= \emptyset$ by assumption. Therefore, the following are in $J_C$: $$\begin{aligned}
& x_1x_2x_3 \\
x_1(1+x_2)(1+x_3) &= x_{1} + x_{1}x_{2} + x_{1}x_{3} + x_{1}x_{2}x_{3} \\
x_2(1+x_1)(1+x_3) &=  x_{2} + x_{1}x_{2} + x_{2}x_{3} + x_{1}x_{2}x_{3} \\
x_3(1+x_1)(1+x_2) &= x_{3} + x_{1}x_{3} + x_{2}x_{3} + x_{1}x_{2}x_{3}. \end{aligned}$$ Adding the four terms again gives $x_1+x_2+x_3 \in J_C$.

[*Converse of Type 6:*]{} Again, when $n=1$, it is impossible to have $m \geq 2$ different indices, so the implication is vacuously true.

Modifying the Type 4-6 Relations
================================

In this section, we discuss modified versions of the Type 4-6 relations and prove that their converses hold. The modifications are marked in bold in the following result.

Let $\mathcal{U} = \{ U_i \}_{i=1}^n$ be a collection of sets in a stimulus space $X$. Let $C = C(\mathcal{U})$ denote the corresponding receptive field code, and let $J_C$ be the neural ideal of $C$. Then for any subsets $\sigma_1$, $\sigma_2$, $\tau_1$, $\tau_2 \subseteq [n]$ [**such that $\sigma_1 \cup \sigma_2$ and $\tau_1 \cup \tau_2$ are disjoint**]{}, and $m$ indices $1 \leq i_1 < i_2 < ... < i_m \leq n$, with $m \geq 2$, we have the following equivalences:

-   $x_{\sigma_1} \prod_{i \in \tau_1} (1+x_i) + x_{\sigma_2} \prod_{j \in \tau_2} 