whose elements we call [*variables*]{}. The free commutative monoid generated by $X$ is denoted ${\mathrm{Mon}}$; its elements are called [*monomials*]{}. Suppose that we have

1.  a term order, i.e., a well-order $\leq$ on ${\mathrm{Mon}}$ such that $m \leq m' \Rightarrow mm'' \leq m'm''$ for all $m,m',m'' \in {\mathrm{Mon}}$; and

2.  a monoid $G$, i.e., a (typically non-commutative) semigroup with identity, acting on ${\mathrm{Mon}}$ by means of monoid homomorphisms ${\mathrm{Mon}}\to {\mathrm{Mon}}$ preserving the strict order: $\pi 1=1,\
    \pi(mm')=(\pi m)(\pi m'),$ and $m < m' \Rightarrow \pi m< \pi m'$ for all $\pi  \in G,\ m,m' \in {\mathrm{Mon}}$.

\[ex:AH\] The setting that Aschenbrenner and Hillar study in [@Aschenbrenner07] fits into this framework, and indeed inspired our set-up. There $X=\{x_1,x_2,\ldots\}$ and $G$ is the monoid ${\mathrm{Inc}({{\mathbb N}})}$ of all increasing maps $\pi: {{\mathbb N}}\to {{\mathbb N}}$ acting on $X$ by $\pi x_i=x_{\pi(i)}$ and on ${\mathrm{Mon}}$ by multiplicativity. As a term order one can choose the lexicographic order with $x_i>x_j$ if $i>j$. Aschenbrenner and Hillar have turned their proof of finite generation of ${\mathrm{Inc}({{\mathbb N}})}$-stable ideals in $K[x_1,x_2,\ldots]$ into an algorithm; see [@Aschenbrenner08].

Let $K$ be a field and let $K[X]=K{\mathrm{Mon}}$ be the polynomial $K$-algebra in the variables $X$, or, equivalently, the monoid $K$-algebra 