multiplication.

Now we are ready to generalize the commutative Gebauer-Möller criteria (see [@CKR04] and [@GM88]) to the non-commutative case. Recall that, in step (B4) of the Buchberger Procedure, when a new generator $g_{s}$ is added, we immediately construct new obstructions $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$. We want to detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ of newly constructed obstructions as well as in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ of previously constructed obstructions. We achieve this goal via the following three steps. Firstly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with the aid of other obstructions also in this set. This step is called a *head reduction step* in [@CKR04]. Secondly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ with the aid of obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$. This step is called a *tail reduction step* in [@CKR04]. Thirdly, we detect unnecessary obstructions in the set $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ with the aid of the new generator $g_{s}$. Indeed, the first step corresponds to the commutative Gebauer-Möller criteria $M$ and $F$, and the last step corresponds to criterion $B_k$ (c.f. [@GM88], Subsection 3.4).

The following 