The number of locally invariant orderings of a group

TL;DR

This paper proves that nontrivial groups with a locally invariant ordering have uncountably many such orderings, providing explicit constructions for left-orderable groups and an existence theorem for general groups.

Contribution

It introduces the concept of locally invariant orderings, explores their properties, and establishes the existence of uncountably many such orderings for nontrivial groups.

Findings

Nontrivial groups with a locally invariant ordering have uncountably many such orderings.

Explicit constructions of uncountable locally invariant orderings are provided for left-orderable groups.

An existence theorem using compactness is established for general groups.

Abstract

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of locally invariant orderings; for a general group we provide an existence theorem that applies compactness to yield uncountably many locally invariant orderings. Along the way, we define and investigate the space of locally invariant orderings of a group, the natural group actions on this space, and their relationship to the space of left-orderings.