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String Rewriting and Gröbner Bases - A General Approach to Monoid and Group Rings

Klaus Madlener, Birgit Reinert
Universität Kaiserslautern
67663 Kaiserslautern

presented at the
Workshop on Symbolic Rewriting Systems
Monte Verita, May 1995

October, 1997


The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Gröbner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some non-commutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Gröbner bases for finitely generated right ideals is shown and a procedure to compute them is given.

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