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Xn ), and It(Xjgi ) ::: T then the left syzygy X j • s einS(gi) had been considered during the execution of Method 2. Hence, there exist gli,j E G and hi,j E A such that It(gli,j) Ip It(Xjgi ) and p(i,j) := Xjg i + hi,jgli,j E J(G,lt(Xjg i » (3) . Let 1 'S i,j 'S l satisfy It(gj) Ip It(gi) and It(gi) ::: T. In the same way as above we deduce the existence of Aj E A such that q(i,j) :=gi+Ajgj E J(G,lt(gi» (4) . e. u = v(i, u) Os w(i, u). We will show the existence of an element r(i, u) by induction on the length of w(i, u).

Definition: Let F ~ J be a subset of the left ideal J ~ A. Then F is called a Grabner basis of J with respect to the graded structure As = (A,5, --<, It) iff LIns(F) = LIns(J). In the particular case Ci,j = 1 and Pi,j = 0 for all 1 :S i < j :S n the ring A is isomorphic to the polynomial ring R = JC[X] and the definitions of It given here and in Section 2 coincide when A and R are identified under the natural 38 Joachim Apel isomorphism. Hence, the following notion is a straight forward generalization of the Pommaret bases of polynomial ideals defined in Section 2.

A parallel Grobner factorizer. In Hong [Hon94], pages 174-180. [GM88] R. M. Moller. On an installation of Buchberger's J. Symbolic Computation, 6(2 & 3):275-286, Octoalgorithm. ber/December 1988. [GMN+91] Alessandro Giovini, Teo Mora, Gianfranco Niesi, Lorenzo Robbiano, and Carlo Traverso. "One sugar cube, please;" or: Selection strategies in the Buchberger algorithm. In Watt [Wat91], pages 49-54. [Haw91] David J. Hawley. A Buchberger algorithm for distributed memory multi-processors. P. Zima, editor, Parallel Computation (1st Internatl.