Diagram groups are totally orderable

TL;DR

This paper proves that all diagram groups are totally orderable by introducing the independence graph of a directed 2-complex and analyzing their structure through graph products and semi-direct products.

Contribution

It introduces the independence graph of a directed 2-complex and establishes that diagram groups are closed under graph products, leading to the proof of total orderability.

Findings

All diagram groups are totally orderable.

Diagram groups can be expressed as semi-direct products involving R. Thompson's group F.

Closure of diagram groups under graph products over independence graphs.

Abstract

In this paper, we introduce the concept of the independence graph of a directed 2-complex. We show that the class of diagram groups is closed under graph products over independence graphs of rooted 2-trees. This allows us to show that a diagram group containing all countable diagram groups is a semi-direct product of a partially commutative group and R. Thompson's group $F$. As a result, we prove that all diagram groups are totally orderable.