other hand, if a non-invertible map is to preserve the restriction of the term order to the set of coordinates, then that set better be infinite, in contrast with the usual set-up in computational commutative algebra.

In fact, there is another, more compelling reason for allowing infinitely many variables: many varieties from applications come in infinite families, and it is convenient to pass to a suitable limit. For example, the variety of symmetric $n \times n$-matrices of rank $2$ has a well-defined projective limit for $n$ tending to infinity, and so does the closely related two-factor model that we study in this paper. In both cases, the limit is not only stable under the union of all symmetric groups $S_n$ simultaneously permuting rows and columns, but also under the [*monoid*]{} ${\mathrm{Inc}({{\mathbb N}})}$ of all strictly increasing maps from ${{\mathbb N}}$ to itself. And while the union of the symmetric groups does not preserve any term order, the monoid ${\mathrm{Inc}({{\mathbb N}})}$ does preserve such an order; this fundamental observation allows us to do computations in Section \[sec:2f\].

This discussion leads to the following set-up, which we believe will have applications to numerous other problems. Let $X$ be a potentially infinite set, 