Details:
Title  On Gröbner Bases in Monoid and Group Rings  Author(s)  Klaus Madlener, Birgit Reinert  Type  Technical Report, Misc  Abstract  Following 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 semiThue 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 spolynomials, confluence criteria are developed and a completion procedure is given. In case T = \emptyset or (\Sigma, T ) is a convergent, 2monadic 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. 
Language  English  Year  1993  Edition  0  Translation 
No  Refereed 
No 
