added consecutively to $B$. Then for all $i<j$ we have ${\mathrm{lm}}(r_i) \not
\preceq {\mathrm{lm}}(r_j)$ since $r_j$ is $G$-reduced modulo $r_i$. Using the fact that decreasing $\preceq$-chains are finite, one finds an infinite subsequence $r_{i_1}, r_{i_2}, \ldots $ with $i_1<i_2<\ldots$ such that ${\mathrm{lm}}(r_{i_a}) \not \preceq {\mathrm{lm}}(r_{i_b})$ holds not only for $a<b$ but also for $a>b$. This sequence contradicts the assumption that $\preceq$ does not have infinite antichains.

An equivariant Gröbner basis for the two-factor model {#sec:2f}
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Theorem \[thm:Main\] will follow from the following result. Let $X=\{y_{ij}
\mid \ i,j \in {{\mathbb N}}, i \geq j\}$ be a set of variables representing the entries of a symmetric matrix. We consider the lexicographic monomial order on ${\mathrm{Mon}}$ in which the diagonal variables $y_{ii}$ are larger than all variables $y_{ij}$ with $i>j$, and apart from that $y_{ij} \geq y_{i'j'}$ if and only if $i>i'$ or $i=i'$ and $j \geq j'$. So for instance we have $$y_{2,2} > y_{1,1} > y_{5,2} > y_{4,3}.$$ Note that this monomial order is compatible with the action of the monoid ${\mathrm{Inc}({{\mathbb N}})}$ of all increasing maps ${{\mathbb N}}\to {{\mathbb N}}$. For any polynomial $p \in
K[X]$ let ${l(p)}$ denote the [*largest index of $p$*]{}, i.e., the largest index 