The standard-expression criterion for left Noetherian differential operator rings

Let $S$ be an affine semigroup with standard expression, and let $S_2$ be the semigroup appearing in the standard expressions of $S$ and $S_2$ referenced in the source. Write $D(S_2,S)$ for the corresponding differential-operator submodule, and let $\tau_i$, ${\boldsymbol b}_i$ and $l$ be the indices and data from those standard expressions. The standard-expression criterion. The following conditions are equivalent:

1. $D(S)$ is left Noetherian.
2. $D(S_2)/D(S_2,S)$ is a Noetherian left $D(S)$-module.
3. $D(S_2)/D(S)$ is a Noetherian left $D(S)$-module.
4. For all $i>l$,

$$\left(\bigcap_{\tau_j\succ\tau_i,\, {\boldsymbol b}_i-{\boldsymbol b}_j\in K\tau_j,\, j\leq l}\tau_j\right)=\tau_i.$$

This conjecture proposes equivalent algebraic and combinatorial tests for left Noetherianity. The preceding theorem establishes non-Noetherianity in a particular configuration, while the general equivalence is presented as a conjecture and is not resolved in the supplied text.