Contribution

It provides a detailed description of the closure of dynamical-lexicographic orderings and proves the existence of orderings beyond this closure.

Findings

Characterization of the closure of dynamical-lexicographic orderings.

Existence of orderings outside the characterized closure.

Advancement in understanding order structures on $ ext{Homeo}_+(R)$.

Abstract

We characterize a closure of the set of dynamical-lexicographic orderings on $\mathrm{Homeo}_{+}{(\mathbb{R})}$ and prove the existence of orders outside of it.