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Topologically simple infinite matrix groups indexed by ordered sets

João V. P. e Silva

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.

Topologically simple infinite matrix groups indexed by ordered sets

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.

Paper Structure

This paper contains 12 sections, 26 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Background
  4. Ordered sets
  5. The units group of a matrix ring
  6. Topological groups
  7. Group of invertible matrices
  8. TEXT and its properties
  9. The center
  10. Building simple topological groups
  11. Matrix groups of finite fields and t.d.l.c.s.c. groups
  12. Relation to the matrix groups defined in TEXT

Key Result

Proposition 2.7

Let $\Lambda$ be a non-empty preordered set. Then there is a collection $\{\Lambda_i\}_{i\in I}$ of irreducible subsets of $\Lambda$ such that $\Lambda=\bigsqcup_{i\in I} \Lambda_i$.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 51 more