with the canonical basis $\{e_{1},\dots,e_{s}\}$, where $e_{i}=(0,\dots,0,1\otimes1,$ $0,\dots,0)$ with $1\otimes1$ occurring in the $i^{\rm th}$ position for $i=1,\dots,s$, and we let $\mathbb{T}(F_{s})$ be the set of terms in $F_{s}$, i.e. $\mathbb{T}(F_{s})=\{we_{i}w'\ |\ i\in\{1,\dots,s\}, w,w'\in\langle X\rangle\}$.

\[sec2def4\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ with $s\geq 1$, and let $i,j\in\{1,\dots,s\}$ such that $i\leq j$.

-   If there exist some words $w_{i},w'_{i},w_{j},w'_{j}\in\langle X\rangle$ such that $w_{i}\lw_{\sigma}(g_{i})w'_{i}=w_{j}\lw_{\sigma}(g_{j})w'_{j}$, then we call the element $${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\frac{1}{\lc_{\sigma}(g_{i})}w_{i}e_{i}w'_{i}-\frac{1}{\lc_{\sigma}(g_{j})}w_{j}e_{j}w'_{j}\in F_{s}\setminus\{0\}$$ an *obstruction* of $g_i$ and $g_j$. If $i=j$, it is called a *self obstruction* of $g_{i}$. We will denote the *set of all obstructions* of $g_{i}$ and $g_{j}$ by ${\rm Obs}(i,j)$.

-   Let ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})\in {\rm Obs}(i,j)$ be an obstruction of $g_{i}$ and $g_{j}$. The polynomial $$S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\frac{1}{\lc_{\sigma}(g_{i})}w_{i}g_{i}w'_{i}-\frac{1}{\lc_{\sigma}(g_{j})}w_{j}g_{j}w'_{j}\in K\langle X\rangle$$ is called the *S-polynomial* of ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$.

Using these definitions, we can characterize Gröbner bases in the following way.

\[sec2pro5\] Let $G=\{g_1,\dots,g_s\}\subseteq K\langle X\rangle\setminus\{0\}$ be a set of polynomials which generate a two-sided ideal $I=\langle G\rangle\subseteq K\langle X\rangle$. Then the following conditions are equivalent.

-   The set $G$ is a ${\sigma}$-Gröbner basis of $I$.

-   For every obstruction ${\rm o}_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ in the set $\bigcup_{1\leq i\leq j\leq s}{\rm Obs}(i,j)$, its S-polynomial $S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})$ has a representation $$S_{i,j}(w_{i},w'_{i};w_{j},w'_{j})=\sum^{\mu}_{k=1}c_{k}w_{k}g_{i_{k}}w'_{k}$$ with $c_{k}\in K, w_{k},w'_{k}\in\langle X\rangle$, and $g_{i_{k}}\in 