# Ordered groups as a tensor category

## TL;DR

This paper constructs a functorial tensor category structure on ordered groups by extending the free product operation, demonstrating continuity and injectivity in the space of orderings.

## Contribution

It introduces a functorial tensor product for ordered groups using Bergman's method, enriching the category with monoidal structure and analyzing the topology of orderings.

## Key findings

- Constructs a functorial tensor product on ordered groups.
- Shows the space of orderings is mapped continuously and injectively.
- Extends results to left-ordered groups.

## Abstract

It is a classical theorem that the free product of ordered groups is orderable. In this note we show that, using a method of G. Bergman, an ordering of the free product can be constructed in a functorial manner, in the category of ordered groups and order-preserving homomorphisms. With this functor interpreted as a tensor product this category becomes a tensor (or monoidal) category. Moreover, if $O(G)$ denotes the space of orderings of the group $G$ with the natural topology, then for fixed groups $F$ and $G$ our construction can be considered a function $O(F) \times O(G) \to O(F * G)$. We show that this function is continuous and injective. Similar results hold for left-ordered groups.