Diagram groups are totally orderable. (Q814772)

The paper is a sequel of the authors' former works on diagram groups and directed 2-complexes: homotopy and homology [see J. Pure Appl. Algebra 205, No. 1, 1-47 (2006; Zbl 1163.20027)]. In the present paper, the authors describe diagram groups of directed 2-complexes glued from other directed 2-complexes in a certain way (Theorem 2.1). They then introduce the concept of the independence graph of a directed 2-complex, and show that if \(\Gamma\) is an independence graph of a rooted 2-tree and \(\{G_i\}\) is a collection of diagram groups then the graph product of \(G_i\) corresponding to \(\Gamma\) coincides with a diagram product of these groups. In particular, the partially commutative groups corresponding to independence graphs of directed 2-trees are diagram groups (Theorem 2.4). From these facts the authors get that the class of diagram groups is closed under countable directed products (Theorem 2.5). The authors also describe expansions of 2-complexes, representations of diagram groups from which they show how to construct families of such representations using ``transition schemes''. Finally, they prove their main result: that all diagram groups of any directed 2-complex are totally orderable.