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

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