not necessary to have ${\rm o}_{i,s}(w_{i},w'_{i};w,w')>_{\tau}{\rm o}_{i,s}(\tilde{w}_{i},\tilde{w}'_{i};\tilde{w},\tilde{w}')$ (compared to Lemmas \[sec3lem3\] and \[sec3lem7\]). The same also holds for the obstruction ${\rm o}_{j,s}(w_{j},w'_{j};w,w')$ in the equation of Lemma \[sec3lem10\].

\[sec3exa11\] Consider polynomials $G=\{g_{1},g_{2},g_{3}\}$ in the non-commutative polynomial ring $K\langle x,y\rangle$ with $\lm_{\sigma}(g_{1})=x^3yx, \lm_{\sigma}(g_{2})=x^2$ and $\lm_{\sigma}(g_{3})=x$. We have ${\rm o}_{1,2}(1,1;x,yx)\in\bigcup_{1\leq i\leq j\leq 2}{\rm NTObs}(i,j)$ and $x\lw_{\sigma}(g_{2})yx=x^3yx=x^3y\lw_{\sigma}(g_{3})$ and $${\rm o}_{1,2}(1,1;x,yx)={\rm o}_{1,3}(1,1;x^3y,1)-{\rm o}_{2,3}(x,yx;x^3y,1).$$ One can check that ${\rm o}_{1,3}(1,1;x^3y,1)$ is a non-trivial obstruction in ${\rm NTObs}(1,3)$ and ${\rm o}_{1,2}(1,1;x,yx)<_{\tau}{\rm o}_{1,3}(1,1;x^3y,1)$. Moreover, ${\rm o}_{2,3}(x,yx;x^3y,1)$ is an obstruction without overlap.

The following is a non-commutative analogue of the Gebauer-Möller criterian $B_k$, which is also known as the *chain criterion* (or *criterion 1*) of Buchberger (cf. [@Bu85]).

\[ncBKCrit\] Suppose that ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ is a non-trivial obstruction. Then in the execution of the Buchberger Procedure ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ can be removed from $\bigcup_{1\leq i\leq j\leq s-1}{\rm NTObs}(i,j)$ if the following three conditions are satisfied.

-   There are two words $w,w'\in\langle X\rangle$ such that $w_{j}\lw_{\sigma}(g_{j})w'_{j}=w\lw_{\sigma}(g_{s})w'$.

-   The obstruction ${\rm o}_{i,s}(w_{i},w'_{i};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction in $\bigcup_{1\leq i\leq s}{\rm NTObs}(i,s)$.

-   The obstruction ${\rm o}_{j,s}(w_{j},w'_{j};w,w')$ is either an obstruction without overlap or a multiple of a non-trivial obstruction 