# Parikh-reducing Church-Rosser representations for some classes of regular languages

## TL;DR

This paper investigates Parikh-reducing Church-Rosser systems for specific regular language classes, providing finite representations and analyzing their complexity, especially for languages with abelian group syntactic monoids.

## Contribution

It demonstrates the existence of finite Parikh-reducing Church-Rosser systems for certain regular languages and constructs monoid representations with abelian subgroups.

## Key findings

- Existence of finite systems for languages with abelian group syntactic monoids
- Construction of monoid representations with all subgroups abelian
- Analysis of the complexity of these representations

## Abstract

In this paper the concept of Parikh-reducing Church-Rosser systems is studied. It is shown that for two classes of regular languages there exist such systems which describe the languages using finitely many equivalence classes of the rewriting system. The two classes are: 1.) the class of all regular languages such that the syntactic monoid contains only abelian groups and 2.) the class of all group languages over a two-letter alphabet. The construction of the systems yield a monoid representation such that all subgroups are abelian. Additionally, the complexity of those representations is studied.