Viktor Levandovskyy pointed out to the second author that our equivariant Gröbner bases are a special case of Gröbner $S$-bases in the sense of [@Drensky06], which were invented for analysing certain two-sided ideals in free associative algebras. The focus of the present article is on getting exactly the right set-up for doing machine computations of equivariant Gröbner bases in the commutative setting.

It is easy to see that if $B$ is a $G$-Gröbner basis of $I$, then $GB$ generates $I$ as an ideal; see Lemma \[lm:GBgenerates\].

\[ex:Rankk\] Let $X=\{y_{ij} \mid i,j \in {{\mathbb N}}\}$, let $k$ be a natural number, and let $I$ be the ideal of all polynomials in the $y_{ij}$ that vanish on all ${{\mathbb N}}\times {{\mathbb N}}$-matrices $y$ of rank at most $k$. Order the variables $y_{ij}$ lexicographically by the pair $(i,j)$, where $i$ is the most significant index; so for instance $y_{3,5}>y_{2,6}>y_{2,4}>y_{1,10}$. The corresponding lexicographic order on monomials in the $y_{ij}$ is a well-order. Let $G:={\mathrm{Inc}({{\mathbb N}})}\times {\mathrm{Inc}({{\mathbb N}})}$ act on $X$ by $(\pi,\sigma)y_{ij}=y_{\pi(i),\sigma(j)};$ this action preserves the strict order. The $G$-orbit of the determinant $D$ of the matrix $(y_{ij})_{i,j=1,\ldots,k+1}$ consists of all $(k+1)\times(k+1)$-minors of $y$, which by the results of [@Sturmfels90] form a Gröbner basis 