[@Hillar09]. However, for doing computations we need that ${\mathrm{Mon}}$ has more structure; see conditions EGB3 and EGB4 below. This is why we have restricted ourselves to free monoids ${\mathrm{Mon}}$.

In the polynomial ring of Example \[ex:Rankk\] the set $\{y_{12}y_{21},y_{12}y_{23}y_{31},y_{12}y_{23}y_{34}y_{41},\ldots\}$ is an infinite $\preceq$-antichain of monomials, hence the ${\mathrm{Inc}({{\mathbb N}})}$-stable ideal generated by it does not have a finite ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis. But even in such a setting where not [*all*]{} $G$-stable ideals have finite $G$-Gröbner bases, ideals of interesting $G$-stable varieties may still have such bases. We will derive an algorithm for computing equivariant Gröbner bases under the following two additional assumptions:

1.  for all $\pi \in G$ and $m,m' \in {\mathrm{Mon}}$ we have ${\operatorname{lcm}}(\pi
    m,\pi m')=\pi {\operatorname{lcm}}(m,m')$; and

2.  for all $f,h \in K[X]$ the set $Gf \times Gh$ is the union of a finite number of $G$-orbits (where $G$ acts diagonally on $K[X]
    \times K[X]$), and generators of these orbits can be computed effectively.

Note that EGB3 is automatically satisfied if $G$ stabilises the set $X$ of variables. Although this is the only setting that we will need for the application to the two-factor model, future applications may need the greater generality where $X$ is not $G$-stable. Condition 