of $b$ for which ${\mathrm{lm}}(f_{b}b)$ equals $m$. This contradicts the minimality of the expression chosen above.

We prove that $GB$ is an ordinary Gröbner basis of the ideal that it generates. By the ordinary Buchberger criterion it suffices to verify that for all $b_0,b_1 \in B$ and $\pi_0,\pi_1 \in G$ the S-polynomial $S(\pi_0 b_0,\pi_1 b_1)$ has $0$ as a remainder modulo $GB$. By assumption there exists a triple $(\sigma_0,\sigma_1,\pi_2)$ for which $(\pi_0 b_0,\pi_1 b_1)=(\pi_2 \sigma_0 b_0,\pi_2 \sigma_1
b_1)$ and for which $S(\sigma_0
b_0,\sigma_1 b_1)$ has $0$ as a $G$-remainder modulo $B$, which means that it has $0$ as an ordinary remainder modulo $GB$. Since $G$ preserves least common multiples (EGB3), we have $$S(\pi_2 \sigma_0 b_0, \pi_2 \sigma_1 b_1)=
\pi_2 S(\sigma_0 b_0,\sigma_1 b_1),$$ and applying $\pi_2$ to the entire reduction of $S(\sigma_0 b_0,\sigma_1 b_1)$ to $0$ modulo $GB$ yields a reduction of $S(\pi_0 b_0,\pi_1 b_1)$ to $0$, as claimed.

The following example shows that EGB3 is not a redundant assumption in Theorem \[thm:Buchberger\].

\[ex:EGB3\] Suppose that $X=\{x,y,z_1,z_2,\ldots\}$ and that the monoid $G$ is generated by ${\mathrm{Inc}({{\mathbb N}})}$ acting by $\pi z_i=z_{\pi i}$ and trivially on $x,y$, together with a single homomorphism $\sigma:{\mathrm{Mon}}\to {\mathrm{Mon}}$ determined by $\sigma x=x$, $\sigma y=x z_1$, and 