Topologically Simple Infinite Matrix Groups Indexed by Ordered Sets

This article focuses on the study of the group of units of incidence rings, which is a class of infinite matrix groups indexed by ordered sets, on a topological perspective. We first show when these groups can inherit the topological structure from the incidence rings. It is later shown that infinite matrix groups of topological fields can be used to build simple topological matrix groups, generalizing a result proven recently by P. Groenhout, C. Reid and G. Willis Topologically simple, totally disconnected, locally compact infinite matrix groups, J. Lie Theory 30 (2020) 965-980]. We finish by relating the structure of these groups with elementary totally disconnected, locally compact groups, an important class for the study of totally disconnected, locally compact groups.