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

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

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Author:Klaus Madlener, Birgit Reinert
URN (permanent link):urn:nbn:de:hbz:386-kluedo-4385
Serie (Series number):Reports on Computer Algebra (ZCA Report) (16)
Document Type:Preprint
Language of publication:English
Year of Completion:1997
Year of Publication:1997
Publishing Institute:Technische Universität Kaiserslautern
Date of the Publication (Server):2000/04/03
Faculties / Organisational entities:Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011