$G=\{g_{1},g_{2},g_{3}\}$ in the non-commutative polynomial ring $K\langle x,y\rangle$.

-   Assume that $\lm_{\sigma}(g_{1})=y^3, \lm_{\sigma}(g_{2})=x^2y^{2}$ and $\lm_{\sigma}(g_{3})=xyx^{2}y$. Then we have ${\rm o}_{1,3}(xyx^{2},1;1,y^{2}), {\rm o}_{2,3}(xy,1;1,y)\in\bigcup_{1\leq i\leq 3}{\rm NTObs}(i,3)$, and $${\rm o}_{1,3}(xyx^{2},1;1,y^{2})={\rm o}_{2,3}(xy,1;1,y)y+{\rm o}_{1,2}(xyx^2,1;xy,y).$$ Observe that ${\rm o}_{1,2}(xyx^2,xy;y)=xy{\rm o}_{1,2}(x^2,1;1,y)$ is a multiple of the non-trivial obstruction ${\rm o}_{1,2}(x^2,1;1,y)$.

-   Now assume that $\lm_{\sigma}(g_{1})=(xy)^{2}, \lm_{\sigma}(g_{2})=y$ and $\lm_{\sigma}(g_{3})=xyx^{2}y$. Then we have ${\rm o}_{1,3}(xyx,1;1,xy), {\rm o}_{2,3}(x,x^{2}y;1,1)\in\bigcup_{1\leq i\leq 3}{\rm NTObs}(i,3)$, and $${\rm o}_{1,3}(xyx,1;1,xy)={\rm o}_{2,3}(x,x^{2}y;1,1)xy+{\rm o}_{1,2}(xyx,1;x,x^{2}yxy).$$ One can check that ${\rm o}_{1,2}(xyx,1;x,x^{2}yxy)$ is an obstruction without overlap.

In the following, we present the non-commutative multiply criterion and the leading word criterion. They are non-commutative analogues of the Gebauer-Möller criteria M and F, respectively.

\[ncMCrit\] Suppose that ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ and ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ are two distinct non-trivial obstructions in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ such that there exist two words $w,w'$ in $\langle X\rangle$ satisfying $u_{s}=wv_{s}$ and $u'_{s}=v'_{s}w'$. Then we can remove the obstruction ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ from $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$ in the execution of the Buchberger Procedure if $ww'\neq 1$.

By the previous lemma, the obstruction ${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})$ can be represented as $${\rm o}_{i,s}(w_{i},w'_{i};u_{s},u'_{s})=w{\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})w'+a{\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$$ with $a\in\{1,-1\}$ and $k=\min\{i,j\},l=\max\{i,j\}$. To prove that ${\rm o}_{i,s}(w_{i},w'_{i};$ $u_{s},u'_{s})$ is strictly larger than ${\rm o}_{j,s}(w_{j},w'_{j};v_{s},v'_{s})$ and ${\rm o}_{k,l}(w_{k},w'_{k};w_{l},w'_{l})$, we consider two cases. If $i>j$, 