*product criterion* (or *criterion 2*) of Buchberger (cf. [@Bu85]).

\[sec2def9\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ with $s\geq 1$.

-   Let $i,j\in\{1,\dots,s\}$ and $i<j$. An obstruction in ${\rm Obs}(i,j)$ is called *non-trivial* if it has an overlap and is of the form ${\rm o}_{i,j}(w_{i},1;1,w'_{j})$, or ${\rm o}_{i,j}(1,w'_{i};$ $w_{j},1)$, or ${\rm o}_{i,j}(w_{i},w'_{i};1,1)$, or ${\rm o}_{i,j}(1,1;w_{j},w'_{j})$ with $w_{i},w'_{i},w_{j},w'_{j}\in\langle X\rangle$.

-   Let $i\in\{1,\dots,s\}$. A self obstruction in ${\rm Obs}(i,i)$ is called *non-trivial* if it has an overlap and is of the form ${\rm o}_{i,i}(1,w'_{i};w_{i},1)$ with $w_{i},w'_{i}\in\langle X\rangle\setminus\{1\}$.

-   Let $i,j\in\{1,\dots,s\}$ and $i\leq j$. The *set of all non-trivial obstructions* of $g_{i}$ and $g_{j}$ will be denoted by ${\rm NTObs}(i,j)$.

In the literature, a non-trivial obstruction of the form ${\rm o}_{i,j}(w_{i},1;1,w'_{j})$ is called a *left obstruction*, a non-trivial obstruction of the form ${\rm o}_{i,j}(1,w'_{i};w_{j},1)$ is called a *right obstruction*, and a non-trivial obstruction of the form ${\rm o}_{i,j}(w_{i},w'_{i};$ $1,1)$ or ${\rm o}_{i,j}(1,1;w_{j},w'_{j})$ is called a *center obstruction*. We picture four types of obstructions as follows.

[\*[3]{}[C]{}]{} &\
&\
&&\
& &\
\
&&

[\*[3]{}[C]{}]{} &\
&\
&&\
\
& &\
&&

At this point we can refine the characterization of Gröbner bases given in Proposition \[sec2pro5\] in the following way.

\[sec2pro10\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle$ be 