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TitleOn Gröbner Bases in Monoid and Group Rings
Author(s) Klaus Madlener, Birgit Reinert
TypeTechnical Report, Misc
AbstractFollowing Buchberger's approach to computing a Gröbner basis of a polynomial ideal in polynomial rings, a completion procedure for finitely
generated right ideals in Z[H] is given, where H is an ordered monoid presented by a finite, convergent semi-Thue system (\Sigma, T). Taking a finite set F \subseteq Z[H] we get a (possibly infinite) basis of the right ideal generated by F , such that using this basis we have unique normal forms for all p \in Z[H] (especially the normal form is 0 in case p is an element of the right ideal generated by F ). As the ordering and multiplication on H need not be compatible, reduction has to be defined carefully in order to make it Noetherian. Further we no longer have p \cdot x \rightarrow p 0 for p \in Z[H], x \in H. Similar to Buchberger's s-polynomials, confluence criteria are developed and a completion procedure is given. In case T = \emptyset or (\Sigma, T ) is a convergent, 2-monadic presentation of a group providing inverses of length 1 for the generators or (\Sigma, T) is a convergent presentation of a commutative monoid, termination can be shown. So in this cases finitely generated right ideals admit finite Gröbner bases. The connection to the subgroup problem is discussed.
LanguageEnglish
Year1993
Edition0
Translation No
Refereed No
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