Bases for counting functions on free monoids and groups

TL;DR

This paper offers a simplified approach to describing bases of counting functions on free monoids and groups, correcting previous errors and enhancing understanding of the Brooks space.

Contribution

It introduces a new, simpler method for establishing bases of counting functions and corrects earlier inaccuracies in the theoretical framework.

Findings

Simplified proofs for bases of counting functions

Correction of previous theorem errors

Enhanced understanding of the Brooks space

Abstract

The focus of this paper is providing a description of the spaces of counting functions on free monoids and groups and of the Brooks space. These results have been obtained in an earlier publication, however we propose an alternative, simpler approach to proving the theorems concerning the bases. Furthermore, one of the theorems at the time of writing this paper contained a slight error, and we correct it here.