Topologically simple infinite matrix groups indexed by ordered sets

TL;DR

This paper investigates the topological properties of infinite matrix groups derived from incidence rings indexed by ordered sets, introducing conditions for their topological structure and linking them to elementary totally disconnected groups.

Contribution

It characterizes when these infinite matrix groups can inherit topologies from incidence rings and constructs simple topological matrix groups using topological fields, extending previous results.

Findings

Conditions for topological inheritance in incidence ring groups

Construction of simple topological matrix groups from topological fields

Connection between these groups and elementary totally disconnected groups

Abstract

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 in ``Topologically simple, totally disconnected, locally compact infinite matrix groups''. 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.