Titel: On Gröbner Bases in Monoid and Group Rings

Language of presentation: English

Promotor: Prof. Dr. Klaus E. Madlener and Prof. Dr. Volker Weispfenning

Date of defense: June 14, 1995

Institution granting degree: Universität Kaiserslautern, Germany

Abstract:

In this thesis the concept of Groebner bases is generalized for finitely generated monoid and group rings. Reduction methods are used to represent the monoid elements as well as to describe the right ideal congruence in the respective rings. Since in general monoids do not allow admissible orderings, in defining suitable reduction relations serious problems arise: on one hand it is difficult to guarantee termination for reduction relations, and on the other hand, reduction does not necessarily capture the right ideal congruence. In this thesis different possible definitions of reduction are given and studied with respect to these problems. For special classes of monoids - e.g. finite, commutative or free - and different classes of groups - e.g. finite, free, plain, context-free or nilpotent - using special structural properties we have succeeded in defining special reduction relations and developing terminating algorithms to compute Groebner bases with respect to these reduction relations.