On Grr Obner Bases in Monoid and Group Rings

Following Buchberger's approach to computing a Grr obner basis of a polynomial ideal in polynomial rings, a completion procedure for nitely generated right ideals in ZH] is given, where H is an ordered monoid presented by a nite, convergent semi{Thue system ((; T). Taking a nite set F ZH] we get a (possibly innnite) basis of the right ideal generated by F , such that using this basis we have unique normal forms for all p 2 ZH] (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 deened carefully in order to make it Noetherian. Further we no longer have p x ! p 0 for p 2 ZH];x 2 H. Similar to Buchberger's s{polynomials, connuence criteria are developed and a completion procedure is given. In case T = ; or ((; T) is a convergent, 2{monadic presentation of a group providing inverses of length 1 for the generators or ((; T) is a convergent presentation of a commutative monoid , termination can be shown. So in this cases nitely generated right ideals admit nite Grr obner bases. The connection to the subgroup problem is discussed.