# Orders on groups, and spectral spaces of lattice-groups

## TL;DR

This paper develops a systematic framework linking spaces of right orders on groups with spectral spaces of free lattice-ordered groups, enhancing the understanding of their interplay and paving the way for future research.

## Contribution

It introduces a new systematic approach connecting right orders on groups with spectral spaces of lattice-ordered groups, expanding the theoretical foundation of both areas.

## Key findings

- Established a correspondence between right orders and spectral spaces.
- Extended prior work by Weinberg, Conrad, and McCleary.
- Facilitated future research in ordered group theory.

## Abstract

Extending pioneering work by Weinberg, Conrad, McCleary, and others, we provide a systematic way of relating spaces of right orders on a partially ordered group, on the one hand, and spectral spaces of free lattice-ordered groups, on the other. The aim of the theory is to pave the way for further fruitful interactions between the study of right orders on groups and that of lattice-groups. Special attention is paid to the important case of orders on groups.