|Group presentation||left GBs||right GBs||two-sided GBs|
Finally we want to sketch how the results of this report can be lifted to group rings over nilpotent-by-finite respectively polycyclic-by-finite groups. Essential in this approach is the use of semi-Thue systems related to extensions of groups as introduced for context-free groups by Cremanns and Otto in [CrOt94]. Details of the lifting process for respective group rings can be found in [Re95] and [MaRe96]. The key idea is to combine a convergent presentation of a finite group with a convergent PCNI-presentation respecitively PCP-presentation of a nilpotent respectively polycyclic group presented by . Assuming , let and let T consist of the set of rules , and the following additional rules:
Then in case is convergent it is called the extension presentation of as an extension of by (see e.g. [Cr95]). Every element in has a representative of the form eu where and . We can specify a total well-founded ordering on our group by combining a total well-founded ordering on and the syllable ordering on : For we define if and only if or (e1 = e2 and . Furthermore, we can lift the tuple ordering to as follows: For two elements eu, ev, we define if and we define . According to this ordering we call ev a (commutative) prefix of eu if and introducing the concept of -closure as in [Re95] or [MaRe96] we can proceed to prove lemmata and theorems similar to those in section 5 and 6.
l rwr for all , where , xa awx for all , for all , where .