On Groebner Basis in Monoid and Group Rings

TL;DR

This thesis extends the concept of Groebner bases to finitely generated monoid and group rings, addressing challenges like non-admissible orderings and non-compatibility with multiplication, and develops algorithms for specific classes.

Contribution

It generalizes Groebner bases to non-commutative, non-Noetherian monoid and group rings without relying on admissible orderings, introducing new reduction relations and algorithms.

Findings

Defined reduction relations for specific monoid and group classes

Developed terminating algorithms for Groebner basis computation

Characterized Groebner bases using saturation and structural properties

Abstract

Developed by Buchberger for commutative polynomial rings, Groebner Bases are frequently applied to solve algorithmic problems, such as the congruence problem for ideals. Until now, these ideas have been transmitted to different in part non-commutative and non-Noetherian algebras. Most of these approaches require an admissible ordering on terms. In this Ph.D.thesis, the concept of Groebner bases is generalized to finitely generated monoid and group rings. Reduction methods are applied to represent monoid and group elements and to describe the right-ideal congruence in the corresponding monoid and group rings, respectively. In general, monoids and especially groups do not offer admissible orderings anymore. Thus, the definition of an appropriate reduction relation is confronted with the following problems: On the one hand, it is difficult to guarantee termination. On the other hand, reduction steps are not compatible with multiplication anymore, so that the reduction relation does not necessarily generate a right-ideal congruence. In this thesis, several possibilities of defining reduction relations are presented and evaluated w.r.t. the described problems. The concept of saturation (i.e. the extension of a set of polynomials in such a way that the right-ideal congruence it generates can be captured by reduction) is applied to characterize Groebner bases relatively to the different reductions by s-polynomials. Using these concepts and special structural properties, it was possible to define special reduction relations for special classes of monoids (e.g., finite, commutative, or free) and for different classes of groups (e.g., finite, free, plain, context-free, or nilpotent) and to develop terminating algorithms for computing Groebner bases w.r.t. these reduction relations.