|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 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.