of the ideal $I$. As a consequence, $\{D\}$ is a $G$-Gröbner basis of $I$.

A $G$-stable ideal need not have a finite $G$-Gröbner basis. Indeed, if one requires that [*every*]{} $G$-stable ideal $I$ in $K[X]$ has a finite $G$-Gröbner basis, then this must in particular be true for [*monomial*]{} ideals. This implies that ${\mathrm{Mon}}$ does not have infinite antichains relative to the [*partial order*]{} on ${\mathrm{Mon}}$ defined by $m \preceq m':\Leftrightarrow \exists \pi \in G: \pi m | m'$. Observe that this is, indeed, a partial order: transitivity is straightforward, and antisymmetry follows from the fact that $\pi m|m \Rightarrow \pi m
\leq m$, while on the other hand $\pi m \geq m$ for all $\pi,m$; see Remark \[re:WhyMonoid\]. Conversely, if $({\mathrm{Mon}}, \preceq)$ does not have infinite anti-chains, then every ideal has a finite $G$-Gröbner basis. This is the case in the set-up of Example \[ex:AH\], which is generalised in [@Hillar09]; there equivariant Gröbner bases are called [*monoidal Gröbner bases*]{}.

We have not yet really used that ${\mathrm{Mon}}$ is the free commutative monoid generated by $X$. So far, we could have taken ${\mathrm{Mon}}$ any commutative monoid equipped with EGB1 and EGB2. This viewpoint, and a generalisation thereof, is adopted in 